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Hindawi Publishing CorporationISRNMathematical PhysicsVolume 2013 Article ID 748613 24 pageshttpdxdoiorg1011552013748613

Research ArticleAir-Aided Shear on a Thin Film Subjected to a TransverseMagnetic Field of Constant Strength Stability and Dynamics

Mohammed Rizwan Sadiq Iqbal

Department of Mathematics and Statistics University of Constance 78457 Constance Germany

Correspondence should be addressed to Mohammed Rizwan Sadiq Iqbal rizwansadiqigmailcom

Received 28 June 2013 Accepted 22 August 2013

Academic Editors L E Oxman and W-H Steeb

Copyright copy 2013 Mohammed Rizwan Sadiq Iqbal This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The effect of air shear on the hydromagnetic instability is studied through (i) linear stability (ii) weakly nonlinear theory (iii)sideband stability of the filtered wave and (iv) numerical integration of the nonlinear equation Additionally a discussion on theequilibria of a truncated bimodal dynamical system is performed While the linear and weakly nonlinear analyses demonstrate thestabilizing (destabilizing) tendency of the uphill (downhill) shear the numerics confirm the stability predictions They show that(a) the downhill shear destabilizes the flow (b) the time taken for the amplitudes corresponding to the uphill shear to be dominatedby the one corresponding to the zero shear increases with magnetic fields strength and (c) among the uphill shear-induced flows ittakes a long time for the wave amplitude corresponding to small shear values to become smaller than the one corresponding to largeshear values when the magnetic field intensity increases Simulations show that the streamwise and transverse velocities increasewhen the downhill shear acts in favor of inertial force to destabilize the flowmechanism However the uphill shear acts oppositelyIt supports the hydrostatic pressure and magnetic field in enhancing films stability Consequently reduced constant flow rates anduniform velocities are observed

1 Introduction

A nonlinear fourth-order degenerate parabolic differentialequation of the form

ℎ119905+A (ℎ) ℎ

119909+

120597

120597119909B (ℎ) ℎ

119909+C (ℎ) ℎ

119909119909119909 = 0 (1)

whereAB andC are arbitrary continuous functions of theinterfacial thickness ℎ(119909 119905) represents a scalar conservationlaw associated to the flow of a thin viscous layer on an inclineunder different conditions The stability and dynamics ofequations of the type of (1) is a subject of major interest [1ndash18] because of their robustness in regimes where viscositydominates inertia [19] Such studies have focused attentionprimarily on the isothermal and nonisothermal instabilityanalysis mainly for nonconducting fluids

Since the investigation of Chandrasekar [20] on thestability of a flow between coaxial rotating cylinders in thepresence of a magnetic field held in the axial direction thelaminar flow of an electrically conducting fluid under the

presence of a magnetic field has been studied extensively Forinstance Stuart [21] has reported on the stability of a pressureflow between parallel plates under the application of a parallelmagnetic field Among other earlier investigations Lock [22]examined the stability when the magnetic field is appliedperpendicular to the flow direction and to the boundaryplanes Hsieh [23] found that the magnetic field stabilizesthe flow through Hartmann number when the electricallyconducting fluid is exposed to a transverse magnetic fieldprovided that the surface-tension effects are negligible ina horizontal film Ladikov [24] studied a problem in thepresence of longitudinal and transverse magnetic fieldsThe author observed that the longitudinal magnetic fieldplays a stabilizing role and that the effect of instability atsmall wave numbers could be removed if the longitudinalmagnetic field satisfies certain conditions Lu and Sarma [25]investigated the transverse effects of the magnetic field inmagnetohydrodynamic gravity-capillary waves The flow ofan electrically conducting fluid over a horizontal plane inthe presence of tangential electric and magnetic fields was

2 ISRNMathematical Physics

reported byGordeev andMurzenko [26]They found that theflow suffers from instability not due to the Reynolds numberbut due to the strength of the external electric field

In applications such as magnetic-field-controlled mate-rial-processing systems aeronautics plasma engineeringMEMS technology andmagnetorheological lubrication tech-nologies the hydromagnetic effects are important Alsoliquid metal film flows are used to protect the solid structuresfrom thermonuclear plasma in magnetic confinement fusionreactors and this application requires a better understandingof the instability mechanism arising in a thin magnetohydro-dynamic flow over planar substrates [27] Furthermore thepresence of an external magnetic field regulates the thicknessof a coating film It prevents any form of direct electrical ormechanical contact with the fluid thereby reducing the riskof contamination [28] Renardy and Sun [29] pointed outthat the magnetic fluid is effective in controlling the flowof ordinary fluids and in reducing hydraulic resistance Thereason is that the magnetic fluids can be easily controlledwith external magnetic fields and that coating streamlinedbodies with a layer of less-viscousmagnetic fluid significantlyreduces the shear stress in flow boundaries Magnetohydro-dynamic flow can be a viable option for transporting weaklyconducting fluids in microscale systems such as flows insidea micro-channel network of a lab-on-a-chip device [30 31]In all of the above applications considering the associatedstability problem is important because it gives guidance inchoosing the flow parameters for practical purposes In thisregard in the past two decades the emerging studies onthe hydromagnetic effects have focused their attention onstability problems and on analyzing the flow characteristics[28 29 32ndash43]

The shearing effect of the surrounding air on the fluidin realistic situations induces stress tangentially on the inter-facial surface The hydrodynamic instability in thin films inthe presence of an external air stream attracted attention inthe mid 1960s which led Craik [44] to conduct laboratoryresearch and study theory He found that the instabilityoccurs regardless of the magnitude of the air stream whenthe film is sufficiently thin Tuck and Vanden-Broeck [45]reported on the effect of air stream in industrial applica-tions of thin films of infinite extent in coating technologySheintuch and Dukler [46] did phase plane and bifurcationanalyses of thin wavy films subjected to shear from counter-current gas flows and found satisfactory agreement of theirresults connected to the wave velocity along the floodingcurvewith the experimental results of Zabaras [47] Althoughthe experimental results related to the substrate thickness didnot match exactly with the theoretical predictions still theresults gave qualitative information about the model Thinliquid layer supported by steady air-flow surface-tractionwas reported by King and Tuck [48] Their study modelsthe surface-traction-supported fluid drops observed on thewindscreen of a moving car on a rainy day Incorporating thesuperficial shear offered by the air Pascal [49] studied a prob-lemwhichmodels themechanismofwind-aided spreading ofoil on the sea and found quantitative information regardingthe maximum upwind spread of the gravity current Wilsonand Duffy [50] studied the steady unidirectional flow of a

thin rivulet on a vertical substrate subjected to a prescribeduniform longitudinal shear stress on the free surface Theycategorized the possible flow patterns and found that thedirection of the prescribed shear stress affects the velocityin the entire rivulet The generation of roll waves on thefree boundary of a non-Newtonian liquid was numericallyassessed using a finite volumemethod by Pascal andDrsquoAlessio[51] revealing the significant effect of air shear on theevolution of the flow Their study showed that the instabilitycriteria was conditional and depends on the directionallyinduced shear Kalpathy et al [52] investigated an idealizedmodel suitable for lithographic printing by examining theshear-induced suppression of two stratified thin liquid filmsconfined between parallel plates taking into account the vander Waals force A film thickness equation for the liquid-liquid interface was derived in their study using lubricationapproximationThey found that the effect of shear affects theimaginary part of the growth rate indicating the existenceof traveling waves Furthermore they also observed a criticalshear rate value beyond which the rupture mechanism couldbe suppressed This study motivated Davis et al [53] toconsider the effect of unidirectional air shear on a singlefluid layer For a two-dimensional ultrathin liquid Daviset al [53] showed that the rupture mechanism induced bythe London van der Waals force could be suppressed whenthe magnitude of the wind shear exceeds a critical value asobserved byKalpathy et al [52] Recently Uma [18]measurednumerically the profound effect of unidirectional wind stresson the stability of a condensateevaporating power-law liquidflowing down an incline

In the present investigation the effects of downhill anduphill air shear on a thin falling film in the presence of a trans-verse magnetic field are studied Such an investigation willillustrate the realistic influence of the natural environmentacting upon the flows Or it may illustrate the need to controlthe flow mechanism through artificial techniques whichblow air when the hydromagnetic effects are consideredThe outline of this paper is organized as follows Section 1presents the introduction In Section 2 mathematical equa-tions governing the physical problem are presented Section 3discusses the long-wave Benney-type equation In Section 4linear stability analysis weakly nonlinear stability analysisand the instability arising due to sideband disturbances areanalyzed The equilibria of a truncated bimodal dynamicalsystem is mathematically presented in Section 5 While theresults in Section 6 discuss nonlinear simulations of the filmthickness evolution Section 7 highlights the main conclu-sions of the study and includes future perspectives

2 Problem Description

A thin Newtonian liquid layer of an infinite extent fallingfreely over a plane under the influence of gravitationalacceleration 119892 is considered The flow is oriented towardsthe 119909-axis and the plane makes an angle 120573 with the horizonProperties of the fluid like density (120588) viscosity (120583) andsurface-tension (120590) are constants The magnetic flux densityis defined by the vectorB = 119861

0119895 where119861

0is themagnitude of

ISRNMathematical Physics 3

Y

120573

h(x t)B0

120582

g

120591a

X

h0

Figure 1 Sketch of the flow configuration When 120591119886gt 0 the air

shears the surface along the downhill direction (in the direction ofthe arrow) However if 120591

119886lt 0 the air shears the surface along the

uphill direction (in this case the arrow would point to the oppositedirection) The air shear effect is zero when 120591

119886= 0

the magnetic field imposed along the 119910-direction (Figure 1)It is assumed that there is no exchange of heat between theliquid and the surrounding air but an air flow (either inthe uphill or in the downhill direction) induces a constantstress ofmagnitude 120591

119886on the interface andmoves tangentially

along the surface The 119910-axis is perpendicular to the planarsubstrate such that at any instant of time 119905 ℎ(119909 119905) measuresthe film thickness

The magnetohydrodynamic phenomena can be modeledby the following equations which express the momentumand mass balance

120588(120597U120597119905

+ U sdot nablaU) = 120588G minus nabla119901 + 120583nabla2U + F (2)

nabla sdot U = 0 (3)

The last term in (2) arises due to the contribution of Lorenzbody force based on Maxwellrsquos generalized electromagneticfield equations [28 54 55]

In a realistic situation corresponding to a three-dimensional flow the total current flow can be defined usingOhmrsquos law as follows

J = Σ (E + U times B) (4)

where J E U and Σ represent the current density electricfield velocity vector and electrical conductivity respectivelyThe Lorenz force acting on the liquid is defined asF = JtimesB Inthe rest of the analysis a short circuited systemcorrespondingto a two-dimensional problem is considered by assuming thatE = 0This assumption simplifies the last term correspondingto the electromagnetic contribution in (2) In this case thepondermotive force acting on the flow (the last term in (2))has only one nonvanishing term in the 119909-direction therefore

F = (119865119909 119865119910) = (minusΣ119861

2

0119906 0) (5)

where G = (119892 cos120573 minus119892 sin120573) and 119901 is the pressure

Boundary conditions on the planar surface and theinterface are added to complete the problem definition of (2)-(3) On the solid substrate the no-slip and the no-penetrationconditions are imposed which read as

U = (119906 V) = 0 (6)

The jump in the normal component of the surface-tractionacross the interface is balanced by the capillary pressure(product of themean surface-tension coefficient and the localcurvature of the interface) which is expressed as

119901119886minus 119901 + (120583 (nablaU + nablaU119879) sdot n) sdot n

= 2120590Π (ℎ) on 119910 = ℎ (119909 119905)

(7)

where Π(ℎ) = minus(12)nabla sdot n is the mean film curvature and119901119886is the pressure afforded by the surrounding air The unit

outward normal vector at any point on the free surface is n =

nabla(119910 minus ℎ)|nabla(119910 minus ℎ)| and t represents the unit vector alongthe tangential direction at that point such that n sdot t = 0 Thetangential component of the surface-traction is influenced bythe air stress 120591

119886and reads as

(120583 (nablaU + nablaU119879) sdot t) sdot n = 120591119886

on 119910 = ℎ (119909 119905) (8)

The location of the interface can be tracked through thefollowing kinematic condition

120597ℎ

120597119905= V minus 119906

120597ℎ

120597119909on 119910 = ℎ (119909 119905) (9)

In order to remove the units associated with the model(2)ndash(9) involving the physical variables reference scales mustbe prescribed In principle one can nondimensionalize thesystem based on the nature of the problem by choosingone of the following scales [9 16] (a) kinematic viscositybased scales (b) gravitational acceleration as the flow agentbased scales (c) mean surface-tension based scales and (d)Marangoni effect as the flow agent based scales Howeverfor very thin falling films the main characteristic time is theviscous one [7ndash9 16] An advantage of choosing the viscousscale is that either both the large and the small inclinationangles could be considered by maintaining the sine of theangle and the Galileo number as separate entities [7 16] orthe Reynolds number as a product of Galileo number andsine of the inclination angle could be defined as a single entity[9] Also such a scale plays a neutral role while comparingthe action of gravity and Marangoni effect in nonisothermalproblems [16] Choosing the viscous scale the horizontaldistance is scaled by 119897 vertical distance by ℎ

0 streamwise

velocity by ]ℎ0 transverse velocity by ]119897 pressure by

120588]2ℎ20 time by 119897ℎ

0] and finally the shear stress offered by

the wind by 120588]2ℎ20 In addition the slenderness parameter

(120576 = ℎ0119897) is considered small and a gradient expansion of

the dependent variables is done [7ndash9] The horizontal lengthscale 119897 is chosen such that 119897 = 120582 where 120582 is a typicalwavelength larger than the film thickness


The dimensionless system is presented in Appendix A(the same symbols have been used to avoid new nota-tions) The set of nondimensional parameters arising duringnondimensionalization procedure are Re = Ga sin120573 (theReynolds number) [9] Ga = 119892ℎ

3

0]2 (the Galileo number)

Ha = Σ1198612

0ℎ2

0120583 (the Hartmann number which measures the

relative importance of the drag force resulting frommagneticinduction to the viscous force arising in the flow) S =

120590ℎ0120588]2 (the surface-tension parameter) and 120591 = 120591

119886ℎ2

0120588]2

(the shear stress parameter) The surface-tension parameteris usually large therefore it is rescaled as 1205762S = 119878 and setas O(1) in accordance to the waves observed in laboratoryexperiments All of the other quantities are considered O(1)The long-wave equation is derived in the next step

3 Long-Wave Equation

The dependent variables are asymptotically expanded interms of the slenderness parameter 120576 to derive the long-wave equation Using the symbolic math toolbox availablein MATLAB the zeroth and the first-order systems aresolved These solutions are then substituted in the kinematiccondition (A5) to derive a Benney-typemodel accurate up toO(120576) of the form (1) as

ℎ119905+ 119860 (ℎ) ℎ

119909+ 120576(119861 (ℎ) ℎ

119909+ 119862 (ℎ) ℎ

119909119909119909)119909+ O (120576

2) = 0 (10)

The standard procedure for the derivation [7 9 14 15 28] isskipped here The expressions for 119860(ℎ) 119861(ℎ) and 119862(ℎ) arethe following

119860 (ℎ) =ReHa

tanh2 (radicHa ℎ)

+120591

radicHatanh (radicHa ℎ) sech (radicHa ℎ)

(11)

119861 (ℎ) = Re cot120573Ha32

tanh (radicHa ℎ) minusRe cot120573ℎ

Ha

+ Re2ℎ2Ha52

tanh (radicHa ℎ) sech4 (radicHa ℎ)

+Re2ℎHa52


minus3Re2

2Ha3tanh2 (radicHa ℎ) sech2 (radicHaℎ)

+ 120591 3Re

2Ha52tanh (radicHa ℎ) sech (radicHa ℎ)

minus3ReHa52


+Re ℎHa2

(sech5 (radicHa ℎ)

+3

2sech3 (radicHa ℎ)

minussech (radicHa ℎ))

+ 1205912ℎ tanh (radicHa ℎ)

Ha32( minus sech2 (radicHa ℎ)

minus1

2sech4 (radicHa ℎ))

+3

2Ha2tanh2 (radicHa) sech2 (radicHa ℎ)

(12)

119862 (ℎ) =119878ℎ

Haminus

119878

Ha32tanh (radicHa ℎ) (13)

It should be remarked that when 120591 = 0 the above termsagree with the long-wave equation derived by Tsai et al[28] when the phase-change effects on the interface areneglected The dimensionless parameters differ from thenondimensional set presented in Tsai et al [28] becauseviscous scales are employed here Although one can considerdifferent scales in principle the structure of the evolutionequation remains unchanged regardless of the dimensionlessparameters appearing in the problem Furthermore the errorin the second term associated with 119861(ℎ) in Tsai et al [28] iscorrected here which as per the convention followed in theirpaper should read as (4120572Re ℎ119898

5)sech2119898ℎ tanh119898ℎ Also

the case corresponding toHa = 0 can be recovered from (11)ndash(13) when Ha rarr 0 In this case the functions in (11)ndash(13)read as

119860 (ℎ) = Re ℎ2+ 120591ℎ

119861 (ℎ) =2

15Re ℎ5(Re ℎ + 120591) minus

1

3Re cot120573ℎ3

119862 (ℎ) = 119878ℎ3

3

(14)

which match with the evolution equation derived by Miladi-nova et al [9] in the absence of Marangoni and air shearingeffect The effect of magnetic field and air shear affectsthe leading order solution through 119860(ℎ) term in (11) Thisterm contributes towards wave propagation and steepeningmechanismThe effect of hydrostatic pressure is measured bythe terms within the first flower bracket in 119861(ℎ) in (12) Therest of the terms in 119861(ℎ) affect the mean flow due to inertialair shear andmagnetic field contributionsThe function119862(ℎ)

corresponds to themean surface-tension effect Although theeffect of 120591 cannot be properly judged based on its appearancein (11)ndash(13) it is obvious that in the absence of the magneticfield both 119860(ℎ) and 119861(ℎ) increase (decrease) when 120591 gt 0 (lt

0) The stability of the long-wave model (10) subject to (11)ndash(13) is investigated next


4 Stability Analysis

The Nusselt solution corresponding to the problem is

ℎ0= 1 (15a)

1199060=

sinh (radicHa119910)radicHa

(ReradicHa

tanhradicHa + 120591 sechradicHa)

+ReHa

(1 minus cosh (radicHa119910)) (15b)

V0= 0 (15c)

1199010= Re cot120573 (1 minus 119910) (15d)

For a parallel shear flow (10) subject to (11)ndash(13) admitsnormal mode solutions of the form ℎ = 1 + 119867(119909 119905) where119867(119909 119905) is the unsteady part of the film thickness representingthe disturbance component such that 119867 ≪ 1 [56 57]Inserting ℎ = 1 + 119867(119909 119905) in (10) and invoking a Taylorseries expansion about ℎ = 1 the unsteady nonlinearequation representing a slight perturbation to the free surfaceis obtained as

119867119905+ 1198601119867119909+ 1205761198611119867119909119909

+ 1205761198621119867119909119909119909119909

= minus[1198601015840

1119867 +

11986010158401015840

1

21198672+119860101584010158401015840

1

61198673]119867119909

minus 120576 [1198611015840

1119867 +

11986110158401015840

1

21198672]119867119909119909

minus 120576(1198621015840

1119867 +

11986210158401015840

1

21198672)119867119909119909119909119909

minus 120576 [1198611015840

1+ 11986110158401015840

1119867] (119867

119909)2

minus 120576 (1198621015840

1+ 11986210158401015840

1119867)119867119909119867119909119909119909

+ O (1205761198674 1198675 1205762119867)

(16)

where (a prime denotes the order of the derivative withrespect to ℎ)

1198601= 119860 (ℎ = 1) 119860

1015840

1= 1198601015840(ℎ = 1)

11986010158401015840

1= 11986010158401015840(ℎ = 1)

1198611= 119861 (ℎ = 1) 119861

1015840

1= 1198611015840(ℎ = 1)

11986110158401015840

1= 11986110158401015840(ℎ = 1)

1198621= 119862 (ℎ = 1) 119862

1015840

1= 1198621015840(ℎ = 1)

11986210158401015840

1= 11986210158401015840(ℎ = 1)

(17)

It should be remarked that while expanding the Taylor seriesthere are two small parameters namely 120576 ≪ 1 and 119867 ≪ 1

whose orders of magnitude should be considered such that120576 ≪ 119867 ≪ 1 When O(1205761198673) terms are retained and since

1205761198673≪ 1198674 (11986010158401015840101584016)1198673119867119909appears as a unique contribution of

order1198674 in (16) This term although present in the unsteadyequation (16) does not contributewhen amultiple-scale anal-ysis is done (refer to Section 421 and Appendix B) whereequations only up to O(1205723) are considered while deriving acomplex Ginzburg-Landau-type equation [12 14 17 56 57]In addition such a term did not appear in earlier studies[12 14 17 56 57] because 119860(ℎ) was a mathematical functionof second degree in ℎ whose higher-order derivatives arezero

Equation (16) forms the starting point for the linear sta-bility analysis and describes the behavior of finite-amplitudedisturbances of the film Such an equation predicts theevolution of timewise behavior of an initially sinusoidaldisturbance given to the film It is important to note thatthe constant film thickness approximation with long-waveperturbations is a reasonable approximation only for certainsegments of flow and implies that (16) is only locally valid

41 Linear Stability To assess the linear stability the linearterms in (16) are considered The unsteady part of the filmthickness is decomposed as (a tilde denotes the complexconjugate)

119867(119909 119905) = 120578119890119894(119896119909minus119862119871119905)+119862119877119905 + 120578119890

minus119894(119896119909minus119862119871119905)minus119862119877119905 (18)

where 120578 (≪ 1) is a complex disturbance amplitude indepen-dent of 119909 and 119905 The complex eigenvalue is given by 120582 =

119862119871+ 119894119862119877such that 119896 isin [0 1] represents the streamwise

wavenumber The linear wave velocity and the linear growthrate (amplification rate) of the disturbance are 119862

119871119896 119862119877

isin

(minusinfininfin) respectively Explicitly they are found as

119862119871119881

=119862119871

119896= 1198601

119862119877= 1205761198962(1198611minus 11989621198621)

(19)

The disturbances grow (decay) when 119862119877gt 0 (119862

119877lt 0)

However when 119862119877

= 0 the curve 119896 = 0 and the positivebranch of 1198962 = 119861

11198621represent the neutral stability curves

Identifying the positive branch of 1198611minus 11989621198621

= 0 as 119896119888=

radic(11986111198621) (119896119888is the critical wavenumber) the wavenumber

119896119898corresponding to the maximal growth rate is obtained

from (120597120597119896)119862119877

= 0 This gives 1198611minus 21198962

1198981198621= 0 such that

119896119898

= 119896119888radic2 The linear amplification of the most unstable

mode is calculated from 119862119877|119896=119896119898

A parametric study considering the elements of the set

S = ReHa 119878 120573 120591 is done in order to trace the neutralstability and linear amplification curves by assuming theslenderness parameter to be 01 Only those curves which arerelevant in drawing an opinion are presented

The influence of the magnetic field on the criticalReynolds number (Re

119888) varying as a function of the shear

parameter is presented in Figure 2 For each Ha there is aRe119888below which the flow is stable The critical Reynolds

number decreases when the angle of inclination increasesand therefore the flow destabilizes The stabilizing effect ofthemagnetic field is also seenwhenHa increasesThere existsa certain 120591 gt 0 such that the flow remains unstable beyond it


0 50

1

2

3

4

5

6

7

8

9

10

Unstable

Stable

minus5

120591

Ha = 05

Ha = 01

Ha = 0

Rec

(a)

0 50

1

2

3

4

5

6

7

8

9

10

Stable

Unstable

minus5

120591

Ha = 05

Ha = 01Ha = 0

Rec

(b)

Figure 2 Critical Reynolds number as a function of 120591 when 119878 = 0 (a) 120573 = 45∘ (b) 120573 = 80

∘

Figures 3 and 4 display the neutral stability curves whichdivide the 119896 minus Re and 119896 minus 120591 planes into regions of stableand unstable domains On the other hand Figure 5 showsthe linear amplification curves The shear stress offered bythe air destabilizes the flow when it flows along the downhilldirection (120591 gt 0) and increases the instability thresholdcompared to the case corresponding to 120591 = 0 (Figures 3and 4) However the flow mechanism is better stabilizedwhen the applied shear stress offered by the air is in theuphill direction (120591 lt 0) than when 120591 = 0 (Figures 3and 4) As seen from Figure 3 the portion of the axiscorresponding to the unstable Reynolds numbers increasesand extends towards the left when the angle of inclinationincreases thereby reducing the stabilizing effect offered by thehydrostatic pressure at small inclination angles From curves2 and 3 corresponding to Figure 3 it is observed that the forceof surface-tension stabilizes the flow mechanism Figure 4supports the information available from Figure 3 When themagnitude of the Hartmann number increases the instabilityregion decreases because the value of the critical wavenumberdecreases Comparing Figure 4(a) with Figure 4(c) it isalso observed that the inertial force destabilizes the flowmechanism The growth rate curves (Figure 5) agree withthe results offered by the neutral stability curves (Figures 3and 4)

The linearly increasing graphs of 119862119871and the decreasing

plots of 119862119871119881

with respect to 119896 and Ha respectively arepresented in Figure 6 The effect of inertia doubles the linearwave speed 119862

119871119881 But when the magnitude of Ha increases

the linear wave speed decreases The downhill effect of theshear stress on the interface makes the linear wave speedlarger than the cases corresponding to 120591 = 0 and 120591 lt 0

The linear stability results give only a firsthand infor-mation about the stability mechanism The influence ofair-induced shear on the stability of the flow under theapplication of a transverse magnetic field will be betterunderstood only when the nonlinear effects are additionallyconsidered To analyze and illustrate the nonlinear effects onthe stability threshold a weakly nonlinear study is performedin the next step

42 Multiple-Scale Analysis

421 Weakly NonlinearTheory In order to domultiple-scaleanalysis the following slow scales are introduced followingSadiq and Usha [14] (the justification for stretching the scalesis provided in Lin [56] and in Krishna and Lin [57])

1199051= 120572119905 119905

2= 1205722119905 119909

1= 120572119909 (20)

such that

120597

120597119905997888rarr

120597

120597119905+ 120572

120597

1205971199051

+ 1205722 120597

1205971199052

120597

120597119909997888rarr

120597

120597119909+ 120572

120597

1205971199091

(21)

Here 120572 is a small parameter independent of 120576 and measuresthe distance from criticality such that 119862

119877sim O(1205722) (or 119861

1minus

11989621198621

sim O(1205722)) The motivation behind such a study liesin deriving the complex Ginzburg-Landau equation (CGLE)which describes the evolution of amplitudes of unstablemodes for any process exhibiting a Hopf bifurcation Usingsuch an analysis it is possible to examine whether the


0 2 4 6 8 100

02

04

06

08

02

04

06

08

1

0 2 4 6 8 100

1

1 2

3

ST

US

1 2 3

ST

US

120591 = 0 120591 = 1

02

04

06

08

0 2 4 6 8 100

1

1 2 3 ST

US

kc

kc

kc

120591 = minus1

Re Re Re

(a)

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

1 1 1 2

2 2 3 3 3

ST

ST

ST

US US US

0

02

04

06

08

1

kc

0

02

04

06

08

1

kc

0

02

04

06

08

1

kc

120591 = 0 120591 = 1 120591 = minus1

Re Re Re

(b)

Figure 3 Neutral stability curves The symbols ST and US indicate the stable and unstable regions Here (1) Ha = 0001 and 119878 = 1 (2)Ha = 02 and 119878 = 1 (3)Ha = 02 and 119878 = 10 (a) 120573 = 30

∘ and (b) 120573 = 60∘

nonlinear waves in the vicinity of criticality attain a finiteheight and remain stable or continue to grow in time andeventually become unstable Refer to Appendix B for thedetailed derivations of the threshold amplitude 120572Γ

0 and the

nonlinear wave speed119873119888

The threshold amplitude subdivides the flow domainaccording to the signs of 119862

119877and 119869

2[14] If 119862

119877lt 0 and

1198692

lt 0 the flow exhibits subcritical instability Howeverwhen 119862

119877gt 0 and 119869

2gt 0 the Landau state is supercritically

stable If on the other hand 119862119877lt 0 and 119869

2gt 0 the flow is

subcritically stable A blow-up supercritical explosive state isobserved when 119862

119877gt 0 and 119869

2lt 0

Different regions of the instability threshold obtainedthroughmultiple-scale analysis are illustrated in Figure 7Thesubcritical unstable region is affected due to the variation of 120591For 120591 lt 0 such a region is larger than the ones correspondingto 120591 ge 0 In addition the subcritical unstable and thestable regions increase whenHa increasesThe explosive stateregion (also called the nonsaturation zone) decreases when120591 lt 0 than when 120591 ge 0 The strip enclosing asterisks is the

supercritical stable region where the flow although linearlyunstable exhibits a finite-amplitude behavior and saturatesas time progresses [8 9 14] The bottom line of the strip isthe curve 119896

119904= 1198961198882 which separates the supercritical stable

region from the explosive region [56 58 59]Within the strip119896119904lt 119896 lt 119896

119888 different possible shapes of the waves exist

[7 9 14]Tables 1 and 2 show the explosive and equilibration

state values for different flow parameters The wavenumbervalue corresponding to the explosive state increases when theinertial effects increase This increases the unstable region0 lt 119896 lt 119896

119904(Table 1) Such a wavenumber decreases either

when the hydromagnetic effect is increased or when the airshear is in the uphill direction It is evident from Table 1 thatthe explosive state occurs at small values of Re and 120573 which isalso true for large values ofHawhen 120591 lt 0 Also it is observedthat either when the Hartmann number or the value of theuphill shear is increased the critical wavenumber becomeszero (Table 2) Therefore 119896

119904= 0 (Table 1)


00

5

ST

US

1

2

3

4

1

08

06

04

02

minus5

kc

120591

(a)

00

5

ST

US

1 2

3

4

1

08

06

04

02

minus5

kc

120591

(b)

00

5

ST

US

1 4

2

3

1

08

06

04

02

minus5

kc

120591

(c)

Figure 4 Neutral stability curves in the 119896 minus 120591 plane for 119878 = 1 (1) Ha = 0 (2) Ha = 005 (3) Ha = 01 and (4) Ha = 015 (a) 120573 = 60∘ and

Re = 1 (b) 120573 = 75∘ and Re = 1 and (c) 120573 = 60

∘ and Re = 25

The threshold amplitude (120572Γ0) profiles display an asym-

metric structure (Figure 8) increasing up to a certainwavenumber (gt119896

119898) and then decreasing beyond it in the

supercritical stable region The 120591 gt 0 induced amplitudesshow larger peak amplitude than the cases corresponding to120591 = 0 and 120591 lt 0 The peak amplitude value decreases whenHa increasesThenonlinearwave speed (119873

119888) curves represent

a 90∘ counterclockwise rotation of the mirror image of the

alphabet 119871 when Ha is small The nonlinear speed decreasesto a particular value in the vertical direction and thereaftertraces a constant value beyond it as thewavenumber increasesin the supercritical stable region However when the effectof the magnetic field is increased the nonlinear wave speeddecreases in magnitude and sketches an almost linear con-stant profile The magnitude of 119873

119888in the supercritical stable

region corresponding to 120591 = 0 remains inbetween the valuescorresponding to 120591 lt 0 and 120591 gt 0

422 Sideband Instability Let us consider the quasi-mono-chromatic wave of (B4) exhibiting no spatial modulation ofthe form

120578 = Γ119904(1199052) = Γ0119890minus1198941198871199052 (22)

where Γ0and 119887 are defined by (B10) and (B11) such that

120596119904+ 119887 is the wave frequency and 119896(gt 0) is the modulation

wavenumberIf one considers a band of frequencies centered around120596

119904

the interaction of one side-mode with the second harmonicwould be resonant with the other side-mode causing thefrequency to amplify This leads to an instability known assideband instability [60 61]

To investigate such an instability Γ119904(1199052) is subjugated

to sideband disturbances of bandwidth 119896120572 [12 56 57]


0 02 04 06 08 1

0

001

002

003

minus001

minus002

minus003

120591 = 1

120591 = 0

120591 = minus1

CR

k

0 1

0

00025

0005

minus00025

minus0005

120591 = 1

120591 = 0

120591 = minus1

CR

02 04 06 08k

Ha = 0001

Re = 2

Ha = 03

Re = 2

(a)

0 02 04 06 08 1

0

001

002

003

004

005

0 02 04 06 08 1

0

0005

001

0015

CR

minus001

minus0005

minus001

minus0015

120591 = 1 120591 = 1

120591 = 0

120591 = 0

120591 = minus1

120591 = minus1

CR

kk

Ha = 0001

Re = 2

Ha = 03

Re = 2

(b)

Figure 5 Growth rate curves when We = 1 (a) 120573 = 75∘ and (b) 120573 = 90

∘

The explicit expression for the eigenvalues is found as (referto Appendix C)

120599 =1

2(tr (M) plusmn radictr2 (M) minus 4 det (M))

= minus119888119894+ 11989621198691119903plusmn1

2radic4 (1198882119894+ 1198962V2) minus

81198941198694119888119894V119896

1198692

= 1206031plusmn 1206032

(23)

where

tr (M) = 2119888119894+ 211989621198691119903minus 41198692

1003816100381610038161003816Γ11990410038161003816100381610038162

= minus2119888119894+ 211989621198691119903

det (M) = 1198692

11198964minus (V2 + 2119888

1198941198691) 1198962minus2119894119888119894V1198694

1198692

119896

(24)

It should be remarked that the above expression for 120599 istrue only if V sim O(120572) However when V = 0 it is easily

seen from (23) that 1205991= 1206031+ 1206032= 11989621198691119903

lt 0 and 1205992=

1206031minus 1206032= minus2119888

119894+ 11989621198691119903

lt 0 (when 119888119894gt 0) implying that

the system is stable to the sideband disturbances as 1199052rarr infin

For nonzero V the eigenvalues depend on the dimensionlessflow parameters If 120603

2gt 0 and is less than the absolute value

of 1206031 the sideband modes stabilize the system as 119905

2rarr infin

However if 1206032gt 0 and is greater than the absolute value of

1206031 only one of the modes is sideband stable On the other

hand if (1198882119894+ 1198962V2) minus (8119894119869

4119888119894V119896 119869

2) lt 0 again one of the

modes is sideband stable

5 Equilibria of a Bimodal Dynamical System

Considering the initial thickness of the amplitude to beone Gjevik [58] analyzed the amplitude equations by rep-resenting the amplitude using a truncated Fourier seriesand by imposing restrictions on its coefficients The velocityalong the mean flow direction and the corresponding surfacedeflection moving with this velocity were deduced by posing


0

1

2

3

4

5

6

0 02 04 06 08 10

05

1

15

2

25

3

k

0 02 04 06 08 1

0 02 04 06 08 1 0 02 04 06 08 1

k

0 02 04 06 08 1

k

0 02 04 06 08 1

k

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

CL

0

05

1

15

2

25

CL

CL

0

1

2

3

4

5

CL

0

1

2

3

4

5

6

CLV

0

1

2

3

4

5

6

CLV

Ha = 0001

Re = 2

Ha = 05

Re = 2

Re = 2

Ha = 05

Re = 5

Re = 5

Ha = 0001

Re = 5

Ha Ha

Figure 6 Variation of the linear wave velocity 119862119871119881 with respect to Ha and 119862

119871versus 119896


lowast lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 0

lowast

lowast

lowast lowast

lowast

lowast

lowast lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast

lowast lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 02120591 = 0

lowast lowast

lowast


lowast lowast


lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = 0

lowast lowast lowast lowast

lowast lowast


lowast


lowast

lowast lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 1


lowast lowast


lowast lowast

lowast lowast


lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 02120591 = 1

lowast lowast

lowast lowast


lowast

lowast

lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

23

Ha = 04120591 = 1


lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

Ha = 005120591 = minus1


lowast

lowast lowast

lowast

lowast

lowast lowast


lowast lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

1

2

2

3

3

Ha = 02120591 = minus1





lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = minus1

Figure 7 Variation of 119896 with respect to Re when 120573 = 50∘ and 119878 = 5 (1) 119862

119877lt 0 and 119869

2lt 0 (2) 119862

119877lt 0 and 119869

2gt 0 (3) 119862

119877gt 0 and 119869

2lt 0 The

strip enclosing asterisks indicates that 119862119877gt 0 and 119869

2gt 0 The broken line within the strip represents 119896

119898= 119896119888radic2

the problem into a dynamical system Gottlieb and Oron[62] and Dandapat and Samanta [12] also expanded theevolution equation using a truncated Fourier series to derivea modal dynamical system The results in Gottlieb and Oron[62] showed that a two-mode model was found to coincidewith the numerical solution along the Hopf bifurcationcurve Based on this confidence the stability of the bimodaldynamical system was assessed in Dandapat and Samanta[12]

In this section the stability of a truncated bimodaldynamical system is analyzed using the approach followed bythe above authors but using the assumptions considered inGjevik [58]The coupled dynamical system and its entries arelisted in Appendix D

It should be remarked that in the 119896 minus Re plane theequations 119887

111= 0 and 120597[119896b

111]120597119896 = 0 give the neutral

stability curve and the curve corresponding to the maximumrate of amplification for linear disturbances The expression

V119888=

119889

1198891199051205791(119905) = 119888

111+ [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]

times 1198862(119905) + 119888

1311198862

1(119905) + 119888

1321198862

2(119905)

(25)

measures the velocity along the direction of themeanflowof asteady finite-amplitude wave The surface deflection ℎ(119909 119905)moving with velocity V

119888along the mean flow direction in a

coordinate system 1199091015840= 119909 + 120579

1is calculated from (D1) and

reads as

ℎ (119909 119905) = 1 + 2 [1198611cos1199091015840 + 119861

2cos (21199091015840 minus 120601)] (26)

The steady solutions of the system (D2a) and (D2b)correspond to the fixed points of (D4a)ndash(D4c) In additionto the trivial solution 119886

1(119905) = 0 119886

2(119905) = 0 and 120601(119905) =

0 (1199111(119905) = 0 119911

2(119905) = 0) system (D4a)ndash(D4c) offers

nontrivial fixed points [62ndash64] These fixed points can beclassified as pure-mode fixed points (where one of the fixedpoints is zero and the others are nonzero) mixed-mode fixedpoints (nonzero fixed points with a zero or nonzero phasedifference) and traveling waves with nonzero constant phasedifference

51 Pure-Mode The fixed points in this case are obtained bysetting 119891

119894(1198861 1198862 120601) = 0 119894 = 1 2 3 in (D4a)ndash(D4c) This

gives 1198861(119905) = 0 and 119886

2

2(119905) = minus119887

211119887232

The solution existsif sgn(119887

211119887232

) = minus1 For convenience 11988622(119905) = 119886

2 is set From


Table 1 Parameter values for which the nonlinear wave explodes

120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119904= 03671 119896

119904= 07933 119896

119904= 04467 119896

119904= 08707

Ha = 02 119896119904= 02309 119896

119904= 06056 119896

119904= 03438 119896

119904= 07045

Ha = 04 119896119904= 00268 119896

119904= 04399 119896

119904= 02561 119896

119904= 05686

120591 = 0

Ha = 0 119896119904= 02733 119896

119904= 07132 119896

119904= 03734 119896

119904= 07992

Ha = 02 119896119904= 01536 119896

119904= 05517 119896

119904= 02974 119896

119904= 06588

Ha = 04 119896119904= 0 119896

119904= 04073 119896

119904= 02319 119896

119904= 05436

120591 = minus1

Ha = 0 119896119904= 01211 119896

119904= 06237 119896

119904= 02823 119896

119904= 07205

Ha = 02 119896119904= 0 119896

119904= 04861 119896

119904= 02301 119896

119904= 06049

Ha = 04 119896119904= 0 119896

119904= 03602 119896

119904= 01835 119896

119904= 05094

Table 2 Parameter values for which the nonlinear wave attains a finite amplitude

120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119888= 07342 119896

119888= 15866 119896

119888= 08933 119896

119888= 17415

Ha = 02 119896119888= 04618 119896

119888= 12112 119896

119888= 06875 119896

119888= 14092

Ha = 04 119896119888= 00536 119896

119888= 08799 119896

119888= 05121 119896

119888= 11371

120591 = 0

Ha = 0 119896119888= 05467 119896

119888= 14263 119896

119888= 07468 119896

119888= 15984

Ha = 02 119896119888= 03071 119896

119888= 11034 119896

119888= 05947 119896

119888= 13177

Ha = 04 119896119888= 0 119896

119888= 08145 119896

119888= 04638 119896

119888= 10872

120591 = minus1

Ha = 0 119896119888= 02421 119896

119888= 12475 119896

119888= 05646 119896

119888= 14411

Ha = 02 119896119888= 0 119896

119888= 09722 119896

119888= 04602 119896

119888= 12099

Ha = 04 119896119888= 0 119896

119888= 07205 119896

119888= 01834 119896

119888= 10187

(D4c) 119888121

cos120601(119905) minus 119887121

sin120601(119905) = (1198962)11986010158401015840

1119886 is obtained

From this condition the fixed point for 120601(119905) is derived as

120601 (119905) = 120601 (119905) = tanminus1 [(4119887121

119888121

plusmn 11989611986010158401015840

1119886

times 41198872

121+ 41198882

121minus 1198962119860101584010158402

1119886212

)

times (41198872

121minus 1198962119860101584010158402

11198862)minus1

]

(27)

provided that the quantity within the square root is realand positive The stability of the nonlinear dynamical system(D4a)ndash(D4c) can be locally evaluated using the eigenvaluesof the matrix obtained after linearizing the system aroundthe fixed points The linear approximation of the dynamicalsystem (D4a)ndash(D4c) can be represented in matrix notationas

((

(

1198891198861(119905)

119889119905

1198891198862(119905)

119889119905

119889120601 (119905)

119889119905

))

)

= (

1199011

0 0

0 1199012

0

0 1199013

1199014

)(

1198861(119905)

1198862(119905)

120601 (119905)

) + (

1199021

1199022

1199023

)

(28)

such that

1199011= 119887111

+ (119887121

cos120601 + 119888121

sin120601) 119886 + 119887132

1198862

1199012= 119887211

+ 3119887232

1198862

1199013= minus2119887

121sin120601 + 2119888

121cos120601 minus 2119896119860

10158401015840

1119886

1199014= minus (2119887

121cos120601 + 2119888

121sin120601) 119886

1199021= 0 119902

2= minus2119887

2321198863

1199023= 11989611986010158401015840

11198862+ (2119887121

cos120601 + 2119888121

sin120601) 120601119886

(29)

The stability of the linear system (28) depends on theeigenvalues of the coefficient matrix with entries 119901

119894 The

eigenvalues are found as

1205821= 119887211

+ 31198862119887232

= minus2119887211

1205822= minus2119886 (119887

121cos120601 + 119888

121sin120601)

1205823= 119887111

+ 119886 (119887121

cos120601 + 119888121

sin120601) + 119887132

1198862

(30)


0

0005

001

0015

002

0025

003

0035

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

0

0005

001

0015

002

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

Nc

Nc

Ha = 005

Re = 5

Ha = 005

Re = 5

Ha = 02

Re = 5

Ha = 02

Re = 5

Figure 8 Threshold amplitude and nonlinear wave speed in the supercritical stable region for 120573 = 50∘ and 119878 = 5 (1) 120591 = minus1 (2) 120591 = 0 and

(3) 120591 = 1

TheHopf bifurcation at the critical threshold is defined by1198611= 11989621198621 and this yields 120582

1= 24120576119896

41198621gt 0 This eigenvalue

being independent of 120591 increases when the effects of surface-tension and Hartmann number increase Therefore at thecritical threshold the fixed points corresponding to thepure-mode are unstable Beyond the neutral stability limitwhere the flow is linearly stable (119862

119877lt 0) and where

the stable wavenumber region decreases when the directionof the shear offered by the wind changes from uphill todownhill direction the eigenvalue may remain positive andstill destabilize the system as shown in Figure 9 The stablewavenumbers corresponding to the linear stability threshold(refer to Figure 5 withHa = 03 and Re = 2) are considered toplot the eigenvalue120582

1 Although themagnitude of120582

1remains

larger for 120591 lt 0 than for 120591 ge 0 being positive it plays adestabilizing role

52 Mixed-Mode When 1198861(119905) 1198862(119905) = 0 the fixed points in

this case correspond to mixed-mode Considering 120601(119905) = 0

and 119886lowast1 119886lowast2 and 120601lowast to be the fixed points of (D4a)ndash(D4c) the

nonlinear system can be linearized around the fixed pointsThen the stability (instability) of the fixed points demandsall of the eigenvalues of the linearized Jacobian matrix to benegative (at least one of them to be positive)The nine entries119905119894119895(119894 119895 = 1 2 3) of the 3 times 3 Jacobian matrix arising due to

linearization are the following

11990511

= 119887111

+ (119887121

cos120601lowast + 119888121

sin120601lowast) 119886lowast2

+ 3119887131

119886lowast2

1+ 119887132

119886lowast2

2

11990512

= (119887121

cos120601lowast + 119888121

sin120601lowast) 119886lowast1+ 2119887132

119886lowast

1119886lowast

2

11990513

= (minus119887121

sin120601lowast + 119888121

cos120601lowast) 119886lowast1119886lowast

2

11990521

= 2 (119887221

cos120601lowast minus 119888221


119886lowast

1119886lowast

2

11990522

= 119887211

+ 119887231

119886lowast2

1+ 3119887232

119886lowast2

2

11990523

= minus (119887221

sin120601lowast + 119888221

cos120601lowast) 119886lowast21


0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

1205821=minus2b

211

k

120591 = 1

120591 = 0

120591 = minus1

120591 = minus2

120591 = 2

Ha = 03

Re = 2

Figure 9 Eigenvalue1205821as a function of 119896whenWe = 1 and120573 = 75

∘

11990531

= 211989611986010158401015840

1119886lowast

1minus2119886lowast

1

119886lowast2

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990532

= minus 211989611986010158401015840

1119886lowast

2+ 2 (119888121


sin120601lowast)

+119886lowast2

1

119886lowast22

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990533

= minus 2 (119888121

sin120601lowast + 119887121


minus (119887221


sin120601lowast)119886lowast2

1

119886lowast2

(31)

It should be remarked that Samanta [65] also discussedthe mixed-mode for the flow of a thin film on a nonuni-formly heated vertical wall However it should be notedthat the Jacobian was computed not by considering a zerophase difference with 119886

1(119905) 1198862(119905) = 0 but by evaluating the

Jacobian first by imposing 1198861(119905) 1198862(119905) 120601(119905) = 0 and then by

substituting 120601 = 0 in the computed Jacobian If one considers1198861(119905) 1198862(119905) 120601(119905) = 0 an overdetermined system is obtained

53 Traveling Waves Fixed points of (D4a)ndash(D4c) with 120601

being a nonzero constant correspond to traveling waves Themodal amplitudes are considered small in the neighborhoodof the neutral stability limit After rescaling 119886

1(119905) rarr 120572119898

1(119905)

and 1198862(119905) rarr 120572

21198982(119905) the following system is obtained

1198891198981(119905)

119889119905= 119887111

1198981(119905) + 120572

2

times [119887121

cos120601 (119905) + 119888121

sin120601 (119905)]1198981(119905)1198982(119905)

+ 1205722119887131

1198983

1(119905) + O (120572

4)

(32a)

1198891198982(119905)

119889119905= 119887211

1198982(119905)

+ [119887221

cos120601 (119905) minus 119888221

sin120601 (119905)]1198982

1(119905)

+ 1205722119887231

1198982

1(119905)1198982(119905) + O (120572

4)

(32b)

119889120601 (119905)

119889119905= minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198982

1(119905)

1198982(119905)

+ 120572211989611986010158401015840

11198982

1(119905) + 2120572

2

times [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]1198982(119905) + O (120572

4)

(32c)

To the O(1205722) there are no nontrivial fixed points in theabove system The phase evolution is governed by equatingthe right-hand side of (32c) to zero by considering terms upto O(1205722)

tan120601 (119905) =

(minus11989711198972plusmn 1198973(1198972

1+ 1198972

2minus 1198972

3)12

)

(11989721minus 11989723)

(33)

where 1198971= (1 + 2120572

2119887121

1198982

2(119905)119887221

1198982

1(119905)) 1198972= (119888221

119887221

) minus

(21205722119888121

1198982

2(119905)119887221

1198982

1(119905)) and 119897

3= 120572211989611986010158401015840

11198982(119905)119887221

In theabove equation if only the leading order effect is consideredthe fixed point is found as

120601lowast(119905) = tanminus1 (minus119888

221

119887221

) + O (1205722) (34)

Equating (32a) to zero and using (34) it is found that

1205722119887131

1198982

1(119905) = minus119887

111minus 120572212059311198982(119905) (35)

Such a solution exists if [119887111

+ 120572212059311198982(119905)]119887131

lt 0 where1205931= (119887121

cos120601lowast + 119888121

sin120601lowast) A quadratic equation in1198982(119905)

is obtained by considering (32b) as

1205724119887231

12059311198982

2(119905) + 120572

2(119887111

119887231

minus 119887211

119887131

+ 12059311205932)

times 1198982(119905) + 120593

2119887111

= 0

(36)

where 1205932

= 119887221


sin120601lowast For this equationthere are two solutions The existence of such solutions tobe a real number demands (119887

111119887231

minus 119887211

119887131

+ 12059311205932)2minus

4120572412059311205932119887111

119887231

gt 0 Since 119887111

= 0 in the neutral stabilitylimit (36) gives the amplitude of the nonzero traveling waveas

1198982(119905) =

119887211

119887131

minus 12059311205932

1205722119887231

1205931

=3119896211986211198692

1205722 (119861101584010158401minus 4119896211986210158401015840

1) 1205931

(37)

When 1198692= 0 119898

2(119905) is zero and this corresponds to finding

the critical Reynolds number in the neutral stability limit [12]


08

10

12

minus120587k x 120587k

t = 15 t = 145

h(xt)

(a)

08

10

12

minus120587k x 120587k

t = 2995

t = 2985

h(xt)

(b)

08

10

12

minus120587k x 120587k

t = 2t = 0

h(xt)

(c)

Figure 10 (a) (b) and (c) represent the surface-wave instability curves reproduced from Joo et al [7] corresponding to Figures 5(b) 7(a)and 9(a) respectively in their study

If119898lowast1and119898lowast

2are the fixed points obtained from (35) and (36)

the stability or instability of the fixed points corresponding totraveling waves depends on the sign of the eigenvalues of thefollowing matrix

119869 = (

21205722119887131

119898lowast2

112057221205931119898lowast

1minus1205722 sin120601lowast (119887

121+119888121

119887221

119888221

)119898lowast

1119898lowast

2

2 (1205932+ 1205722119887231

119898lowast

2)119898lowast

11205722119887211

119898lowast2

10

2120572211989611986010158401015840

1119898lowast

1minus21205722(119887121

+119888121

119887221

119888221

) sin120601lowast minus(212057221205931119898lowast

2+ 1205932

119898lowast2

1

119898lowast2

)

) (38)

6 Nonlinear Development ofthe Interfacial Surface

The nonlinear interactions are studied numerically beyondthe linear stability threshold in a periodic domain D =

(119909 119905) 119909 isin (minus120587119896119898

120587119896119898

) 119905 isin [0infin) by solving (10)subject to the initial condition ℎ(119909 0) = 1 minus 01 cos(119896

119898119909)

A central difference scheme which is second-order accuratein space and a backward Euler method which is implicit inforward time are adopted to solve the problem in MATLAB


0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03

Figure 11 Free surface profiles at Re = 5 120573 = 45∘ and 119878 = 5 when 120591 = 0 at time 119905 = 250

0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)

Figure 12 Maximum film thickness profiles at Re = 5 120573 = 45∘ and 119878 = 5 (a) Ha = 005 and (b) Ha = 02


0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t

Figure 13Maximumandminimumamplitude profiles correspond-ing to Re = 5 120573 = 45

∘ 119878 = 5 and Ha = 03

The local truncation error for this numerical scheme is T =

O(Δ119905 Δ1199092) The computations are performed with a smalltime increment Δ119905 = 10

minus3 The numerical scheme is alwaysstable (since a backward Euler method is used) and doesnot build up errors The number of nodes along the spatialdirection is 119873 = 800 such that the spatial step length isΔ119909 = 2120587119873119896

119898 An error tolerance of 10

minus10 is set andthe simulations are stopped once the absolute value of theerror becomes smaller than this value Also the numericalsimulations show no particular deviation from the resultsobtained neither when the spatial grid points are doublednor when the time step is further reduced Figures 5(b) 7(a)and 9(a) available in Joo et al [7] are reproduced in Figure 10to show the correctness of the numerical scheme This givesconfidence in applying it to the evolution equation consideredhere

The effect of the transverse magnetic field is illustratedin Figure 11 The wave structure when the magnetic fieldstrength is zero displays a steep curvier wave than whenHa gt

0 The surface-wave instability decreases when the strengthof the applied magnetic field increases This reveals thestabilizing mechanism of the transversely applied magneticfield

Figure 12 displays maximum amplitude profiles of thesheared flow for two different values of Ha on a semi-inclinedplane The maximum amplitude initially increases and thendecreases Due to saturation of the nonlinear interactions theinitial perturbation of the free surface is damped after a longtime For a short time the amplitude profile correspondingto 120591 gt 0 remains smaller than the ones corresponding to120591 le 0 Also for different Ha the amplitudes correspondingto 120591 lt 0 remain larger than those corresponding to 120591 = 0 upto a certain time (see also Figure 13) However these trendschange over time When the magnitude of the Hartmannnumber increases the time required for the amplitudescorresponding to 120591 lt 0 to decrease below the amplitudecorresponding to 120591 = 0 increases (Figures 12(a) 12(b) and13) In addition it is also observed that the time needed for

the amplitude profiles corresponding to 120591 = minus1 to dominatethe amplitude profiles corresponding to 120591 = minus15 increaseswhen the value of Ha increases (Figures 12(a) and 12(b))

The surface waves emerging at the free surface arecaptured and presented in Figure 14 For 120591 = 0 a wave whichhas a stretched front is observed after a long time When120591 = minus1 a one-humped solitary-like wave is formed whena large amplitude wave and a small capillary ripple coalescetogether (Figure 14(b)) at the time of saturation (a certaintime after which all of the waves have the same structure andshape) Figure 14 demonstrates that the instability measuredby the wave height decreases when the shear is induced alongthe uphill direction

Tsai et al [28] pointed out that when the intensity of themagnetic field increases the flow retards and stabilizes thesystem The interfacial surface subjected to air shear affectsthe velocity and flow rate 119902(119909 119905) = int

ℎ

0119906(119909 119905)119889119910 for different

values of 120591 and Ha (Figure 15) The numerical interactionsreveal that the velocity and the flow rate can either increaseor decrease depending on the direction the air shears thedeformable free interface In conjunction to the nonlinearwave speed traced in Figure 8 the velocity profiles traced inFigure 15 show a similar response For Ha = 0 and 120591 gt 0the streamwise velocity and the flow rate increase Also thetransverse velocity is larger for Ha = 0 than when Ha gt 0Furthermore the velocity and the flow rate decrease whenthe strength of the applied magnetic field increases Whenthe magnitude of the shear induced in the uphill directionis increased (by considering a small negative 120591 value) forlarge values of Ha constant velocity and flow rate profiles areobserved

It is inferred from the nonlinear simulations that theeffect of magnetic field and wind shear greatly affects thethickness of the interfacial free surface When the strength ofthe magnetic field increases the flow has a uniform velocityand constant discharge For Ha gt 0 the transverse velocityshows a small growth in the positive directionThe growth ofthe unwanted development of the free surface is retarded bythemagnetic field when it acts opposite to the flow directionSuch a growth-reducing mechanism of the amplitude inparticular can be further enhanced by applying uphill shearon the free surface This inclusion causes further reductionin the values of streamwise and transverse velocities therebyleading to constant flux situation and improved stability

7 Conclusions and Perspectives

Linear and nonlinear analyses on the stability of a thin filmsubjected to air shear on a free surface in the presence ofa transversely applied external magnetic field of constantstrength have been studied Instead of obtaining the solutionsat different orders of approximation using a power seriesapproach [37ndash39 43] the solutions of equations arising atvarious orders have been straightforwardly solved [28]

The linear stability results gave firsthand informationabout the stability mechanism The modal interaction


0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)

Figure 14 Evolution of the surface-wave instability at Re = 5 120573 = 45∘ 119878 = 5 and Ha = 03 (a) 120591 = 0 and (b) 120591 = minus1 The waves are plotted

at a gap of 01 time units in a periodic domain

phenomenon was studied by deriving a complex Ginzburg-Landau-type equation using the method of multiple scalesThe stability regimes identified by the linear theory werefurther categorized using the weakly nonlinear theory byconsidering a filtered wave to be the solution of the complexGinzburg-Landau equation When the nonlinear amplifica-tion rate (119869

2) is positive an infinitesimal disturbance in the

linearly unstable region attains a finite-amplitude equilibriumstate The threshold amplitude and the nonlinear wave speedexist in the supercritical stable region Stability of a fil-tered wave subject to sideband disturbances was consideredand the conditions under which the eigenvalues decay intime were mathematically identified Considering the initialthickness to be one and imposing restrictions upon theamplitude coefficients a truncated Fourier series was used toderive a bimodal dynamical systemThe stability of the pure-mode mixed-mode and the traveling wave solution has beenmathematically discussed

Although there are several investigations based onweaklynonlinear theory [28 37ndash40 43] the complete nonlinearevolution equation was not studied numerically beyond thelinear stability threshold in the presence of an externalmagnetic field especially in a short-circuited system In thisregard the nonlinear interactions have been numericallyassessed using finite-difference technique with an implicittime-stepping procedure Using such a scheme the evolutionof disturbances to a small monochromatic perturbation was

analyzedThe destabilizing mechanism of the downhill shearwas identified with the help of computer simulation Also ittakes a long time for the amplitude profiles corresponding tothe zero-shear-induced effect to dominate those correspond-ing to the uphill shear when the strength of the magneticfield increases Furthermore among the uphill shear-inducedflows the smaller the uphill shear-induced effect is the longerit takes for the respective amplitude to be taken over bythe amplitude corresponding to large values of uphill shearprovided that the magnetic fields intensity increases

The velocity and the flow rate profiles gave a clearunderstanding of the physical mechanism involved in theprocess When the magnetic field effect and the air shearare not considered the gravitational acceleration the inertialforce and the hydrostatic pressure determine the mechanismof long-wave instability by competing against each otherThese forces trigger the flow amplify the film thicknessand prevent the wave formation respectively [66] Whenthe magnetic field effect is included it competes with otherforces to decide the stability threshold The applied magneticfield retards the flow considerably by reducing the transversevelocity This phenomenon reduces the wave thickness butit favors hydrostatic pressure and surface-tension to promotestability on a semi-inclined plane However when the airshear is considered it can either stabilize or destabilizethe system depending on the direction it shears the freesurface For downhill shear the transverse velocity increases


minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02

Figure 15 (1) Streamwise velocity (2) flow rate and (3) transverse velocity profiles at Re = 5 120573 = 45∘ and 119878 = 5 in a periodic domain

resulting in increased accumulation of the fluid under atypical cusp thereby favoring inertial force Overall thismechanism causes the film thickness to increase For uphillshear the results are opposite

The present investigation suggests the inclusion of super-ficial shear stress on the interfacial surface to enhance thestability of the film or on the other hand shows the natural

effect of air shear on the films stability under natural condi-tions in particular to those films subjected to a transverselyapplied uniform external magnetic field in a short-circuitedsystem Future research activities may include (i) assessingnonisothermal effects (ii) studying non-Newtonian filmsand (iii) considering the contribution of electric field Thelast point would demand considering Poisson equation to


describe electric potential together with appropriate bound-ary conditions

Appendices

A Dimensionless Equations

The dimensionless equations are given by

119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)

and the respective boundary conditions are

119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)

B Multiple Scale Analysis

The film thickness119867 is expanded as

119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)

Exploiting (16) using (20) and (B1) the following equation isobtained

(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)

where the operators 1198710 1198711 and 119871

2and the quantities119873

2and

1198733are given as follows

1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)

The equations are solved order-by-order up to 119874(1205723) [12

14 56 57] to arrive at an equation related to the com-plex Ginzburg-Landau type for the perturbation amplitude120578(1199091 1199051 1199052)

120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)

It should be remarked that in (B4) an imaginary apparentlyconvective term of the form

119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)

could be additionally considered on the left-hand side of(B4) as done in Dandapat and Samanta [12] and Samanta[65] provided that such a term is of O(120572) and V lt 0 Such aterm arises in the secular condition at119874(120572

3) as a contribution

arising from one of the terms from 11987111198671at 119874(120572

2) The

solution of (B4) for a filtered wave in which the spatialmodulation does not exist and the diffusion term in (B4)becomes zero is obtained by considering 120578 = Γ

0(1199052)119890[minus119894119887(1199052)1199052]


This leads to a nonlinear ordinary differential equation for Γ0

namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)

where the real and imaginary parts when separated from(37) give

119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692

= 0 in (B8) the Ginzburg-Landau equation isreduced to a linear ordinary differential equation for theamplitude of a filtered wave The second term on the right-hand side in (B8) induced by the effect of nonlinearitycan either accelerate or decelerate the exponential growthof the linear disturbance depending on the signs of 119862

119877and

1198692 The perturbed wave speed caused by the infinitesimal

disturbances appearing in the nonlinear system can bemodified using (B8)The threshold amplitude 120572Γ

0from (B8)

is obtained as (when Γ0is nonzero and independent of 119905

2)

120572Γ0= radic

119862119877

1198692

(B10)

Equation (B9) with the use of (B10) gives

119887 =1198621198771198694

12057221198692

(B11)

The speed119873119888of the nonlinearwave is nowobtained from (26)

by substituting 120578 = Γ0119890[minus1198941198871199052] and by using 119905

2= 1205722119905 This gives

the nonlinear wave speed as

119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)

C Sideband Stability

The sideband instability is analyzed by subjecting Γ119904(1199052) to

sideband disturbances in the form of (Δ ≪ 1) [12 56 57]

120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)

in (B4) Neglecting the nonlinear terms and separating thecoefficients of Δ119890119894(119896119909minus1198871199052) and Δ119890

minus119894(119896119909minus1198871199052) the following set ofequations are derived (V = (2120576119896120572)(119861

1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γminus(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γ+(1199052)

(C2b)

where the barred quantities are the complex conjugatescorresponding to their counterparts For convenience theabove system is represented as

(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)

whereM is a 2 times 2 matrix whose entries are

11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222

= minus119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692minus 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890

1205991199052 the eigenvaluesare obtained from |M minus 120599119868| = 0 and the stability of (22)subject to sideband disturbances as 119905

2rarr infin demands that

120599 lt 0 The eigenvalues are found as follows

120599 =1

2(tr (M) plusmn radictr2 (M) minus 4 det (M)) (C5)

D Bimodal Dynamical System

In order to analyze the stability of the bimodal dynamicalsystem the film thickness is expanded as a truncated Fourierseries (119911

119899(119905) is complex and a bar above it designates its

complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)

and is substituted in the evolution equation (10) A Taylorseries expansion is then sought to be about ldquo1rdquo and the


following coupled dynamical system is obtained from thecoefficients of 119890119894119896119909 and 119890

2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)

It should be remarked that the higher-order terms arisingwhile deducing (D2a) and (D2b) are neglected by assumingthat |119911

1| ≪ 1 and |119911

2| ≪ |119911

1| The coefficients of the above

system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)

The complex conjugates occurring in system (D3) canbe avoided by representing 119911

119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579

119899(119905) are real functions such that 119899 =

1 2 Using polar notation the following three equations areobtained

1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)

While the coefficient of 119890119894120579119899(119905) term yields (D4a) and

(D4b) (D4c) is obtained from the two equations whichresult as the coefficient of 119894119890119894120579119899(119905) term with the impositionthat the phase relationship is defined as 120601(119905) = 2120579

1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896

are real numbers and represent the realand imaginary parts of a typical 119886

119894119895119896term in (D3) such that

119894 119896 = 1 2 and 119895 = 1 2 3

References

[1] D J Benney ldquoLongwaves on liquid filmsrdquo Journal ofMathemat-ical Physics vol 45 pp 150ndash155 1966

[2] B Gjevik ldquoOccurrence of finite-amplitude surface waves onfalling liquid filmsrdquoPhysics of Fluids vol 13 no 8 pp 1918ndash19251970

[3] R W Atherton and G M Homsy ldquoOn the derivation ofevolution equations for interfacial wavesrdquoChemical EngineeringCommunications vol 2 no 2 pp 57ndash77 1976

[4] S P Lin and M V G Krishna ldquoStability of a liquid filmwith respect to initially finite three-dimensional disturbancesrdquoPhysics of Fluids vol 20 no 12 pp 2005ndash2011 1977

[5] A Pumir P Manneville and T Pomeau ldquoOn solitary wavesrunning down an inclined planerdquo Journal of Fluid Mechanicsvol 135 pp 27ndash50 1983

[6] J P Burelbach S G Bankoff and S H Davis ldquoNonlinearstability of evaporatingcondensing liquid filmsrdquo Journal ofFluid Mechanics vol 195 pp 463ndash494 1988

[7] SW Joo SHDavis and SG Bankoff ldquoLong-wave instabilitiesof heated falling films Two-dimensional theory of uniformlayersrdquo Journal of Fluid Mechanics vol 230 pp 117ndash146 1991

[8] S W Joo and S H Davis ldquoInstabilities of three-dimensionalviscous falling filmsrdquo Journal of Fluid Mechanics vol 242 pp529ndash547 1992

[9] S Miladinova S Slavtchev G Lebon and J-C Legros ldquoLong-wave instabilities of non-uniformly heated falling filmsrdquo Journalof Fluid Mechanics vol 453 pp 153ndash175 2002

[10] S Miladinova D Staykova G Lebon and B Scheid ldquoEffect ofnonuniform wall heating on the three-dimensional secondaryinstability of falling filmsrdquo Acta Mechanica vol 156 no 1-2 pp79ndash91 2002

[11] B Scheid A Oron P Colinet U Thiele and J C LegrosldquoNonlinear evolution of nonuniformly heated falling liquidfilmsrdquo Physics of Fluids vol 15 no 2 p 583 2003 Erratum inPhysics of Fluids vol 14 pp 4130 2002

[12] B S Dandapat andA Samanta ldquoBifurcation analysis of first andsecond order Benney equations for viscoelastic fluid flowingdown a vertical planerdquo Journal of Physics D vol 41 no 9 ArticleID 095501 2008

[13] R Usha and I M R Sadiq ldquoWeakly nonlinear stability analysisof a non-uniformly heated non-Newtonian falling filmrdquo inProceedings of the 5th Joint ASMEJSME Fluids EngineeringSummer Conference (FEDSM rsquo07) pp 1661ndash1670 August 2007


[14] I M R Sadiq and R Usha ldquoThin Newtonian film flow down aporous inclined plane stability analysisrdquo Physics of Fluids vol20 no 2 Article ID 022105 2008

[15] I M R Sadiq and R Usha ldquoEffect of permeability on theinstability of a non-Newtonian film down a porous inclinedplanerdquo Journal of Non-Newtonian Fluid Mechanics vol 165 no19-20 pp 1171ndash1188 2010

[16] U Thiele B Goyeau and M G Velarde ldquoStability analysis ofthin film flow along a heated porous wallrdquo Physics of Fluids vol21 no 1 Article ID 014103 2009

[17] I M R Sadiq R Usha and S W Joo ldquoInstabilities in a liquidfilm flow over an inclined heated porous substraterdquo ChemicalEngineering Science vol 65 no 15 pp 4443ndash4459 2010

[18] B Uma ldquoEffect of wind stress on the dynamics and stability ofnon-isothermal power-law film down an inclined planerdquo ISRNMathematical Physics vol 2012 Article ID 732675 31 pages2012

[19] B Scheid C Ruyer-Quil U Thiele O A Kabov J C Legrosand P Colinet ldquoValidity domain of the Benney equationincluding the Marangoni effect for closed and open flowsrdquoJournal of Fluid Mechanics vol 527 pp 303ndash335 2005

[20] S Chandrasekhar ldquoThe stability of viscous flow between rotat-ing cylinders in the presence of a magnetic fieldrdquo Proceedings ofthe Royal Society of London A vol 216 pp 293ndash309 1953

[21] J T Stuart ldquoOn the stability of viscous flow between parallelplanes in the presence of a co-planar magnetic fieldrdquo Proceed-ings of the Royal Society of London A vol 221 pp 189ndash206 1954

[22] R C Lock ldquoThe stability of the flow of an electrically conduct-ing fluid between parallel planes under a transverse magneticfieldrdquo Proceedings of the Royal Society of London A vol 233 pp105ndash125 1955

[23] D-Y Hsieh ldquoStability of a conducting fluid flowing down aninclined plane in a magnetic fieldrdquo Physics of Fluids vol 8 no10 pp 1785ndash1791 1965

[24] Y P Ladikov ldquoFlow stability of a conducting liquid flowingdown an inclined plane in the presence of a magnetic fieldrdquoFluid Dynamics vol 1 no 1 pp 1ndash4 1966

[25] P C Lu and G S R Sarma ldquoMagnetohydrodynamic gravitymdashcapillary waves in a liquid filmrdquo Physics of Fluids vol 10 no 11p 2339 1967

[26] Yu N Gordeev and V V Murzenko ldquoWave film flows of aconducting viscous fluid film in the tangential magnetic fieldrdquoApplied Mathematics andTheoretical Physics vol 3 p 96 1990

[27] D Gao and N B Morley ldquoNumerical investigation of surfacedisturbance on liquidmetal falling film flows in amagnetic fieldgradientrdquoMagnitnaya Gidrodinamika vol 40 no 1 pp 99ndash1132004

[28] J-S Tsai C-I Hung and C-K Chen ldquoNonlinear hydromag-netic stability analysis of condensation film flow down a verticalplaterdquo Acta Mechanica vol 118 pp 197ndash212 1996

[29] Y Renardy and S M Sun ldquoStability of a layer of viscousmagnetic fluid flow down an inclined planerdquo Physics of Fluidsvol 6 no 10 pp 3235ndash3248 1994

[30] H H Bau J Z Zhu S Qian and Y Xiang ldquoA magneto-hydrodynamically controlled fluidic networkrdquo Sensors andActuators B vol 88 no 2 pp 205ndash216 2003

[31] H Kabbani MMarc SW Joo and S Qian ldquoAnalytical predic-tion of flow field in magnetohydrodynamic-based microfluidicdevicesrdquo Journal of Fluids Engineering vol 130 no 9 Article ID091204 6 pages 2008

[32] A E Radwan ldquoHydromagnetic stability of a gas-liquid interfaceof coaxial cylindersrdquoAstrophysics and Space Science vol 161 no2 pp 279ndash293 1989

[33] M C Shen S M Sun and R E Meyer ldquoSurface waves onviscous magnetic fluid flow down an inclined planerdquo Physics ofFluids A vol 3 no 3 pp 439ndash445 1991

[34] M H Chang and C-K Chen ldquoHydromagnetic stability ofcurrent-inducedflow in a small gap between concentric rotatingcylindersrdquo Proceedings of the Royal Society A vol 454 no 1975pp 1857ndash1873 1998

[35] S Korsunsky ldquoLong waves on a thin layer of conducting fluidflowing down an inclined plane in an electromagnetic fieldrdquoEuropean Journal ofMechanics B vol 18 no 2 pp 295ndash313 1999

[36] A J Chamkha ldquoOn two-dimensional laminar hydromagneticfluid-particle flow over a surface in the presence of a gravityfieldrdquo Journal of Fluids Engineering Transactions of the ASMEvol 123 no 1 pp 43ndash49 2001

[37] P-J Cheng and D T W Lin ldquoSurface waves on viscoelasticmagnetic fluid film flow down a vertical columnrdquo InternationalJournal of Engineering Science vol 45 no 11 pp 905ndash922 2007

[38] P-J Cheng ldquoHydromagnetic stability of a thin electricallyconducting fluid film flowing down the outside surface of along vertically aligned columnrdquo Proceedings of the Institution ofMechanical Engineers C vol 223 no 2 pp 415ndash426 2009

[39] P-J Cheng and K-C Liu ldquoHydromagnetic instability of apower-law liquid film flowing down a vertical cylinder usingnumerical approximation approach techniquesrdquo Applied Math-ematical Modelling vol 33 no 4 pp 1904ndash1914 2009

[40] C-K Chen and D-Y Lai ldquoWeakly nonlinear stability analysisof a thin magnetic fluid during spin coatingrdquo MathematicalProblems in Engineering vol 2010 Article ID 987891 17 pages2010

[41] S K Ghosh O A Beg and J Zueco ldquoHydromagnetic freeconvection flowwith inducedmagnetic field effectsrdquoMeccanicavol 45 no 2 pp 175ndash185 2010

[42] O A Beg J Zueco and T B Chang ldquoNumerical analysis ofhydromagnetic gravity-driven thin film micropolar flow alongan inclined planerdquo Chemical Engineering Communications vol198 no 3 pp 312ndash331 2011

[43] P-J Cheng K-C Liu and D T W Lin ldquoHydromagneticstability analysis of a film coating flow down a rotating verticalcylinderrdquo Journal of Mechanics vol 27 no 1 pp 27ndash36 2011

[44] A D D Craik ldquoWind-generated waves in thin liquid filmsrdquoJournal of Fluid Mechanics vol 26 no 2 pp 369ndash392 1966

[45] E O Tuck and J-M Vanden-Broeck ldquoInfluence of surfacetension on jet-stripped continuous coating of sheet materialsrdquoAIChE Journal vol 30 pp 808ndash811 1984

[46] M Sheintuch and A E Dukler ldquoPhase plane and bifurcationanalysis of thin wavy films under shearrdquo AIChE Journal vol 35no 2 pp 177ndash186 1989

[47] G J Zabaras Jr Studies of vertical cocurrent and countercurrentannular gas-liquid flows [PhD thesis] University of Houston1985

[48] A C King and E O Tuck ldquoThin liquid layers supported bysteady air-flow surface tractionrdquo Journal of FluidMechanics vol251 pp 709ndash718 1993

[49] J P Pascal ldquoA two-layer model for a non-Newtonian gravitycurrent subjected to wind shearrdquo Acta Mechanica vol 162 no1ndash4 pp 83ndash98 2003

[50] S K Wilson and B R Duffy ldquoUnidirectional flow of a thinrivulet on a vertical substrate subject to a prescribed uniform


shear stress at its free surfacerdquo Physics of Fluids vol 17 no 10Article ID 108105 2005

[51] J P Pascal and S J D DrsquoAlessio ldquoInstability of power-lawfluid flows down an incline subjected to wind stressrdquo AppliedMathematical Modelling vol 31 no 7 pp 1229ndash1248 2007

[52] S K Kalpathy L F Francis and S Kumar ldquoShear-inducedsuppression of rupture in two-layer thin liquid filmsrdquo Journal ofColloid and Interface Science vol 348 no 1 pp 271ndash279 2010

[53] M J Davis M B Gratton and S H Davis ldquoSuppressingvan der Waals driven rupture through shearrdquo Journal of FluidMechanics vol 661 pp 522ndash529 2010

[54] G W Sutton and A Sherman Engineering Magnetohydrody-namics McGraw-Hill New York NY USA 1965

[55] S Kakac R K Shah and W Aung Handbook of Single-PhaseHeat Transfer John Wiley amp Sons New York NY USA 1987

[56] S P Lin ldquoFinite amplitude side-band stability of a viscous filmrdquoJournal of Fluid Mechanics vol 63 no 3 pp 417ndash429 1974

[57] M V G Krishna and S P Lin ldquoNonlinear stability of a viscousfilmwith respect to three-dimensional side-band disturbancesrdquoPhysics of Fluids vol 20 no 7 pp 1039ndash1044 1977


[59] H-C Chang ldquoOnset of nonlinear waves on falling filmsrdquoPhysics of Fluids A vol 1 no 8 pp 1314ndash1327 1989

[60] J T Stuart and R C DiPrima ldquoThe Eckhaus and Benjamin-Feir resonance mechanismsrdquo Proceedings of the Royal Society ofLondon A vol 362 no 1708 pp 27ndash41 1978

[61] C Godreche and P Manneville Hydrodynamics and NonlinearInstabilities Cambridge University Press New York NY USA1998

[62] O Gottlieb and A Oron ldquoStability and bifurcations of para-metrically excited thin liquid filmsrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 12 pp 4117ndash4141 2004

[63] D Armbruster J Guckenheimer and P Holmes ldquoHeterocliniccycles and modulated travelling waves in systems with O(2)symmetryrdquo Physica D vol 29 no 3 pp 257ndash282 1988

[64] J Mizushima and K Fujimura ldquoHigher harmonic resonance oftwo-dimensional disturbances in Rayleigh-Benard convectionrdquoJournal of Fluid Mechanics vol 234 pp 651ndash667 1992

[65] A Samanta ldquoStability of liquid film falling down a vertical non-uniformly heated wallrdquo Physica D vol 237 no 20 pp 2587ndash2598 2008

[66] M K Smith ldquoThe mechanism for the long-wave instability inthin liquid filmsrdquo Journal of Fluid Mechanics vol 217 pp 469ndash485 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


reported byGordeev andMurzenko [26]They found that theflow suffers from instability not due to the Reynolds numberbut due to the strength of the external electric field

In applications such as magnetic-field-controlled mate-rial-processing systems aeronautics plasma engineeringMEMS technology andmagnetorheological lubrication tech-nologies the hydromagnetic effects are important Alsoliquid metal film flows are used to protect the solid structuresfrom thermonuclear plasma in magnetic confinement fusionreactors and this application requires a better understandingof the instability mechanism arising in a thin magnetohydro-dynamic flow over planar substrates [27] Furthermore thepresence of an external magnetic field regulates the thicknessof a coating film It prevents any form of direct electrical ormechanical contact with the fluid thereby reducing the riskof contamination [28] Renardy and Sun [29] pointed outthat the magnetic fluid is effective in controlling the flowof ordinary fluids and in reducing hydraulic resistance Thereason is that the magnetic fluids can be easily controlledwith external magnetic fields and that coating streamlinedbodies with a layer of less-viscousmagnetic fluid significantlyreduces the shear stress in flow boundaries Magnetohydro-dynamic flow can be a viable option for transporting weaklyconducting fluids in microscale systems such as flows insidea micro-channel network of a lab-on-a-chip device [30 31]In all of the above applications considering the associatedstability problem is important because it gives guidance inchoosing the flow parameters for practical purposes In thisregard in the past two decades the emerging studies onthe hydromagnetic effects have focused their attention onstability problems and on analyzing the flow characteristics[28 29 32ndash43]

The shearing effect of the surrounding air on the fluidin realistic situations induces stress tangentially on the inter-facial surface The hydrodynamic instability in thin films inthe presence of an external air stream attracted attention inthe mid 1960s which led Craik [44] to conduct laboratoryresearch and study theory He found that the instabilityoccurs regardless of the magnitude of the air stream whenthe film is sufficiently thin Tuck and Vanden-Broeck [45]reported on the effect of air stream in industrial applica-tions of thin films of infinite extent in coating technologySheintuch and Dukler [46] did phase plane and bifurcationanalyses of thin wavy films subjected to shear from counter-current gas flows and found satisfactory agreement of theirresults connected to the wave velocity along the floodingcurvewith the experimental results of Zabaras [47] Althoughthe experimental results related to the substrate thickness didnot match exactly with the theoretical predictions still theresults gave qualitative information about the model Thinliquid layer supported by steady air-flow surface-tractionwas reported by King and Tuck [48] Their study modelsthe surface-traction-supported fluid drops observed on thewindscreen of a moving car on a rainy day Incorporating thesuperficial shear offered by the air Pascal [49] studied a prob-lemwhichmodels themechanismofwind-aided spreading ofoil on the sea and found quantitative information regardingthe maximum upwind spread of the gravity current Wilsonand Duffy [50] studied the steady unidirectional flow of a

thin rivulet on a vertical substrate subjected to a prescribeduniform longitudinal shear stress on the free surface Theycategorized the possible flow patterns and found that thedirection of the prescribed shear stress affects the velocityin the entire rivulet The generation of roll waves on thefree boundary of a non-Newtonian liquid was numericallyassessed using a finite volumemethod by Pascal andDrsquoAlessio[51] revealing the significant effect of air shear on theevolution of the flow Their study showed that the instabilitycriteria was conditional and depends on the directionallyinduced shear Kalpathy et al [52] investigated an idealizedmodel suitable for lithographic printing by examining theshear-induced suppression of two stratified thin liquid filmsconfined between parallel plates taking into account the vander Waals force A film thickness equation for the liquid-liquid interface was derived in their study using lubricationapproximationThey found that the effect of shear affects theimaginary part of the growth rate indicating the existenceof traveling waves Furthermore they also observed a criticalshear rate value beyond which the rupture mechanism couldbe suppressed This study motivated Davis et al [53] toconsider the effect of unidirectional air shear on a singlefluid layer For a two-dimensional ultrathin liquid Daviset al [53] showed that the rupture mechanism induced bythe London van der Waals force could be suppressed whenthe magnitude of the wind shear exceeds a critical value asobserved byKalpathy et al [52] Recently Uma [18]measurednumerically the profound effect of unidirectional wind stresson the stability of a condensateevaporating power-law liquidflowing down an incline

In the present investigation the effects of downhill anduphill air shear on a thin falling film in the presence of a trans-verse magnetic field are studied Such an investigation willillustrate the realistic influence of the natural environmentacting upon the flows Or it may illustrate the need to controlthe flow mechanism through artificial techniques whichblow air when the hydromagnetic effects are consideredThe outline of this paper is organized as follows Section 1presents the introduction In Section 2 mathematical equa-tions governing the physical problem are presented Section 3discusses the long-wave Benney-type equation In Section 4linear stability analysis weakly nonlinear stability analysisand the instability arising due to sideband disturbances areanalyzed The equilibria of a truncated bimodal dynamicalsystem is mathematically presented in Section 5 While theresults in Section 6 discuss nonlinear simulations of the filmthickness evolution Section 7 highlights the main conclu-sions of the study and includes future perspectives

2 Problem Description

A thin Newtonian liquid layer of an infinite extent fallingfreely over a plane under the influence of gravitationalacceleration 119892 is considered The flow is oriented towardsthe 119909-axis and the plane makes an angle 120573 with the horizonProperties of the fluid like density (120588) viscosity (120583) andsurface-tension (120590) are constants The magnetic flux densityis defined by the vectorB = 119861

0119895 where119861

0is themagnitude of


Y

120573

h(x t)B0

120582

g

120591a

X

h0





119886= 0





120588(120597U120597119905







F = (119865119909 119865119910) = (minusΣ119861

2

0119906 0) (5)



U = (119906 V) = 0 (6)



= 2120590Π (ℎ) on 119910 = ℎ (119909 119905)

(7)




119886and reads as


on 119910 = ℎ (119909 119905) (8)


120597ℎ

120597119905= V minus 119906

120597ℎ

120597119909on 119910 = ℎ (119909 119905) (9)


0 streamwise


120588]2ℎ20 time by 119897ℎ







3


Ha = Σ1198612

0ℎ2




119886ℎ2

0120588]2




ℎ119905+ 119860 (ℎ) ℎ

119909+ 120576(119861 (ℎ) ℎ

119909+ 119862 (ℎ) ℎ

119909119909119909)119909+ O (120576

2) = 0 (10)


119860 (ℎ) =ReHa

tanh2 (radicHa ℎ)

+120591


(11)

119861 (ℎ) = Re cot120573Ha32


Ha

+ Re2ℎ2Ha52


+Re2ℎHa52


minus3Re2


+ 120591 3Re


minus3ReHa52


+Re ℎHa2


+3





minus1


+3


(12)

119862 (ℎ) =119878ℎ

Haminus

119878





119860 (ℎ) = Re ℎ2+ 120591ℎ

119861 (ℎ) =2

15Re ℎ5(Re ℎ + 120591) minus

1

3Re cot120573ℎ3

119862 (ℎ) = 119878ℎ3

3

(14)







ℎ0= 1 (15a)

1199060=


(ReradicHa


+ReHa


V0= 0 (15c)

1199010= Re cot120573 (1 minus 119910) (15d)


119867119905+ 1198601119867119909+ 1205761198611119867119909119909

+ 1205761198621119867119909119909119909119909

= minus[1198601015840

1119867 +

11986010158401015840

1

21198672+119860101584010158401015840

1

61198673]119867119909

minus 120576 [1198611015840

1119867 +

11986110158401015840

1

21198672]119867119909119909

minus 120576(1198621015840

1119867 +

11986210158401015840

1

21198672)119867119909119909119909119909

minus 120576 [1198611015840

1+ 11986110158401015840

1119867] (119867

119909)2

minus 120576 (1198621015840

1+ 11986210158401015840

1119867)119867119909119867119909119909119909

+ O (1205761198674 1198675 1205762119867)

(16)


1198601= 119860 (ℎ = 1) 119860

1015840

1= 1198601015840(ℎ = 1)

11986010158401015840

1= 11986010158401015840(ℎ = 1)

1198611= 119861 (ℎ = 1) 119861

1015840

1= 1198611015840(ℎ = 1)

11986110158401015840

1= 11986110158401015840(ℎ = 1)

1198621= 119862 (ℎ = 1) 119862

1015840

1= 1198621015840(ℎ = 1)

11986210158401015840

1= 11986210158401015840(ℎ = 1)

(17)







119867(119909 119905) = 120578119890119894(119896119909minus119862119871119905)+119862119877119905 + 120578119890





119871119896 119862119877

isin


119862119871119881

=119862119871

119896= 1198601

119862119877= 1205761198962(1198611minus 11989621198621)

(19)


119877lt 0)





= 0 as 119896119888=



from (120597120597119896)119862119877

= 0 This gives 1198611minus 21198962

1198981198621= 0 such that

119896119898










0 50

1

2

3

4

5

6

7

8

9

10

Unstable

Stable

minus5

120591

Ha = 05

Ha = 01

Ha = 0

Rec

(a)

0 50

1

2

3

4

5

6

7

8

9

10

Stable

Unstable

minus5

120591

Ha = 05

Ha = 01Ha = 0

Rec

(b)


∘



plots of 119862119871119881







1199051= 120572119905 119905

2= 1205722119905 119909

1= 120572119909 (20)

such that

120597

120597119905997888rarr

120597

120597119905+ 120572

120597

1205971199051

+ 1205722 120597

1205971199052

120597

120597119909997888rarr

120597

120597119909+ 120572

120597

1205971199091

(21)


119877sim O(1205722) (or 119861

1minus

11989621198621



0 2 4 6 8 100

02

04

06

08

02

04

06

08

1

0 2 4 6 8 100

1

1 2

3

ST

US

1 2 3

ST

US

120591 = 0 120591 = 1

02

04

06

08

0 2 4 6 8 100

1

1 2 3 ST

US

kc

kc

kc

120591 = minus1

Re Re Re

(a)

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

1 1 1 2

2 2 3 3 3

ST

ST

ST

US US US

0

02

04

06

08

1

kc

0

02

04

06

08

1

kc

0

02

04

06

08

1

kc

120591 = 0 120591 = 1 120591 = minus1

Re Re Re

(b)


∘ and (b) 120573 = 60∘


0 and the



119877and 119869

2[14] If 119862

119877lt 0 and

1198692


119877gt 0 and 119869



2gt 0 the flow is


119877gt 0 and 119869

2lt 0










119904= 0 (Table 1)


00

5

ST

US

1

2

3

4

1

08

06

04

02

minus5

kc

120591

(a)

00

5

ST

US

1 2

3

4

1

08

06

04

02

minus5

kc

120591

(b)

00

5

ST

US

1 4

2

3

1

08

06

04

02

minus5

kc

120591

(c)


Re = 1 (b) 120573 = 75∘ and Re = 1 and (c) 120573 = 60

∘ and Re = 25











120578 = Γ119904(1199052) = Γ0119890minus1198941198871199052 (22)




119904





0 02 04 06 08 1

0

001

002

003

minus001

minus002

minus003

120591 = 1

120591 = 0

120591 = minus1

CR

k

0 1

0

00025

0005

minus00025

minus0005

120591 = 1

120591 = 0

120591 = minus1

CR

02 04 06 08k

Ha = 0001

Re = 2

Ha = 03

Re = 2

(a)

0 02 04 06 08 1

0

001

002

003

004

005

0 02 04 06 08 1

0

0005

001

0015

CR

minus001

minus0005

minus001

minus0015

120591 = 1 120591 = 1

120591 = 0

120591 = 0

120591 = minus1

120591 = minus1

CR

kk

Ha = 0001

Re = 2

Ha = 03

Re = 2

(b)


∘


120599 =1


= minus119888119894+ 11989621198691119903plusmn1

2radic4 (1198882119894+ 1198962V2) minus

81198941198694119888119894V119896

1198692

= 1206031plusmn 1206032

(23)

where

tr (M) = 2119888119894+ 211989621198691119903minus 41198692

1003816100381610038161003816Γ11990410038161003816100381610038162

= minus2119888119894+ 211989621198691119903

det (M) = 1198692

11198964minus (V2 + 2119888

1198941198691) 1198962minus2119894119888119894V1198694

1198692

119896

(24)


seen from (23) that 1205991= 1206031+ 1206032= 11989621198691119903

lt 0 and 1205992=

1206031minus 1206032= minus2119888

119894+ 11989621198691119903






2rarr infin



hand if (1198882119894+ 1198962V2) minus (8119894119869

4119888119894V119896 119869






0

1

2

3

4

5

6

0 02 04 06 08 10

05

1

15

2

25

3

k

0 02 04 06 08 1

0 02 04 06 08 1 0 02 04 06 08 1

k

0 02 04 06 08 1

k

0 02 04 06 08 1

k

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

CL

0

05

1

15

2

25

CL

CL

0

1

2

3

4

5

CL

0

1

2

3

4

5

6

CLV

0

1

2

3

4

5

6

CLV

Ha = 0001

Re = 2

Ha = 05

Re = 2

Re = 2

Ha = 05

Re = 5

Re = 5

Ha = 0001

Re = 5

Ha Ha


119871versus 119896


lowast lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 0

lowast

lowast

lowast lowast

lowast

lowast


lowast lowast

lowast lowast

lowast lowast

lowast

lowast lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 02120591 = 0

lowast lowast

lowast


lowast lowast


lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = 0


lowast lowast


lowast


lowast

lowast lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 1


lowast lowast


lowast lowast

lowast lowast


lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 02120591 = 1

lowast lowast

lowast lowast


lowast

lowast

lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

23

Ha = 04120591 = 1


lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

Ha = 005120591 = minus1


lowast

lowast lowast

lowast

lowast

lowast lowast


lowast lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

1

2

2

3

3

Ha = 02120591 = minus1





lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = minus1


119877lt 0 and 119869

2lt 0 (2) 119862

119877lt 0 and 119869

2gt 0 (3) 119862

119877gt 0 and 119869

2lt 0 The



119898= 119896119888radic2




111= 0 and 120597[119896b

111]120597119896 = 0 give the neutral


V119888=

119889

1198891199051205791(119905) = 119888

111+ [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]

times 1198862(119905) + 119888

1311198862

1(119905) + 119888

1321198862

2(119905)

(25)





reads as

ℎ (119909 119905) = 1 + 2 [1198611cos1199091015840 + 119861

2cos (21199091015840 minus 120601)] (26)


1(119905) = 0 119886

2(119905) = 0 and 120601(119905) =

0 (1199111(119905) = 0 119911





gives 1198861(119905) = 0 and 119886

2

2(119905) = minus119887

211119887232


211119887232


2 is set From



120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119904= 03671 119896

119904= 07933 119896

119904= 04467 119896

119904= 08707

Ha = 02 119896119904= 02309 119896

119904= 06056 119896

119904= 03438 119896

119904= 07045

Ha = 04 119896119904= 00268 119896

119904= 04399 119896

119904= 02561 119896

119904= 05686

120591 = 0

Ha = 0 119896119904= 02733 119896

119904= 07132 119896

119904= 03734 119896

119904= 07992

Ha = 02 119896119904= 01536 119896

119904= 05517 119896

119904= 02974 119896

119904= 06588

Ha = 04 119896119904= 0 119896

119904= 04073 119896

119904= 02319 119896

119904= 05436

120591 = minus1

Ha = 0 119896119904= 01211 119896

119904= 06237 119896

119904= 02823 119896

119904= 07205

Ha = 02 119896119904= 0 119896

119904= 04861 119896

119904= 02301 119896

119904= 06049

Ha = 04 119896119904= 0 119896

119904= 03602 119896

119904= 01835 119896

119904= 05094


120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119888= 07342 119896

119888= 15866 119896

119888= 08933 119896

119888= 17415

Ha = 02 119896119888= 04618 119896

119888= 12112 119896

119888= 06875 119896

119888= 14092

Ha = 04 119896119888= 00536 119896

119888= 08799 119896

119888= 05121 119896

119888= 11371

120591 = 0

Ha = 0 119896119888= 05467 119896

119888= 14263 119896

119888= 07468 119896

119888= 15984

Ha = 02 119896119888= 03071 119896

119888= 11034 119896

119888= 05947 119896

119888= 13177

Ha = 04 119896119888= 0 119896

119888= 08145 119896

119888= 04638 119896

119888= 10872

120591 = minus1

Ha = 0 119896119888= 02421 119896

119888= 12475 119896

119888= 05646 119896

119888= 14411

Ha = 02 119896119888= 0 119896

119888= 09722 119896

119888= 04602 119896

119888= 12099

Ha = 04 119896119888= 0 119896

119888= 07205 119896

119888= 01834 119896

119888= 10187

(D4c) 119888121

cos120601(119905) minus 119887121

sin120601(119905) = (1198962)11986010158401015840

1119886 is obtained


120601 (119905) = 120601 (119905) = tanminus1 [(4119887121

119888121

plusmn 11989611986010158401015840

1119886

times 41198872

121+ 41198882

121minus 1198962119860101584010158402

1119886212

)

times (41198872

121minus 1198962119860101584010158402

11198862)minus1

]

(27)


((

(

1198891198861(119905)

119889119905

1198891198862(119905)

119889119905

119889120601 (119905)

119889119905

))

)

= (

1199011

0 0

0 1199012

0

0 1199013

1199014

)(

1198861(119905)

1198862(119905)

120601 (119905)

) + (

1199021

1199022

1199023

)

(28)

such that

1199011= 119887111

+ (119887121

cos120601 + 119888121

sin120601) 119886 + 119887132

1198862

1199012= 119887211

+ 3119887232

1198862

1199013= minus2119887

121sin120601 + 2119888

121cos120601 minus 2119896119860

10158401015840

1119886

1199014= minus (2119887

121cos120601 + 2119888

121sin120601) 119886

1199021= 0 119902

2= minus2119887

2321198863

1199023= 11989611986010158401015840

11198862+ (2119887121

cos120601 + 2119888121

sin120601) 120601119886

(29)


119894 The


1205821= 119887211

+ 31198862119887232

= minus2119887211

1205822= minus2119886 (119887

121cos120601 + 119888

121sin120601)

1205823= 119887111

+ 119886 (119887121

cos120601 + 119888121

sin120601) + 119887132

1198862

(30)


0

0005

001

0015

002

0025

003

0035

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

0

0005

001

0015

002

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

Nc

Nc

Ha = 005

Re = 5

Ha = 005

Re = 5

Ha = 02

Re = 5

Ha = 02

Re = 5


(3) 120591 = 1


1= 24120576119896






1remains







11990511

= 119887111

+ (119887121

cos120601lowast + 119888121


+ 3119887131

119886lowast2

1+ 119887132

119886lowast2

2

11990512

= (119887121

cos120601lowast + 119888121


119886lowast

1119886lowast

2

11990513

= (minus119887121

sin120601lowast + 119888121


2

11990521

= 2 (119887221



119886lowast

1119886lowast

2

11990522

= 119887211

+ 119887231

119886lowast2

1+ 3119887232

119886lowast2

2

11990523

= minus (119887221

sin120601lowast + 119888221



0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

1205821=minus2b

211

k

120591 = 1

120591 = 0

120591 = minus1

120591 = minus2

120591 = 2

Ha = 03

Re = 2


∘

11990531

= 211989611986010158401015840

1119886lowast

1minus2119886lowast

1

119886lowast2

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990532

= minus 211989611986010158401015840

1119886lowast

2+ 2 (119888121


sin120601lowast)

+119886lowast2

1

119886lowast22

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990533

= minus 2 (119888121

sin120601lowast + 119887121


minus (119887221



1

119886lowast2

(31)







1(119905) rarr 120572119898

1(119905)

and 1198862(119905) rarr 120572


1198891198981(119905)

119889119905= 119887111

1198981(119905) + 120572

2

times [119887121

cos120601 (119905) + 119888121

sin120601 (119905)]1198981(119905)1198982(119905)

+ 1205722119887131

1198983

1(119905) + O (120572

4)

(32a)

1198891198982(119905)

119889119905= 119887211

1198982(119905)

+ [119887221

cos120601 (119905) minus 119888221

sin120601 (119905)]1198982

1(119905)

+ 1205722119887231

1198982

1(119905)1198982(119905) + O (120572

4)

(32b)

119889120601 (119905)

119889119905= minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198982

1(119905)

1198982(119905)

+ 120572211989611986010158401015840

11198982

1(119905) + 2120572

2

times [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]1198982(119905) + O (120572

4)

(32c)


tan120601 (119905) =

(minus11989711198972plusmn 1198973(1198972

1+ 1198972

2minus 1198972

3)12

)

(11989721minus 11989723)

(33)

where 1198971= (1 + 2120572

2119887121

1198982

2(119905)119887221

1198982

1(119905)) 1198972= (119888221

119887221

) minus

(21205722119888121

1198982

2(119905)119887221

1198982

1(119905)) and 119897

3= 120572211989611986010158401015840

11198982(119905)119887221



221

119887221

) + O (1205722) (34)


1205722119887131

1198982

1(119905) = minus119887

111minus 120572212059311198982(119905) (35)


+ 120572212059311198982(119905)]119887131

lt 0 where1205931= (119887121

cos120601lowast + 119888121



1205724119887231

12059311198982

2(119905) + 120572

2(119887111

119887231

minus 119887211

119887131

+ 12059311205932)

times 1198982(119905) + 120593

2119887111

= 0

(36)

where 1205932

= 119887221



111119887231

minus 119887211

119887131

+ 12059311205932)2minus

4120572412059311205932119887111

119887231

gt 0 Since 119887111


1198982(119905) =

119887211

119887131

minus 12059311205932

1205722119887231

1205931

=3119896211986211198692

1205722 (119861101584010158401minus 4119896211986210158401015840

1) 1205931

(37)

When 1198692= 0 119898




08

10

12

minus120587k x 120587k

t = 15 t = 145

h(xt)

(a)

08

10

12

minus120587k x 120587k

t = 2995

t = 2985

h(xt)

(b)

08

10

12

minus120587k x 120587k

t = 2t = 0

h(xt)

(c)





119869 = (

21205722119887131

119898lowast2

112057221205931119898lowast

1minus1205722 sin120601lowast (119887

121+119888121

119887221

119888221

)119898lowast

1119898lowast

2

2 (1205932+ 1205722119887231

119898lowast

2)119898lowast

11205722119887211

119898lowast2

10

2120572211989611986010158401015840

1119898lowast

1minus21205722(119887121

+119888121

119887221

119888221


2+ 1205932

119898lowast2

1

119898lowast2

)

) (38)



(119909 119905) 119909 isin (minus120587119896119898

120587119896119898


119898119909)



0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03


0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)



0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Y

120573

h(x t)B0

120582

g

120591a

X

h0





119886= 0





120588(120597U120597119905







F = (119865119909 119865119910) = (minusΣ119861

2

0119906 0) (5)



U = (119906 V) = 0 (6)



= 2120590Π (ℎ) on 119910 = ℎ (119909 119905)

(7)




119886and reads as


on 119910 = ℎ (119909 119905) (8)


120597ℎ

120597119905= V minus 119906

120597ℎ

120597119909on 119910 = ℎ (119909 119905) (9)


0 streamwise


120588]2ℎ20 time by 119897ℎ







3


Ha = Σ1198612

0ℎ2




119886ℎ2

0120588]2




ℎ119905+ 119860 (ℎ) ℎ

119909+ 120576(119861 (ℎ) ℎ

119909+ 119862 (ℎ) ℎ

119909119909119909)119909+ O (120576

2) = 0 (10)


119860 (ℎ) =ReHa

tanh2 (radicHa ℎ)

+120591


(11)

119861 (ℎ) = Re cot120573Ha32


Ha

+ Re2ℎ2Ha52


+Re2ℎHa52


minus3Re2


+ 120591 3Re


minus3ReHa52


+Re ℎHa2


+3





minus1


+3


(12)

119862 (ℎ) =119878ℎ

Haminus

119878





119860 (ℎ) = Re ℎ2+ 120591ℎ

119861 (ℎ) =2

15Re ℎ5(Re ℎ + 120591) minus

1

3Re cot120573ℎ3

119862 (ℎ) = 119878ℎ3

3

(14)







ℎ0= 1 (15a)

1199060=


(ReradicHa


+ReHa


V0= 0 (15c)

1199010= Re cot120573 (1 minus 119910) (15d)


119867119905+ 1198601119867119909+ 1205761198611119867119909119909

+ 1205761198621119867119909119909119909119909

= minus[1198601015840

1119867 +

11986010158401015840

1

21198672+119860101584010158401015840

1

61198673]119867119909

minus 120576 [1198611015840

1119867 +

11986110158401015840

1

21198672]119867119909119909

minus 120576(1198621015840

1119867 +

11986210158401015840

1

21198672)119867119909119909119909119909

minus 120576 [1198611015840

1+ 11986110158401015840

1119867] (119867

119909)2

minus 120576 (1198621015840

1+ 11986210158401015840

1119867)119867119909119867119909119909119909

+ O (1205761198674 1198675 1205762119867)

(16)


1198601= 119860 (ℎ = 1) 119860

1015840

1= 1198601015840(ℎ = 1)

11986010158401015840

1= 11986010158401015840(ℎ = 1)

1198611= 119861 (ℎ = 1) 119861

1015840

1= 1198611015840(ℎ = 1)

11986110158401015840

1= 11986110158401015840(ℎ = 1)

1198621= 119862 (ℎ = 1) 119862

1015840

1= 1198621015840(ℎ = 1)

11986210158401015840

1= 11986210158401015840(ℎ = 1)

(17)







119867(119909 119905) = 120578119890119894(119896119909minus119862119871119905)+119862119877119905 + 120578119890





119871119896 119862119877

isin


119862119871119881

=119862119871

119896= 1198601

119862119877= 1205761198962(1198611minus 11989621198621)

(19)


119877lt 0)





= 0 as 119896119888=



from (120597120597119896)119862119877

= 0 This gives 1198611minus 21198962

1198981198621= 0 such that

119896119898










0 50

1

2

3

4

5

6

7

8

9

10

Unstable

Stable

minus5

120591

Ha = 05

Ha = 01

Ha = 0

Rec

(a)

0 50

1

2

3

4

5

6

7

8

9

10

Stable

Unstable

minus5

120591

Ha = 05

Ha = 01Ha = 0

Rec

(b)


∘



plots of 119862119871119881







1199051= 120572119905 119905

2= 1205722119905 119909

1= 120572119909 (20)

such that

120597

120597119905997888rarr

120597

120597119905+ 120572

120597

1205971199051

+ 1205722 120597

1205971199052

120597

120597119909997888rarr

120597

120597119909+ 120572

120597

1205971199091

(21)


119877sim O(1205722) (or 119861

1minus

11989621198621



0 2 4 6 8 100

02

04

06

08

02

04

06

08

1

0 2 4 6 8 100

1

1 2

3

ST

US

1 2 3

ST

US

120591 = 0 120591 = 1

02

04

06

08

0 2 4 6 8 100

1

1 2 3 ST

US

kc

kc

kc

120591 = minus1

Re Re Re

(a)

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

1 1 1 2

2 2 3 3 3

ST

ST

ST

US US US

0

02

04

06

08

1

kc

0

02

04

06

08

1

kc

0

02

04

06

08

1

kc

120591 = 0 120591 = 1 120591 = minus1

Re Re Re

(b)


∘ and (b) 120573 = 60∘


0 and the



119877and 119869

2[14] If 119862

119877lt 0 and

1198692


119877gt 0 and 119869



2gt 0 the flow is


119877gt 0 and 119869

2lt 0










119904= 0 (Table 1)


00

5

ST

US

1

2

3

4

1

08

06

04

02

minus5

kc

120591

(a)

00

5

ST

US

1 2

3

4

1

08

06

04

02

minus5

kc

120591

(b)

00

5

ST

US

1 4

2

3

1

08

06

04

02

minus5

kc

120591

(c)


Re = 1 (b) 120573 = 75∘ and Re = 1 and (c) 120573 = 60

∘ and Re = 25











120578 = Γ119904(1199052) = Γ0119890minus1198941198871199052 (22)




119904





0 02 04 06 08 1

0

001

002

003

minus001

minus002

minus003

120591 = 1

120591 = 0

120591 = minus1

CR

k

0 1

0

00025

0005

minus00025

minus0005

120591 = 1

120591 = 0

120591 = minus1

CR

02 04 06 08k

Ha = 0001

Re = 2

Ha = 03

Re = 2

(a)

0 02 04 06 08 1

0

001

002

003

004

005

0 02 04 06 08 1

0

0005

001

0015

CR

minus001

minus0005

minus001

minus0015

120591 = 1 120591 = 1

120591 = 0

120591 = 0

120591 = minus1

120591 = minus1

CR

kk

Ha = 0001

Re = 2

Ha = 03

Re = 2

(b)


∘


120599 =1


= minus119888119894+ 11989621198691119903plusmn1

2radic4 (1198882119894+ 1198962V2) minus

81198941198694119888119894V119896

1198692

= 1206031plusmn 1206032

(23)

where

tr (M) = 2119888119894+ 211989621198691119903minus 41198692

1003816100381610038161003816Γ11990410038161003816100381610038162

= minus2119888119894+ 211989621198691119903

det (M) = 1198692

11198964minus (V2 + 2119888

1198941198691) 1198962minus2119894119888119894V1198694

1198692

119896

(24)


seen from (23) that 1205991= 1206031+ 1206032= 11989621198691119903

lt 0 and 1205992=

1206031minus 1206032= minus2119888

119894+ 11989621198691119903






2rarr infin



hand if (1198882119894+ 1198962V2) minus (8119894119869

4119888119894V119896 119869






0

1

2

3

4

5

6

0 02 04 06 08 10

05

1

15

2

25

3

k

0 02 04 06 08 1

0 02 04 06 08 1 0 02 04 06 08 1

k

0 02 04 06 08 1

k

0 02 04 06 08 1

k

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

CL

0

05

1

15

2

25

CL

CL

0

1

2

3

4

5

CL

0

1

2

3

4

5

6

CLV

0

1

2

3

4

5

6

CLV

Ha = 0001

Re = 2

Ha = 05

Re = 2

Re = 2

Ha = 05

Re = 5

Re = 5

Ha = 0001

Re = 5

Ha Ha


119871versus 119896


lowast lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 0

lowast

lowast

lowast lowast

lowast

lowast


lowast lowast

lowast lowast

lowast lowast

lowast

lowast lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 02120591 = 0

lowast lowast

lowast


lowast lowast


lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = 0


lowast lowast


lowast


lowast

lowast lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 1


lowast lowast


lowast lowast

lowast lowast


lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 02120591 = 1

lowast lowast

lowast lowast


lowast

lowast

lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

23

Ha = 04120591 = 1


lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

Ha = 005120591 = minus1


lowast

lowast lowast

lowast

lowast

lowast lowast


lowast lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

1

2

2

3

3

Ha = 02120591 = minus1





lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = minus1


119877lt 0 and 119869

2lt 0 (2) 119862

119877lt 0 and 119869

2gt 0 (3) 119862

119877gt 0 and 119869

2lt 0 The



119898= 119896119888radic2




111= 0 and 120597[119896b

111]120597119896 = 0 give the neutral


V119888=

119889

1198891199051205791(119905) = 119888

111+ [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]

times 1198862(119905) + 119888

1311198862

1(119905) + 119888

1321198862

2(119905)

(25)





reads as

ℎ (119909 119905) = 1 + 2 [1198611cos1199091015840 + 119861

2cos (21199091015840 minus 120601)] (26)


1(119905) = 0 119886

2(119905) = 0 and 120601(119905) =

0 (1199111(119905) = 0 119911





gives 1198861(119905) = 0 and 119886

2

2(119905) = minus119887

211119887232


211119887232


2 is set From



120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119904= 03671 119896

119904= 07933 119896

119904= 04467 119896

119904= 08707

Ha = 02 119896119904= 02309 119896

119904= 06056 119896

119904= 03438 119896

119904= 07045

Ha = 04 119896119904= 00268 119896

119904= 04399 119896

119904= 02561 119896

119904= 05686

120591 = 0

Ha = 0 119896119904= 02733 119896

119904= 07132 119896

119904= 03734 119896

119904= 07992

Ha = 02 119896119904= 01536 119896

119904= 05517 119896

119904= 02974 119896

119904= 06588

Ha = 04 119896119904= 0 119896

119904= 04073 119896

119904= 02319 119896

119904= 05436

120591 = minus1

Ha = 0 119896119904= 01211 119896

119904= 06237 119896

119904= 02823 119896

119904= 07205

Ha = 02 119896119904= 0 119896

119904= 04861 119896

119904= 02301 119896

119904= 06049

Ha = 04 119896119904= 0 119896

119904= 03602 119896

119904= 01835 119896

119904= 05094


120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119888= 07342 119896

119888= 15866 119896

119888= 08933 119896

119888= 17415

Ha = 02 119896119888= 04618 119896

119888= 12112 119896

119888= 06875 119896

119888= 14092

Ha = 04 119896119888= 00536 119896

119888= 08799 119896

119888= 05121 119896

119888= 11371

120591 = 0

Ha = 0 119896119888= 05467 119896

119888= 14263 119896

119888= 07468 119896

119888= 15984

Ha = 02 119896119888= 03071 119896

119888= 11034 119896

119888= 05947 119896

119888= 13177

Ha = 04 119896119888= 0 119896

119888= 08145 119896

119888= 04638 119896

119888= 10872

120591 = minus1

Ha = 0 119896119888= 02421 119896

119888= 12475 119896

119888= 05646 119896

119888= 14411

Ha = 02 119896119888= 0 119896

119888= 09722 119896

119888= 04602 119896

119888= 12099

Ha = 04 119896119888= 0 119896

119888= 07205 119896

119888= 01834 119896

119888= 10187

(D4c) 119888121

cos120601(119905) minus 119887121

sin120601(119905) = (1198962)11986010158401015840

1119886 is obtained


120601 (119905) = 120601 (119905) = tanminus1 [(4119887121

119888121

plusmn 11989611986010158401015840

1119886

times 41198872

121+ 41198882

121minus 1198962119860101584010158402

1119886212

)

times (41198872

121minus 1198962119860101584010158402

11198862)minus1

]

(27)


((

(

1198891198861(119905)

119889119905

1198891198862(119905)

119889119905

119889120601 (119905)

119889119905

))

)

= (

1199011

0 0

0 1199012

0

0 1199013

1199014

)(

1198861(119905)

1198862(119905)

120601 (119905)

) + (

1199021

1199022

1199023

)

(28)

such that

1199011= 119887111

+ (119887121

cos120601 + 119888121

sin120601) 119886 + 119887132

1198862

1199012= 119887211

+ 3119887232

1198862

1199013= minus2119887

121sin120601 + 2119888

121cos120601 minus 2119896119860

10158401015840

1119886

1199014= minus (2119887

121cos120601 + 2119888

121sin120601) 119886

1199021= 0 119902

2= minus2119887

2321198863

1199023= 11989611986010158401015840

11198862+ (2119887121

cos120601 + 2119888121

sin120601) 120601119886

(29)


119894 The


1205821= 119887211

+ 31198862119887232

= minus2119887211

1205822= minus2119886 (119887

121cos120601 + 119888

121sin120601)

1205823= 119887111

+ 119886 (119887121

cos120601 + 119888121

sin120601) + 119887132

1198862

(30)


0

0005

001

0015

002

0025

003

0035

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

0

0005

001

0015

002

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

Nc

Nc

Ha = 005

Re = 5

Ha = 005

Re = 5

Ha = 02

Re = 5

Ha = 02

Re = 5


(3) 120591 = 1


1= 24120576119896






1remains







11990511

= 119887111

+ (119887121

cos120601lowast + 119888121


+ 3119887131

119886lowast2

1+ 119887132

119886lowast2

2

11990512

= (119887121

cos120601lowast + 119888121


119886lowast

1119886lowast

2

11990513

= (minus119887121

sin120601lowast + 119888121


2

11990521

= 2 (119887221



119886lowast

1119886lowast

2

11990522

= 119887211

+ 119887231

119886lowast2

1+ 3119887232

119886lowast2

2

11990523

= minus (119887221

sin120601lowast + 119888221



0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

1205821=minus2b

211

k

120591 = 1

120591 = 0

120591 = minus1

120591 = minus2

120591 = 2

Ha = 03

Re = 2


∘

11990531

= 211989611986010158401015840

1119886lowast

1minus2119886lowast

1

119886lowast2

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990532

= minus 211989611986010158401015840

1119886lowast

2+ 2 (119888121


sin120601lowast)

+119886lowast2

1

119886lowast22

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990533

= minus 2 (119888121

sin120601lowast + 119887121


minus (119887221



1

119886lowast2

(31)







1(119905) rarr 120572119898

1(119905)

and 1198862(119905) rarr 120572


1198891198981(119905)

119889119905= 119887111

1198981(119905) + 120572

2

times [119887121

cos120601 (119905) + 119888121

sin120601 (119905)]1198981(119905)1198982(119905)

+ 1205722119887131

1198983

1(119905) + O (120572

4)

(32a)

1198891198982(119905)

119889119905= 119887211

1198982(119905)

+ [119887221

cos120601 (119905) minus 119888221

sin120601 (119905)]1198982

1(119905)

+ 1205722119887231

1198982

1(119905)1198982(119905) + O (120572

4)

(32b)

119889120601 (119905)

119889119905= minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198982

1(119905)

1198982(119905)

+ 120572211989611986010158401015840

11198982

1(119905) + 2120572

2

times [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]1198982(119905) + O (120572

4)

(32c)


tan120601 (119905) =

(minus11989711198972plusmn 1198973(1198972

1+ 1198972

2minus 1198972

3)12

)

(11989721minus 11989723)

(33)

where 1198971= (1 + 2120572

2119887121

1198982

2(119905)119887221

1198982

1(119905)) 1198972= (119888221

119887221

) minus

(21205722119888121

1198982

2(119905)119887221

1198982

1(119905)) and 119897

3= 120572211989611986010158401015840

11198982(119905)119887221



221

119887221

) + O (1205722) (34)


1205722119887131

1198982

1(119905) = minus119887

111minus 120572212059311198982(119905) (35)


+ 120572212059311198982(119905)]119887131

lt 0 where1205931= (119887121

cos120601lowast + 119888121



1205724119887231

12059311198982

2(119905) + 120572

2(119887111

119887231

minus 119887211

119887131

+ 12059311205932)

times 1198982(119905) + 120593

2119887111

= 0

(36)

where 1205932

= 119887221



111119887231

minus 119887211

119887131

+ 12059311205932)2minus

4120572412059311205932119887111

119887231

gt 0 Since 119887111


1198982(119905) =

119887211

119887131

minus 12059311205932

1205722119887231

1205931

=3119896211986211198692

1205722 (119861101584010158401minus 4119896211986210158401015840

1) 1205931

(37)

When 1198692= 0 119898




08

10

12

minus120587k x 120587k

t = 15 t = 145

h(xt)

(a)

08

10

12

minus120587k x 120587k

t = 2995

t = 2985

h(xt)

(b)

08

10

12

minus120587k x 120587k

t = 2t = 0

h(xt)

(c)





119869 = (

21205722119887131

119898lowast2

112057221205931119898lowast

1minus1205722 sin120601lowast (119887

121+119888121

119887221

119888221

)119898lowast

1119898lowast

2

2 (1205932+ 1205722119887231

119898lowast

2)119898lowast

11205722119887211

119898lowast2

10

2120572211989611986010158401015840

1119898lowast

1minus21205722(119887121

+119888121

119887221

119888221


2+ 1205932

119898lowast2

1

119898lowast2

)

) (38)



(119909 119905) 119909 isin (minus120587119896119898

120587119896119898


119898119909)



0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03


0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)



0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











3


Ha = Σ1198612

0ℎ2




119886ℎ2

0120588]2




ℎ119905+ 119860 (ℎ) ℎ

119909+ 120576(119861 (ℎ) ℎ

119909+ 119862 (ℎ) ℎ

119909119909119909)119909+ O (120576

2) = 0 (10)


119860 (ℎ) =ReHa

tanh2 (radicHa ℎ)

+120591


(11)

119861 (ℎ) = Re cot120573Ha32


Ha

+ Re2ℎ2Ha52


+Re2ℎHa52


minus3Re2


+ 120591 3Re


minus3ReHa52


+Re ℎHa2


+3





minus1


+3


(12)

119862 (ℎ) =119878ℎ

Haminus

119878





119860 (ℎ) = Re ℎ2+ 120591ℎ

119861 (ℎ) =2

15Re ℎ5(Re ℎ + 120591) minus

1

3Re cot120573ℎ3

119862 (ℎ) = 119878ℎ3

3

(14)







ℎ0= 1 (15a)

1199060=


(ReradicHa


+ReHa


V0= 0 (15c)

1199010= Re cot120573 (1 minus 119910) (15d)


119867119905+ 1198601119867119909+ 1205761198611119867119909119909

+ 1205761198621119867119909119909119909119909

= minus[1198601015840

1119867 +

11986010158401015840

1

21198672+119860101584010158401015840

1

61198673]119867119909

minus 120576 [1198611015840

1119867 +

11986110158401015840

1

21198672]119867119909119909

minus 120576(1198621015840

1119867 +

11986210158401015840

1

21198672)119867119909119909119909119909

minus 120576 [1198611015840

1+ 11986110158401015840

1119867] (119867

119909)2

minus 120576 (1198621015840

1+ 11986210158401015840

1119867)119867119909119867119909119909119909

+ O (1205761198674 1198675 1205762119867)

(16)


1198601= 119860 (ℎ = 1) 119860

1015840

1= 1198601015840(ℎ = 1)

11986010158401015840

1= 11986010158401015840(ℎ = 1)

1198611= 119861 (ℎ = 1) 119861

1015840

1= 1198611015840(ℎ = 1)

11986110158401015840

1= 11986110158401015840(ℎ = 1)

1198621= 119862 (ℎ = 1) 119862

1015840

1= 1198621015840(ℎ = 1)

11986210158401015840

1= 11986210158401015840(ℎ = 1)

(17)







119867(119909 119905) = 120578119890119894(119896119909minus119862119871119905)+119862119877119905 + 120578119890





119871119896 119862119877

isin


119862119871119881

=119862119871

119896= 1198601

119862119877= 1205761198962(1198611minus 11989621198621)

(19)


119877lt 0)





= 0 as 119896119888=



from (120597120597119896)119862119877

= 0 This gives 1198611minus 21198962

1198981198621= 0 such that

119896119898










0 50

1

2

3

4

5

6

7

8

9

10

Unstable

Stable

minus5

120591

Ha = 05

Ha = 01

Ha = 0

Rec

(a)

0 50

1

2

3

4

5

6

7

8

9

10

Stable

Unstable

minus5

120591

Ha = 05

Ha = 01Ha = 0

Rec

(b)


∘



plots of 119862119871119881







1199051= 120572119905 119905

2= 1205722119905 119909

1= 120572119909 (20)

such that

120597

120597119905997888rarr

120597

120597119905+ 120572

120597

1205971199051

+ 1205722 120597

1205971199052

120597

120597119909997888rarr

120597

120597119909+ 120572

120597

1205971199091

(21)


119877sim O(1205722) (or 119861

1minus

11989621198621



0 2 4 6 8 100

02

04

06

08

02

04

06

08

1

0 2 4 6 8 100

1

1 2

3

ST

US

1 2 3

ST

US

120591 = 0 120591 = 1

02

04

06

08

0 2 4 6 8 100

1

1 2 3 ST

US

kc

kc

kc

120591 = minus1

Re Re Re

(a)

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

1 1 1 2

2 2 3 3 3

ST

ST

ST

US US US

0

02

04

06

08

1

kc

0

02

04

06

08

1

kc

0

02

04

06

08

1

kc

120591 = 0 120591 = 1 120591 = minus1

Re Re Re

(b)


∘ and (b) 120573 = 60∘


0 and the



119877and 119869

2[14] If 119862

119877lt 0 and

1198692


119877gt 0 and 119869



2gt 0 the flow is


119877gt 0 and 119869

2lt 0










119904= 0 (Table 1)


00

5

ST

US

1

2

3

4

1

08

06

04

02

minus5

kc

120591

(a)

00

5

ST

US

1 2

3

4

1

08

06

04

02

minus5

kc

120591

(b)

00

5

ST

US

1 4

2

3

1

08

06

04

02

minus5

kc

120591

(c)


Re = 1 (b) 120573 = 75∘ and Re = 1 and (c) 120573 = 60

∘ and Re = 25











120578 = Γ119904(1199052) = Γ0119890minus1198941198871199052 (22)




119904





0 02 04 06 08 1

0

001

002

003

minus001

minus002

minus003

120591 = 1

120591 = 0

120591 = minus1

CR

k

0 1

0

00025

0005

minus00025

minus0005

120591 = 1

120591 = 0

120591 = minus1

CR

02 04 06 08k

Ha = 0001

Re = 2

Ha = 03

Re = 2

(a)

0 02 04 06 08 1

0

001

002

003

004

005

0 02 04 06 08 1

0

0005

001

0015

CR

minus001

minus0005

minus001

minus0015

120591 = 1 120591 = 1

120591 = 0

120591 = 0

120591 = minus1

120591 = minus1

CR

kk

Ha = 0001

Re = 2

Ha = 03

Re = 2

(b)


∘


120599 =1


= minus119888119894+ 11989621198691119903plusmn1

2radic4 (1198882119894+ 1198962V2) minus

81198941198694119888119894V119896

1198692

= 1206031plusmn 1206032

(23)

where

tr (M) = 2119888119894+ 211989621198691119903minus 41198692

1003816100381610038161003816Γ11990410038161003816100381610038162

= minus2119888119894+ 211989621198691119903

det (M) = 1198692

11198964minus (V2 + 2119888

1198941198691) 1198962minus2119894119888119894V1198694

1198692

119896

(24)


seen from (23) that 1205991= 1206031+ 1206032= 11989621198691119903

lt 0 and 1205992=

1206031minus 1206032= minus2119888

119894+ 11989621198691119903






2rarr infin



hand if (1198882119894+ 1198962V2) minus (8119894119869

4119888119894V119896 119869






0

1

2

3

4

5

6

0 02 04 06 08 10

05

1

15

2

25

3

k

0 02 04 06 08 1

0 02 04 06 08 1 0 02 04 06 08 1

k

0 02 04 06 08 1

k

0 02 04 06 08 1

k

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

CL

0

05

1

15

2

25

CL

CL

0

1

2

3

4

5

CL

0

1

2

3

4

5

6

CLV

0

1

2

3

4

5

6

CLV

Ha = 0001

Re = 2

Ha = 05

Re = 2

Re = 2

Ha = 05

Re = 5

Re = 5

Ha = 0001

Re = 5

Ha Ha


119871versus 119896


lowast lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 0

lowast

lowast

lowast lowast

lowast

lowast


lowast lowast

lowast lowast

lowast lowast

lowast

lowast lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 02120591 = 0

lowast lowast

lowast


lowast lowast


lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = 0


lowast lowast


lowast


lowast

lowast lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 1


lowast lowast


lowast lowast

lowast lowast


lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 02120591 = 1

lowast lowast

lowast lowast


lowast

lowast

lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

23

Ha = 04120591 = 1


lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

Ha = 005120591 = minus1


lowast

lowast lowast

lowast

lowast

lowast lowast


lowast lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

1

2

2

3

3

Ha = 02120591 = minus1





lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = minus1


119877lt 0 and 119869

2lt 0 (2) 119862

119877lt 0 and 119869

2gt 0 (3) 119862

119877gt 0 and 119869

2lt 0 The



119898= 119896119888radic2




111= 0 and 120597[119896b

111]120597119896 = 0 give the neutral


V119888=

119889

1198891199051205791(119905) = 119888

111+ [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]

times 1198862(119905) + 119888

1311198862

1(119905) + 119888

1321198862

2(119905)

(25)





reads as

ℎ (119909 119905) = 1 + 2 [1198611cos1199091015840 + 119861

2cos (21199091015840 minus 120601)] (26)


1(119905) = 0 119886

2(119905) = 0 and 120601(119905) =

0 (1199111(119905) = 0 119911





gives 1198861(119905) = 0 and 119886

2

2(119905) = minus119887

211119887232


211119887232


2 is set From



120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119904= 03671 119896

119904= 07933 119896

119904= 04467 119896

119904= 08707

Ha = 02 119896119904= 02309 119896

119904= 06056 119896

119904= 03438 119896

119904= 07045

Ha = 04 119896119904= 00268 119896

119904= 04399 119896

119904= 02561 119896

119904= 05686

120591 = 0

Ha = 0 119896119904= 02733 119896

119904= 07132 119896

119904= 03734 119896

119904= 07992

Ha = 02 119896119904= 01536 119896

119904= 05517 119896

119904= 02974 119896

119904= 06588

Ha = 04 119896119904= 0 119896

119904= 04073 119896

119904= 02319 119896

119904= 05436

120591 = minus1

Ha = 0 119896119904= 01211 119896

119904= 06237 119896

119904= 02823 119896

119904= 07205

Ha = 02 119896119904= 0 119896

119904= 04861 119896

119904= 02301 119896

119904= 06049

Ha = 04 119896119904= 0 119896

119904= 03602 119896

119904= 01835 119896

119904= 05094


120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119888= 07342 119896

119888= 15866 119896

119888= 08933 119896

119888= 17415

Ha = 02 119896119888= 04618 119896

119888= 12112 119896

119888= 06875 119896

119888= 14092

Ha = 04 119896119888= 00536 119896

119888= 08799 119896

119888= 05121 119896

119888= 11371

120591 = 0

Ha = 0 119896119888= 05467 119896

119888= 14263 119896

119888= 07468 119896

119888= 15984

Ha = 02 119896119888= 03071 119896

119888= 11034 119896

119888= 05947 119896

119888= 13177

Ha = 04 119896119888= 0 119896

119888= 08145 119896

119888= 04638 119896

119888= 10872

120591 = minus1

Ha = 0 119896119888= 02421 119896

119888= 12475 119896

119888= 05646 119896

119888= 14411

Ha = 02 119896119888= 0 119896

119888= 09722 119896

119888= 04602 119896

119888= 12099

Ha = 04 119896119888= 0 119896

119888= 07205 119896

119888= 01834 119896

119888= 10187

(D4c) 119888121

cos120601(119905) minus 119887121

sin120601(119905) = (1198962)11986010158401015840

1119886 is obtained


120601 (119905) = 120601 (119905) = tanminus1 [(4119887121

119888121

plusmn 11989611986010158401015840

1119886

times 41198872

121+ 41198882

121minus 1198962119860101584010158402

1119886212

)

times (41198872

121minus 1198962119860101584010158402

11198862)minus1

]

(27)


((

(

1198891198861(119905)

119889119905

1198891198862(119905)

119889119905

119889120601 (119905)

119889119905

))

)

= (

1199011

0 0

0 1199012

0

0 1199013

1199014

)(

1198861(119905)

1198862(119905)

120601 (119905)

) + (

1199021

1199022

1199023

)

(28)

such that

1199011= 119887111

+ (119887121

cos120601 + 119888121

sin120601) 119886 + 119887132

1198862

1199012= 119887211

+ 3119887232

1198862

1199013= minus2119887

121sin120601 + 2119888

121cos120601 minus 2119896119860

10158401015840

1119886

1199014= minus (2119887

121cos120601 + 2119888

121sin120601) 119886

1199021= 0 119902

2= minus2119887

2321198863

1199023= 11989611986010158401015840

11198862+ (2119887121

cos120601 + 2119888121

sin120601) 120601119886

(29)


119894 The


1205821= 119887211

+ 31198862119887232

= minus2119887211

1205822= minus2119886 (119887

121cos120601 + 119888

121sin120601)

1205823= 119887111

+ 119886 (119887121

cos120601 + 119888121

sin120601) + 119887132

1198862

(30)


0

0005

001

0015

002

0025

003

0035

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

0

0005

001

0015

002

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

Nc

Nc

Ha = 005

Re = 5

Ha = 005

Re = 5

Ha = 02

Re = 5

Ha = 02

Re = 5


(3) 120591 = 1


1= 24120576119896






1remains







11990511

= 119887111

+ (119887121

cos120601lowast + 119888121


+ 3119887131

119886lowast2

1+ 119887132

119886lowast2

2

11990512

= (119887121

cos120601lowast + 119888121


119886lowast

1119886lowast

2

11990513

= (minus119887121

sin120601lowast + 119888121


2

11990521

= 2 (119887221



119886lowast

1119886lowast

2

11990522

= 119887211

+ 119887231

119886lowast2

1+ 3119887232

119886lowast2

2

11990523

= minus (119887221

sin120601lowast + 119888221



0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

1205821=minus2b

211

k

120591 = 1

120591 = 0

120591 = minus1

120591 = minus2

120591 = 2

Ha = 03

Re = 2


∘

11990531

= 211989611986010158401015840

1119886lowast

1minus2119886lowast

1

119886lowast2

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990532

= minus 211989611986010158401015840

1119886lowast

2+ 2 (119888121


sin120601lowast)

+119886lowast2

1

119886lowast22

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990533

= minus 2 (119888121

sin120601lowast + 119887121


minus (119887221



1

119886lowast2

(31)







1(119905) rarr 120572119898

1(119905)

and 1198862(119905) rarr 120572


1198891198981(119905)

119889119905= 119887111

1198981(119905) + 120572

2

times [119887121

cos120601 (119905) + 119888121

sin120601 (119905)]1198981(119905)1198982(119905)

+ 1205722119887131

1198983

1(119905) + O (120572

4)

(32a)

1198891198982(119905)

119889119905= 119887211

1198982(119905)

+ [119887221

cos120601 (119905) minus 119888221

sin120601 (119905)]1198982

1(119905)

+ 1205722119887231

1198982

1(119905)1198982(119905) + O (120572

4)

(32b)

119889120601 (119905)

119889119905= minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198982

1(119905)

1198982(119905)

+ 120572211989611986010158401015840

11198982

1(119905) + 2120572

2

times [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]1198982(119905) + O (120572

4)

(32c)


tan120601 (119905) =

(minus11989711198972plusmn 1198973(1198972

1+ 1198972

2minus 1198972

3)12

)

(11989721minus 11989723)

(33)

where 1198971= (1 + 2120572

2119887121

1198982

2(119905)119887221

1198982

1(119905)) 1198972= (119888221

119887221

) minus

(21205722119888121

1198982

2(119905)119887221

1198982

1(119905)) and 119897

3= 120572211989611986010158401015840

11198982(119905)119887221



221

119887221

) + O (1205722) (34)


1205722119887131

1198982

1(119905) = minus119887

111minus 120572212059311198982(119905) (35)


+ 120572212059311198982(119905)]119887131

lt 0 where1205931= (119887121

cos120601lowast + 119888121



1205724119887231

12059311198982

2(119905) + 120572

2(119887111

119887231

minus 119887211

119887131

+ 12059311205932)

times 1198982(119905) + 120593

2119887111

= 0

(36)

where 1205932

= 119887221



111119887231

minus 119887211

119887131

+ 12059311205932)2minus

4120572412059311205932119887111

119887231

gt 0 Since 119887111


1198982(119905) =

119887211

119887131

minus 12059311205932

1205722119887231

1205931

=3119896211986211198692

1205722 (119861101584010158401minus 4119896211986210158401015840

1) 1205931

(37)

When 1198692= 0 119898




08

10

12

minus120587k x 120587k

t = 15 t = 145

h(xt)

(a)

08

10

12

minus120587k x 120587k

t = 2995

t = 2985

h(xt)

(b)

08

10

12

minus120587k x 120587k

t = 2t = 0

h(xt)

(c)





119869 = (

21205722119887131

119898lowast2

112057221205931119898lowast

1minus1205722 sin120601lowast (119887

121+119888121

119887221

119888221

)119898lowast

1119898lowast

2

2 (1205932+ 1205722119887231

119898lowast

2)119898lowast

11205722119887211

119898lowast2

10

2120572211989611986010158401015840

1119898lowast

1minus21205722(119887121

+119888121

119887221

119888221


2+ 1205932

119898lowast2

1

119898lowast2

)

) (38)



(119909 119905) 119909 isin (minus120587119896119898

120587119896119898


119898119909)



0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03


0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)



0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












ℎ0= 1 (15a)

1199060=


(ReradicHa


+ReHa


V0= 0 (15c)

1199010= Re cot120573 (1 minus 119910) (15d)


119867119905+ 1198601119867119909+ 1205761198611119867119909119909

+ 1205761198621119867119909119909119909119909

= minus[1198601015840

1119867 +

11986010158401015840

1

21198672+119860101584010158401015840

1

61198673]119867119909

minus 120576 [1198611015840

1119867 +

11986110158401015840

1

21198672]119867119909119909

minus 120576(1198621015840

1119867 +

11986210158401015840

1

21198672)119867119909119909119909119909

minus 120576 [1198611015840

1+ 11986110158401015840

1119867] (119867

119909)2

minus 120576 (1198621015840

1+ 11986210158401015840

1119867)119867119909119867119909119909119909

+ O (1205761198674 1198675 1205762119867)

(16)


1198601= 119860 (ℎ = 1) 119860

1015840

1= 1198601015840(ℎ = 1)

11986010158401015840

1= 11986010158401015840(ℎ = 1)

1198611= 119861 (ℎ = 1) 119861

1015840

1= 1198611015840(ℎ = 1)

11986110158401015840

1= 11986110158401015840(ℎ = 1)

1198621= 119862 (ℎ = 1) 119862

1015840

1= 1198621015840(ℎ = 1)

11986210158401015840

1= 11986210158401015840(ℎ = 1)

(17)







119867(119909 119905) = 120578119890119894(119896119909minus119862119871119905)+119862119877119905 + 120578119890





119871119896 119862119877

isin


119862119871119881

=119862119871

119896= 1198601

119862119877= 1205761198962(1198611minus 11989621198621)

(19)


119877lt 0)





= 0 as 119896119888=



from (120597120597119896)119862119877

= 0 This gives 1198611minus 21198962

1198981198621= 0 such that

119896119898










0 50

1

2

3

4

5

6

7

8

9

10

Unstable

Stable

minus5

120591

Ha = 05

Ha = 01

Ha = 0

Rec

(a)

0 50

1

2

3

4

5

6

7

8

9

10

Stable

Unstable

minus5

120591

Ha = 05

Ha = 01Ha = 0

Rec

(b)


∘



plots of 119862119871119881







1199051= 120572119905 119905

2= 1205722119905 119909

1= 120572119909 (20)

such that

120597

120597119905997888rarr

120597

120597119905+ 120572

120597

1205971199051

+ 1205722 120597

1205971199052

120597

120597119909997888rarr

120597

120597119909+ 120572

120597

1205971199091

(21)


119877sim O(1205722) (or 119861

1minus

11989621198621



0 2 4 6 8 100

02

04

06

08

02

04

06

08

1

0 2 4 6 8 100

1

1 2

3

ST

US

1 2 3

ST

US

120591 = 0 120591 = 1

02

04

06

08

0 2 4 6 8 100

1

1 2 3 ST

US

kc

kc

kc

120591 = minus1

Re Re Re

(a)

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

1 1 1 2

2 2 3 3 3

ST

ST

ST

US US US

0

02

04

06

08

1

kc

0

02

04

06

08

1

kc

0

02

04

06

08

1

kc

120591 = 0 120591 = 1 120591 = minus1

Re Re Re

(b)


∘ and (b) 120573 = 60∘


0 and the



119877and 119869

2[14] If 119862

119877lt 0 and

1198692


119877gt 0 and 119869



2gt 0 the flow is


119877gt 0 and 119869

2lt 0










119904= 0 (Table 1)


00

5

ST

US

1

2

3

4

1

08

06

04

02

minus5

kc

120591

(a)

00

5

ST

US

1 2

3

4

1

08

06

04

02

minus5

kc

120591

(b)

00

5

ST

US

1 4

2

3

1

08

06

04

02

minus5

kc

120591

(c)


Re = 1 (b) 120573 = 75∘ and Re = 1 and (c) 120573 = 60

∘ and Re = 25











120578 = Γ119904(1199052) = Γ0119890minus1198941198871199052 (22)




119904





0 02 04 06 08 1

0

001

002

003

minus001

minus002

minus003

120591 = 1

120591 = 0

120591 = minus1

CR

k

0 1

0

00025

0005

minus00025

minus0005

120591 = 1

120591 = 0

120591 = minus1

CR

02 04 06 08k

Ha = 0001

Re = 2

Ha = 03

Re = 2

(a)

0 02 04 06 08 1

0

001

002

003

004

005

0 02 04 06 08 1

0

0005

001

0015

CR

minus001

minus0005

minus001

minus0015

120591 = 1 120591 = 1

120591 = 0

120591 = 0

120591 = minus1

120591 = minus1

CR

kk

Ha = 0001

Re = 2

Ha = 03

Re = 2

(b)


∘


120599 =1


= minus119888119894+ 11989621198691119903plusmn1

2radic4 (1198882119894+ 1198962V2) minus

81198941198694119888119894V119896

1198692

= 1206031plusmn 1206032

(23)

where

tr (M) = 2119888119894+ 211989621198691119903minus 41198692

1003816100381610038161003816Γ11990410038161003816100381610038162

= minus2119888119894+ 211989621198691119903

det (M) = 1198692

11198964minus (V2 + 2119888

1198941198691) 1198962minus2119894119888119894V1198694

1198692

119896

(24)


seen from (23) that 1205991= 1206031+ 1206032= 11989621198691119903

lt 0 and 1205992=

1206031minus 1206032= minus2119888

119894+ 11989621198691119903






2rarr infin



hand if (1198882119894+ 1198962V2) minus (8119894119869

4119888119894V119896 119869






0

1

2

3

4

5

6

0 02 04 06 08 10

05

1

15

2

25

3

k

0 02 04 06 08 1

0 02 04 06 08 1 0 02 04 06 08 1

k

0 02 04 06 08 1

k

0 02 04 06 08 1

k

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

CL

0

05

1

15

2

25

CL

CL

0

1

2

3

4

5

CL

0

1

2

3

4

5

6

CLV

0

1

2

3

4

5

6

CLV

Ha = 0001

Re = 2

Ha = 05

Re = 2

Re = 2

Ha = 05

Re = 5

Re = 5

Ha = 0001

Re = 5

Ha Ha


119871versus 119896


lowast lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 0

lowast

lowast

lowast lowast

lowast

lowast


lowast lowast

lowast lowast

lowast lowast

lowast

lowast lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 02120591 = 0

lowast lowast

lowast


lowast lowast


lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = 0


lowast lowast


lowast


lowast

lowast lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 1


lowast lowast


lowast lowast

lowast lowast


lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 02120591 = 1

lowast lowast

lowast lowast


lowast

lowast

lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

23

Ha = 04120591 = 1


lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

Ha = 005120591 = minus1


lowast

lowast lowast

lowast

lowast

lowast lowast


lowast lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

1

2

2

3

3

Ha = 02120591 = minus1





lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = minus1


119877lt 0 and 119869

2lt 0 (2) 119862

119877lt 0 and 119869

2gt 0 (3) 119862

119877gt 0 and 119869

2lt 0 The



119898= 119896119888radic2




111= 0 and 120597[119896b

111]120597119896 = 0 give the neutral


V119888=

119889

1198891199051205791(119905) = 119888

111+ [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]

times 1198862(119905) + 119888

1311198862

1(119905) + 119888

1321198862

2(119905)

(25)





reads as

ℎ (119909 119905) = 1 + 2 [1198611cos1199091015840 + 119861

2cos (21199091015840 minus 120601)] (26)


1(119905) = 0 119886

2(119905) = 0 and 120601(119905) =

0 (1199111(119905) = 0 119911





gives 1198861(119905) = 0 and 119886

2

2(119905) = minus119887

211119887232


211119887232


2 is set From



120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119904= 03671 119896

119904= 07933 119896

119904= 04467 119896

119904= 08707

Ha = 02 119896119904= 02309 119896

119904= 06056 119896

119904= 03438 119896

119904= 07045

Ha = 04 119896119904= 00268 119896

119904= 04399 119896

119904= 02561 119896

119904= 05686

120591 = 0

Ha = 0 119896119904= 02733 119896

119904= 07132 119896

119904= 03734 119896

119904= 07992

Ha = 02 119896119904= 01536 119896

119904= 05517 119896

119904= 02974 119896

119904= 06588

Ha = 04 119896119904= 0 119896

119904= 04073 119896

119904= 02319 119896

119904= 05436

120591 = minus1

Ha = 0 119896119904= 01211 119896

119904= 06237 119896

119904= 02823 119896

119904= 07205

Ha = 02 119896119904= 0 119896

119904= 04861 119896

119904= 02301 119896

119904= 06049

Ha = 04 119896119904= 0 119896

119904= 03602 119896

119904= 01835 119896

119904= 05094


120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119888= 07342 119896

119888= 15866 119896

119888= 08933 119896

119888= 17415

Ha = 02 119896119888= 04618 119896

119888= 12112 119896

119888= 06875 119896

119888= 14092

Ha = 04 119896119888= 00536 119896

119888= 08799 119896

119888= 05121 119896

119888= 11371

120591 = 0

Ha = 0 119896119888= 05467 119896

119888= 14263 119896

119888= 07468 119896

119888= 15984

Ha = 02 119896119888= 03071 119896

119888= 11034 119896

119888= 05947 119896

119888= 13177

Ha = 04 119896119888= 0 119896

119888= 08145 119896

119888= 04638 119896

119888= 10872

120591 = minus1

Ha = 0 119896119888= 02421 119896

119888= 12475 119896

119888= 05646 119896

119888= 14411

Ha = 02 119896119888= 0 119896

119888= 09722 119896

119888= 04602 119896

119888= 12099

Ha = 04 119896119888= 0 119896

119888= 07205 119896

119888= 01834 119896

119888= 10187

(D4c) 119888121

cos120601(119905) minus 119887121

sin120601(119905) = (1198962)11986010158401015840

1119886 is obtained


120601 (119905) = 120601 (119905) = tanminus1 [(4119887121

119888121

plusmn 11989611986010158401015840

1119886

times 41198872

121+ 41198882

121minus 1198962119860101584010158402

1119886212

)

times (41198872

121minus 1198962119860101584010158402

11198862)minus1

]

(27)


((

(

1198891198861(119905)

119889119905

1198891198862(119905)

119889119905

119889120601 (119905)

119889119905

))

)

= (

1199011

0 0

0 1199012

0

0 1199013

1199014

)(

1198861(119905)

1198862(119905)

120601 (119905)

) + (

1199021

1199022

1199023

)

(28)

such that

1199011= 119887111

+ (119887121

cos120601 + 119888121

sin120601) 119886 + 119887132

1198862

1199012= 119887211

+ 3119887232

1198862

1199013= minus2119887

121sin120601 + 2119888

121cos120601 minus 2119896119860

10158401015840

1119886

1199014= minus (2119887

121cos120601 + 2119888

121sin120601) 119886

1199021= 0 119902

2= minus2119887

2321198863

1199023= 11989611986010158401015840

11198862+ (2119887121

cos120601 + 2119888121

sin120601) 120601119886

(29)


119894 The


1205821= 119887211

+ 31198862119887232

= minus2119887211

1205822= minus2119886 (119887

121cos120601 + 119888

121sin120601)

1205823= 119887111

+ 119886 (119887121

cos120601 + 119888121

sin120601) + 119887132

1198862

(30)


0

0005

001

0015

002

0025

003

0035

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

0

0005

001

0015

002

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

Nc

Nc

Ha = 005

Re = 5

Ha = 005

Re = 5

Ha = 02

Re = 5

Ha = 02

Re = 5


(3) 120591 = 1


1= 24120576119896






1remains







11990511

= 119887111

+ (119887121

cos120601lowast + 119888121


+ 3119887131

119886lowast2

1+ 119887132

119886lowast2

2

11990512

= (119887121

cos120601lowast + 119888121


119886lowast

1119886lowast

2

11990513

= (minus119887121

sin120601lowast + 119888121


2

11990521

= 2 (119887221



119886lowast

1119886lowast

2

11990522

= 119887211

+ 119887231

119886lowast2

1+ 3119887232

119886lowast2

2

11990523

= minus (119887221

sin120601lowast + 119888221



0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

1205821=minus2b

211

k

120591 = 1

120591 = 0

120591 = minus1

120591 = minus2

120591 = 2

Ha = 03

Re = 2


∘

11990531

= 211989611986010158401015840

1119886lowast

1minus2119886lowast

1

119886lowast2

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990532

= minus 211989611986010158401015840

1119886lowast

2+ 2 (119888121


sin120601lowast)

+119886lowast2

1

119886lowast22

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990533

= minus 2 (119888121

sin120601lowast + 119887121


minus (119887221



1

119886lowast2

(31)







1(119905) rarr 120572119898

1(119905)

and 1198862(119905) rarr 120572


1198891198981(119905)

119889119905= 119887111

1198981(119905) + 120572

2

times [119887121

cos120601 (119905) + 119888121

sin120601 (119905)]1198981(119905)1198982(119905)

+ 1205722119887131

1198983

1(119905) + O (120572

4)

(32a)

1198891198982(119905)

119889119905= 119887211

1198982(119905)

+ [119887221

cos120601 (119905) minus 119888221

sin120601 (119905)]1198982

1(119905)

+ 1205722119887231

1198982

1(119905)1198982(119905) + O (120572

4)

(32b)

119889120601 (119905)

119889119905= minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198982

1(119905)

1198982(119905)

+ 120572211989611986010158401015840

11198982

1(119905) + 2120572

2

times [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]1198982(119905) + O (120572

4)

(32c)


tan120601 (119905) =

(minus11989711198972plusmn 1198973(1198972

1+ 1198972

2minus 1198972

3)12

)

(11989721minus 11989723)

(33)

where 1198971= (1 + 2120572

2119887121

1198982

2(119905)119887221

1198982

1(119905)) 1198972= (119888221

119887221

) minus

(21205722119888121

1198982

2(119905)119887221

1198982

1(119905)) and 119897

3= 120572211989611986010158401015840

11198982(119905)119887221



221

119887221

) + O (1205722) (34)


1205722119887131

1198982

1(119905) = minus119887

111minus 120572212059311198982(119905) (35)


+ 120572212059311198982(119905)]119887131

lt 0 where1205931= (119887121

cos120601lowast + 119888121



1205724119887231

12059311198982

2(119905) + 120572

2(119887111

119887231

minus 119887211

119887131

+ 12059311205932)

times 1198982(119905) + 120593

2119887111

= 0

(36)

where 1205932

= 119887221



111119887231

minus 119887211

119887131

+ 12059311205932)2minus

4120572412059311205932119887111

119887231

gt 0 Since 119887111


1198982(119905) =

119887211

119887131

minus 12059311205932

1205722119887231

1205931

=3119896211986211198692

1205722 (119861101584010158401minus 4119896211986210158401015840

1) 1205931

(37)

When 1198692= 0 119898




08

10

12

minus120587k x 120587k

t = 15 t = 145

h(xt)

(a)

08

10

12

minus120587k x 120587k

t = 2995

t = 2985

h(xt)

(b)

08

10

12

minus120587k x 120587k

t = 2t = 0

h(xt)

(c)





119869 = (

21205722119887131

119898lowast2

112057221205931119898lowast

1minus1205722 sin120601lowast (119887

121+119888121

119887221

119888221

)119898lowast

1119898lowast

2

2 (1205932+ 1205722119887231

119898lowast

2)119898lowast

11205722119887211

119898lowast2

10

2120572211989611986010158401015840

1119898lowast

1minus21205722(119887121

+119888121

119887221

119888221


2+ 1205932

119898lowast2

1

119898lowast2

)

) (38)



(119909 119905) 119909 isin (minus120587119896119898

120587119896119898


119898119909)



0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03


0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)



0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50

1

2

3

4

5

6

7

8

9

10

Unstable

Stable

minus5

120591

Ha = 05

Ha = 01

Ha = 0

Rec

(a)

0 50

1

2

3

4

5

6

7

8

9

10

Stable

Unstable

minus5

120591

Ha = 05

Ha = 01Ha = 0

Rec

(b)


∘



plots of 119862119871119881







1199051= 120572119905 119905

2= 1205722119905 119909

1= 120572119909 (20)

such that

120597

120597119905997888rarr

120597

120597119905+ 120572

120597

1205971199051

+ 1205722 120597

1205971199052

120597

120597119909997888rarr

120597

120597119909+ 120572

120597

1205971199091

(21)


119877sim O(1205722) (or 119861

1minus

11989621198621



0 2 4 6 8 100

02

04

06

08

02

04

06

08

1

0 2 4 6 8 100

1

1 2

3

ST

US

1 2 3

ST

US

120591 = 0 120591 = 1

02

04

06

08

0 2 4 6 8 100

1

1 2 3 ST

US

kc

kc

kc

120591 = minus1

Re Re Re

(a)

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

1 1 1 2

2 2 3 3 3

ST

ST

ST

US US US

0

02

04

06

08

1

kc

0

02

04

06

08

1

kc

0

02

04

06

08

1

kc

120591 = 0 120591 = 1 120591 = minus1

Re Re Re

(b)


∘ and (b) 120573 = 60∘


0 and the



119877and 119869

2[14] If 119862

119877lt 0 and

1198692


119877gt 0 and 119869



2gt 0 the flow is


119877gt 0 and 119869

2lt 0










119904= 0 (Table 1)


00

5

ST

US

1

2

3

4

1

08

06

04

02

minus5

kc

120591

(a)

00

5

ST

US

1 2

3

4

1

08

06

04

02

minus5

kc

120591

(b)

00

5

ST

US

1 4

2

3

1

08

06

04

02

minus5

kc

120591

(c)


Re = 1 (b) 120573 = 75∘ and Re = 1 and (c) 120573 = 60

∘ and Re = 25











120578 = Γ119904(1199052) = Γ0119890minus1198941198871199052 (22)




119904





0 02 04 06 08 1

0

001

002

003

minus001

minus002

minus003

120591 = 1

120591 = 0

120591 = minus1

CR

k

0 1

0

00025

0005

minus00025

minus0005

120591 = 1

120591 = 0

120591 = minus1

CR

02 04 06 08k

Ha = 0001

Re = 2

Ha = 03

Re = 2

(a)

0 02 04 06 08 1

0

001

002

003

004

005

0 02 04 06 08 1

0

0005

001

0015

CR

minus001

minus0005

minus001

minus0015

120591 = 1 120591 = 1

120591 = 0

120591 = 0

120591 = minus1

120591 = minus1

CR

kk

Ha = 0001

Re = 2

Ha = 03

Re = 2

(b)


∘


120599 =1


= minus119888119894+ 11989621198691119903plusmn1

2radic4 (1198882119894+ 1198962V2) minus

81198941198694119888119894V119896

1198692

= 1206031plusmn 1206032

(23)

where

tr (M) = 2119888119894+ 211989621198691119903minus 41198692

1003816100381610038161003816Γ11990410038161003816100381610038162

= minus2119888119894+ 211989621198691119903

det (M) = 1198692

11198964minus (V2 + 2119888

1198941198691) 1198962minus2119894119888119894V1198694

1198692

119896

(24)


seen from (23) that 1205991= 1206031+ 1206032= 11989621198691119903

lt 0 and 1205992=

1206031minus 1206032= minus2119888

119894+ 11989621198691119903






2rarr infin



hand if (1198882119894+ 1198962V2) minus (8119894119869

4119888119894V119896 119869






0

1

2

3

4

5

6

0 02 04 06 08 10

05

1

15

2

25

3

k

0 02 04 06 08 1

0 02 04 06 08 1 0 02 04 06 08 1

k

0 02 04 06 08 1

k

0 02 04 06 08 1

k

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

CL

0

05

1

15

2

25

CL

CL

0

1

2

3

4

5

CL

0

1

2

3

4

5

6

CLV

0

1

2

3

4

5

6

CLV

Ha = 0001

Re = 2

Ha = 05

Re = 2

Re = 2

Ha = 05

Re = 5

Re = 5

Ha = 0001

Re = 5

Ha Ha


119871versus 119896


lowast lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 0

lowast

lowast

lowast lowast

lowast

lowast


lowast lowast

lowast lowast

lowast lowast

lowast

lowast lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 02120591 = 0

lowast lowast

lowast


lowast lowast


lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = 0


lowast lowast


lowast


lowast

lowast lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 1


lowast lowast


lowast lowast

lowast lowast


lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 02120591 = 1

lowast lowast

lowast lowast


lowast

lowast

lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

23

Ha = 04120591 = 1


lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

Ha = 005120591 = minus1


lowast

lowast lowast

lowast

lowast

lowast lowast


lowast lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

1

2

2

3

3

Ha = 02120591 = minus1





lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = minus1


119877lt 0 and 119869

2lt 0 (2) 119862

119877lt 0 and 119869

2gt 0 (3) 119862

119877gt 0 and 119869

2lt 0 The



119898= 119896119888radic2




111= 0 and 120597[119896b

111]120597119896 = 0 give the neutral


V119888=

119889

1198891199051205791(119905) = 119888

111+ [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]

times 1198862(119905) + 119888

1311198862

1(119905) + 119888

1321198862

2(119905)

(25)





reads as

ℎ (119909 119905) = 1 + 2 [1198611cos1199091015840 + 119861

2cos (21199091015840 minus 120601)] (26)


1(119905) = 0 119886

2(119905) = 0 and 120601(119905) =

0 (1199111(119905) = 0 119911





gives 1198861(119905) = 0 and 119886

2

2(119905) = minus119887

211119887232


211119887232


2 is set From



120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119904= 03671 119896

119904= 07933 119896

119904= 04467 119896

119904= 08707

Ha = 02 119896119904= 02309 119896

119904= 06056 119896

119904= 03438 119896

119904= 07045

Ha = 04 119896119904= 00268 119896

119904= 04399 119896

119904= 02561 119896

119904= 05686

120591 = 0

Ha = 0 119896119904= 02733 119896

119904= 07132 119896

119904= 03734 119896

119904= 07992

Ha = 02 119896119904= 01536 119896

119904= 05517 119896

119904= 02974 119896

119904= 06588

Ha = 04 119896119904= 0 119896

119904= 04073 119896

119904= 02319 119896

119904= 05436

120591 = minus1

Ha = 0 119896119904= 01211 119896

119904= 06237 119896

119904= 02823 119896

119904= 07205

Ha = 02 119896119904= 0 119896

119904= 04861 119896

119904= 02301 119896

119904= 06049

Ha = 04 119896119904= 0 119896

119904= 03602 119896

119904= 01835 119896

119904= 05094


120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119888= 07342 119896

119888= 15866 119896

119888= 08933 119896

119888= 17415

Ha = 02 119896119888= 04618 119896

119888= 12112 119896

119888= 06875 119896

119888= 14092

Ha = 04 119896119888= 00536 119896

119888= 08799 119896

119888= 05121 119896

119888= 11371

120591 = 0

Ha = 0 119896119888= 05467 119896

119888= 14263 119896

119888= 07468 119896

119888= 15984

Ha = 02 119896119888= 03071 119896

119888= 11034 119896

119888= 05947 119896

119888= 13177

Ha = 04 119896119888= 0 119896

119888= 08145 119896

119888= 04638 119896

119888= 10872

120591 = minus1

Ha = 0 119896119888= 02421 119896

119888= 12475 119896

119888= 05646 119896

119888= 14411

Ha = 02 119896119888= 0 119896

119888= 09722 119896

119888= 04602 119896

119888= 12099

Ha = 04 119896119888= 0 119896

119888= 07205 119896

119888= 01834 119896

119888= 10187

(D4c) 119888121

cos120601(119905) minus 119887121

sin120601(119905) = (1198962)11986010158401015840

1119886 is obtained


120601 (119905) = 120601 (119905) = tanminus1 [(4119887121

119888121

plusmn 11989611986010158401015840

1119886

times 41198872

121+ 41198882

121minus 1198962119860101584010158402

1119886212

)

times (41198872

121minus 1198962119860101584010158402

11198862)minus1

]

(27)


((

(

1198891198861(119905)

119889119905

1198891198862(119905)

119889119905

119889120601 (119905)

119889119905

))

)

= (

1199011

0 0

0 1199012

0

0 1199013

1199014

)(

1198861(119905)

1198862(119905)

120601 (119905)

) + (

1199021

1199022

1199023

)

(28)

such that

1199011= 119887111

+ (119887121

cos120601 + 119888121

sin120601) 119886 + 119887132

1198862

1199012= 119887211

+ 3119887232

1198862

1199013= minus2119887

121sin120601 + 2119888

121cos120601 minus 2119896119860

10158401015840

1119886

1199014= minus (2119887

121cos120601 + 2119888

121sin120601) 119886

1199021= 0 119902

2= minus2119887

2321198863

1199023= 11989611986010158401015840

11198862+ (2119887121

cos120601 + 2119888121

sin120601) 120601119886

(29)


119894 The


1205821= 119887211

+ 31198862119887232

= minus2119887211

1205822= minus2119886 (119887

121cos120601 + 119888

121sin120601)

1205823= 119887111

+ 119886 (119887121

cos120601 + 119888121

sin120601) + 119887132

1198862

(30)


0

0005

001

0015

002

0025

003

0035

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

0

0005

001

0015

002

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

Nc

Nc

Ha = 005

Re = 5

Ha = 005

Re = 5

Ha = 02

Re = 5

Ha = 02

Re = 5


(3) 120591 = 1


1= 24120576119896






1remains







11990511

= 119887111

+ (119887121

cos120601lowast + 119888121


+ 3119887131

119886lowast2

1+ 119887132

119886lowast2

2

11990512

= (119887121

cos120601lowast + 119888121


119886lowast

1119886lowast

2

11990513

= (minus119887121

sin120601lowast + 119888121


2

11990521

= 2 (119887221



119886lowast

1119886lowast

2

11990522

= 119887211

+ 119887231

119886lowast2

1+ 3119887232

119886lowast2

2

11990523

= minus (119887221

sin120601lowast + 119888221



0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

1205821=minus2b

211

k

120591 = 1

120591 = 0

120591 = minus1

120591 = minus2

120591 = 2

Ha = 03

Re = 2


∘

11990531

= 211989611986010158401015840

1119886lowast

1minus2119886lowast

1

119886lowast2

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990532

= minus 211989611986010158401015840

1119886lowast

2+ 2 (119888121


sin120601lowast)

+119886lowast2

1

119886lowast22

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990533

= minus 2 (119888121

sin120601lowast + 119887121


minus (119887221



1

119886lowast2

(31)







1(119905) rarr 120572119898

1(119905)

and 1198862(119905) rarr 120572


1198891198981(119905)

119889119905= 119887111

1198981(119905) + 120572

2

times [119887121

cos120601 (119905) + 119888121

sin120601 (119905)]1198981(119905)1198982(119905)

+ 1205722119887131

1198983

1(119905) + O (120572

4)

(32a)

1198891198982(119905)

119889119905= 119887211

1198982(119905)

+ [119887221

cos120601 (119905) minus 119888221

sin120601 (119905)]1198982

1(119905)

+ 1205722119887231

1198982

1(119905)1198982(119905) + O (120572

4)

(32b)

119889120601 (119905)

119889119905= minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198982

1(119905)

1198982(119905)

+ 120572211989611986010158401015840

11198982

1(119905) + 2120572

2

times [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]1198982(119905) + O (120572

4)

(32c)


tan120601 (119905) =

(minus11989711198972plusmn 1198973(1198972

1+ 1198972

2minus 1198972

3)12

)

(11989721minus 11989723)

(33)

where 1198971= (1 + 2120572

2119887121

1198982

2(119905)119887221

1198982

1(119905)) 1198972= (119888221

119887221

) minus

(21205722119888121

1198982

2(119905)119887221

1198982

1(119905)) and 119897

3= 120572211989611986010158401015840

11198982(119905)119887221



221

119887221

) + O (1205722) (34)


1205722119887131

1198982

1(119905) = minus119887

111minus 120572212059311198982(119905) (35)


+ 120572212059311198982(119905)]119887131

lt 0 where1205931= (119887121

cos120601lowast + 119888121



1205724119887231

12059311198982

2(119905) + 120572

2(119887111

119887231

minus 119887211

119887131

+ 12059311205932)

times 1198982(119905) + 120593

2119887111

= 0

(36)

where 1205932

= 119887221



111119887231

minus 119887211

119887131

+ 12059311205932)2minus

4120572412059311205932119887111

119887231

gt 0 Since 119887111


1198982(119905) =

119887211

119887131

minus 12059311205932

1205722119887231

1205931

=3119896211986211198692

1205722 (119861101584010158401minus 4119896211986210158401015840

1) 1205931

(37)

When 1198692= 0 119898




08

10

12

minus120587k x 120587k

t = 15 t = 145

h(xt)

(a)

08

10

12

minus120587k x 120587k

t = 2995

t = 2985

h(xt)

(b)

08

10

12

minus120587k x 120587k

t = 2t = 0

h(xt)

(c)





119869 = (

21205722119887131

119898lowast2

112057221205931119898lowast

1minus1205722 sin120601lowast (119887

121+119888121

119887221

119888221

)119898lowast

1119898lowast

2

2 (1205932+ 1205722119887231

119898lowast

2)119898lowast

11205722119887211

119898lowast2

10

2120572211989611986010158401015840

1119898lowast

1minus21205722(119887121

+119888121

119887221

119888221


2+ 1205932

119898lowast2

1

119898lowast2

)

) (38)



(119909 119905) 119909 isin (minus120587119896119898

120587119896119898


119898119909)



0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03


0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)



0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 100

02

04

06

08

02

04

06

08

1

0 2 4 6 8 100

1

1 2

3

ST

US

1 2 3

ST

US

120591 = 0 120591 = 1

02

04

06

08

0 2 4 6 8 100

1

1 2 3 ST

US

kc

kc

kc

120591 = minus1

Re Re Re

(a)

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

1 1 1 2

2 2 3 3 3

ST

ST

ST

US US US

0

02

04

06

08

1

kc

0

02

04

06

08

1

kc

0

02

04

06

08

1

kc

120591 = 0 120591 = 1 120591 = minus1

Re Re Re

(b)


∘ and (b) 120573 = 60∘


0 and the



119877and 119869

2[14] If 119862

119877lt 0 and

1198692


119877gt 0 and 119869



2gt 0 the flow is


119877gt 0 and 119869

2lt 0










119904= 0 (Table 1)


00

5

ST

US

1

2

3

4

1

08

06

04

02

minus5

kc

120591

(a)

00

5

ST

US

1 2

3

4

1

08

06

04

02

minus5

kc

120591

(b)

00

5

ST

US

1 4

2

3

1

08

06

04

02

minus5

kc

120591

(c)


Re = 1 (b) 120573 = 75∘ and Re = 1 and (c) 120573 = 60

∘ and Re = 25











120578 = Γ119904(1199052) = Γ0119890minus1198941198871199052 (22)




119904





0 02 04 06 08 1

0

001

002

003

minus001

minus002

minus003

120591 = 1

120591 = 0

120591 = minus1

CR

k

0 1

0

00025

0005

minus00025

minus0005

120591 = 1

120591 = 0

120591 = minus1

CR

02 04 06 08k

Ha = 0001

Re = 2

Ha = 03

Re = 2

(a)

0 02 04 06 08 1

0

001

002

003

004

005

0 02 04 06 08 1

0

0005

001

0015

CR

minus001

minus0005

minus001

minus0015

120591 = 1 120591 = 1

120591 = 0

120591 = 0

120591 = minus1

120591 = minus1

CR

kk

Ha = 0001

Re = 2

Ha = 03

Re = 2

(b)


∘


120599 =1


= minus119888119894+ 11989621198691119903plusmn1

2radic4 (1198882119894+ 1198962V2) minus

81198941198694119888119894V119896

1198692

= 1206031plusmn 1206032

(23)

where

tr (M) = 2119888119894+ 211989621198691119903minus 41198692

1003816100381610038161003816Γ11990410038161003816100381610038162

= minus2119888119894+ 211989621198691119903

det (M) = 1198692

11198964minus (V2 + 2119888

1198941198691) 1198962minus2119894119888119894V1198694

1198692

119896

(24)


seen from (23) that 1205991= 1206031+ 1206032= 11989621198691119903

lt 0 and 1205992=

1206031minus 1206032= minus2119888

119894+ 11989621198691119903






2rarr infin



hand if (1198882119894+ 1198962V2) minus (8119894119869

4119888119894V119896 119869






0

1

2

3

4

5

6

0 02 04 06 08 10

05

1

15

2

25

3

k

0 02 04 06 08 1

0 02 04 06 08 1 0 02 04 06 08 1

k

0 02 04 06 08 1

k

0 02 04 06 08 1

k

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

CL

0

05

1

15

2

25

CL

CL

0

1

2

3

4

5

CL

0

1

2

3

4

5

6

CLV

0

1

2

3

4

5

6

CLV

Ha = 0001

Re = 2

Ha = 05

Re = 2

Re = 2

Ha = 05

Re = 5

Re = 5

Ha = 0001

Re = 5

Ha Ha


119871versus 119896


lowast lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 0

lowast

lowast

lowast lowast

lowast

lowast


lowast lowast

lowast lowast

lowast lowast

lowast

lowast lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 02120591 = 0

lowast lowast

lowast


lowast lowast


lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = 0


lowast lowast


lowast


lowast

lowast lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 1


lowast lowast


lowast lowast

lowast lowast


lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 02120591 = 1

lowast lowast

lowast lowast


lowast

lowast

lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

23

Ha = 04120591 = 1


lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

Ha = 005120591 = minus1


lowast

lowast lowast

lowast

lowast

lowast lowast


lowast lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

1

2

2

3

3

Ha = 02120591 = minus1





lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = minus1


119877lt 0 and 119869

2lt 0 (2) 119862

119877lt 0 and 119869

2gt 0 (3) 119862

119877gt 0 and 119869

2lt 0 The



119898= 119896119888radic2




111= 0 and 120597[119896b

111]120597119896 = 0 give the neutral


V119888=

119889

1198891199051205791(119905) = 119888

111+ [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]

times 1198862(119905) + 119888

1311198862

1(119905) + 119888

1321198862

2(119905)

(25)





reads as

ℎ (119909 119905) = 1 + 2 [1198611cos1199091015840 + 119861

2cos (21199091015840 minus 120601)] (26)


1(119905) = 0 119886

2(119905) = 0 and 120601(119905) =

0 (1199111(119905) = 0 119911





gives 1198861(119905) = 0 and 119886

2

2(119905) = minus119887

211119887232


211119887232


2 is set From



120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119904= 03671 119896

119904= 07933 119896

119904= 04467 119896

119904= 08707

Ha = 02 119896119904= 02309 119896

119904= 06056 119896

119904= 03438 119896

119904= 07045

Ha = 04 119896119904= 00268 119896

119904= 04399 119896

119904= 02561 119896

119904= 05686

120591 = 0

Ha = 0 119896119904= 02733 119896

119904= 07132 119896

119904= 03734 119896

119904= 07992

Ha = 02 119896119904= 01536 119896

119904= 05517 119896

119904= 02974 119896

119904= 06588

Ha = 04 119896119904= 0 119896

119904= 04073 119896

119904= 02319 119896

119904= 05436

120591 = minus1

Ha = 0 119896119904= 01211 119896

119904= 06237 119896

119904= 02823 119896

119904= 07205

Ha = 02 119896119904= 0 119896

119904= 04861 119896

119904= 02301 119896

119904= 06049

Ha = 04 119896119904= 0 119896

119904= 03602 119896

119904= 01835 119896

119904= 05094


120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119888= 07342 119896

119888= 15866 119896

119888= 08933 119896

119888= 17415

Ha = 02 119896119888= 04618 119896

119888= 12112 119896

119888= 06875 119896

119888= 14092

Ha = 04 119896119888= 00536 119896

119888= 08799 119896

119888= 05121 119896

119888= 11371

120591 = 0

Ha = 0 119896119888= 05467 119896

119888= 14263 119896

119888= 07468 119896

119888= 15984

Ha = 02 119896119888= 03071 119896

119888= 11034 119896

119888= 05947 119896

119888= 13177

Ha = 04 119896119888= 0 119896

119888= 08145 119896

119888= 04638 119896

119888= 10872

120591 = minus1

Ha = 0 119896119888= 02421 119896

119888= 12475 119896

119888= 05646 119896

119888= 14411

Ha = 02 119896119888= 0 119896

119888= 09722 119896

119888= 04602 119896

119888= 12099

Ha = 04 119896119888= 0 119896

119888= 07205 119896

119888= 01834 119896

119888= 10187

(D4c) 119888121

cos120601(119905) minus 119887121

sin120601(119905) = (1198962)11986010158401015840

1119886 is obtained


120601 (119905) = 120601 (119905) = tanminus1 [(4119887121

119888121

plusmn 11989611986010158401015840

1119886

times 41198872

121+ 41198882

121minus 1198962119860101584010158402

1119886212

)

times (41198872

121minus 1198962119860101584010158402

11198862)minus1

]

(27)


((

(

1198891198861(119905)

119889119905

1198891198862(119905)

119889119905

119889120601 (119905)

119889119905

))

)

= (

1199011

0 0

0 1199012

0

0 1199013

1199014

)(

1198861(119905)

1198862(119905)

120601 (119905)

) + (

1199021

1199022

1199023

)

(28)

such that

1199011= 119887111

+ (119887121

cos120601 + 119888121

sin120601) 119886 + 119887132

1198862

1199012= 119887211

+ 3119887232

1198862

1199013= minus2119887

121sin120601 + 2119888

121cos120601 minus 2119896119860

10158401015840

1119886

1199014= minus (2119887

121cos120601 + 2119888

121sin120601) 119886

1199021= 0 119902

2= minus2119887

2321198863

1199023= 11989611986010158401015840

11198862+ (2119887121

cos120601 + 2119888121

sin120601) 120601119886

(29)


119894 The


1205821= 119887211

+ 31198862119887232

= minus2119887211

1205822= minus2119886 (119887

121cos120601 + 119888

121sin120601)

1205823= 119887111

+ 119886 (119887121

cos120601 + 119888121

sin120601) + 119887132

1198862

(30)


0

0005

001

0015

002

0025

003

0035

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

0

0005

001

0015

002

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

Nc

Nc

Ha = 005

Re = 5

Ha = 005

Re = 5

Ha = 02

Re = 5

Ha = 02

Re = 5


(3) 120591 = 1


1= 24120576119896






1remains







11990511

= 119887111

+ (119887121

cos120601lowast + 119888121


+ 3119887131

119886lowast2

1+ 119887132

119886lowast2

2

11990512

= (119887121

cos120601lowast + 119888121


119886lowast

1119886lowast

2

11990513

= (minus119887121

sin120601lowast + 119888121


2

11990521

= 2 (119887221



119886lowast

1119886lowast

2

11990522

= 119887211

+ 119887231

119886lowast2

1+ 3119887232

119886lowast2

2

11990523

= minus (119887221

sin120601lowast + 119888221



0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

1205821=minus2b

211

k

120591 = 1

120591 = 0

120591 = minus1

120591 = minus2

120591 = 2

Ha = 03

Re = 2


∘

11990531

= 211989611986010158401015840

1119886lowast

1minus2119886lowast

1

119886lowast2

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990532

= minus 211989611986010158401015840

1119886lowast

2+ 2 (119888121


sin120601lowast)

+119886lowast2

1

119886lowast22

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990533

= minus 2 (119888121

sin120601lowast + 119887121


minus (119887221



1

119886lowast2

(31)







1(119905) rarr 120572119898

1(119905)

and 1198862(119905) rarr 120572


1198891198981(119905)

119889119905= 119887111

1198981(119905) + 120572

2

times [119887121

cos120601 (119905) + 119888121

sin120601 (119905)]1198981(119905)1198982(119905)

+ 1205722119887131

1198983

1(119905) + O (120572

4)

(32a)

1198891198982(119905)

119889119905= 119887211

1198982(119905)

+ [119887221

cos120601 (119905) minus 119888221

sin120601 (119905)]1198982

1(119905)

+ 1205722119887231

1198982

1(119905)1198982(119905) + O (120572

4)

(32b)

119889120601 (119905)

119889119905= minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198982

1(119905)

1198982(119905)

+ 120572211989611986010158401015840

11198982

1(119905) + 2120572

2

times [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]1198982(119905) + O (120572

4)

(32c)


tan120601 (119905) =

(minus11989711198972plusmn 1198973(1198972

1+ 1198972

2minus 1198972

3)12

)

(11989721minus 11989723)

(33)

where 1198971= (1 + 2120572

2119887121

1198982

2(119905)119887221

1198982

1(119905)) 1198972= (119888221

119887221

) minus

(21205722119888121

1198982

2(119905)119887221

1198982

1(119905)) and 119897

3= 120572211989611986010158401015840

11198982(119905)119887221



221

119887221

) + O (1205722) (34)


1205722119887131

1198982

1(119905) = minus119887

111minus 120572212059311198982(119905) (35)


+ 120572212059311198982(119905)]119887131

lt 0 where1205931= (119887121

cos120601lowast + 119888121



1205724119887231

12059311198982

2(119905) + 120572

2(119887111

119887231

minus 119887211

119887131

+ 12059311205932)

times 1198982(119905) + 120593

2119887111

= 0

(36)

where 1205932

= 119887221



111119887231

minus 119887211

119887131

+ 12059311205932)2minus

4120572412059311205932119887111

119887231

gt 0 Since 119887111


1198982(119905) =

119887211

119887131

minus 12059311205932

1205722119887231

1205931

=3119896211986211198692

1205722 (119861101584010158401minus 4119896211986210158401015840

1) 1205931

(37)

When 1198692= 0 119898




08

10

12

minus120587k x 120587k

t = 15 t = 145

h(xt)

(a)

08

10

12

minus120587k x 120587k

t = 2995

t = 2985

h(xt)

(b)

08

10

12

minus120587k x 120587k

t = 2t = 0

h(xt)

(c)





119869 = (

21205722119887131

119898lowast2

112057221205931119898lowast

1minus1205722 sin120601lowast (119887

121+119888121

119887221

119888221

)119898lowast

1119898lowast

2

2 (1205932+ 1205722119887231

119898lowast

2)119898lowast

11205722119887211

119898lowast2

10

2120572211989611986010158401015840

1119898lowast

1minus21205722(119887121

+119888121

119887221

119888221


2+ 1205932

119898lowast2

1

119898lowast2

)

) (38)



(119909 119905) 119909 isin (minus120587119896119898

120587119896119898


119898119909)



0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03


0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)



0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00

5

ST

US

1

2

3

4

1

08

06

04

02

minus5

kc

120591

(a)

00

5

ST

US

1 2

3

4

1

08

06

04

02

minus5

kc

120591

(b)

00

5

ST

US

1 4

2

3

1

08

06

04

02

minus5

kc

120591

(c)


Re = 1 (b) 120573 = 75∘ and Re = 1 and (c) 120573 = 60

∘ and Re = 25











120578 = Γ119904(1199052) = Γ0119890minus1198941198871199052 (22)




119904





0 02 04 06 08 1

0

001

002

003

minus001

minus002

minus003

120591 = 1

120591 = 0

120591 = minus1

CR

k

0 1

0

00025

0005

minus00025

minus0005

120591 = 1

120591 = 0

120591 = minus1

CR

02 04 06 08k

Ha = 0001

Re = 2

Ha = 03

Re = 2

(a)

0 02 04 06 08 1

0

001

002

003

004

005

0 02 04 06 08 1

0

0005

001

0015

CR

minus001

minus0005

minus001

minus0015

120591 = 1 120591 = 1

120591 = 0

120591 = 0

120591 = minus1

120591 = minus1

CR

kk

Ha = 0001

Re = 2

Ha = 03

Re = 2

(b)


∘


120599 =1


= minus119888119894+ 11989621198691119903plusmn1

2radic4 (1198882119894+ 1198962V2) minus

81198941198694119888119894V119896

1198692

= 1206031plusmn 1206032

(23)

where

tr (M) = 2119888119894+ 211989621198691119903minus 41198692

1003816100381610038161003816Γ11990410038161003816100381610038162

= minus2119888119894+ 211989621198691119903

det (M) = 1198692

11198964minus (V2 + 2119888

1198941198691) 1198962minus2119894119888119894V1198694

1198692

119896

(24)


seen from (23) that 1205991= 1206031+ 1206032= 11989621198691119903

lt 0 and 1205992=

1206031minus 1206032= minus2119888

119894+ 11989621198691119903






2rarr infin



hand if (1198882119894+ 1198962V2) minus (8119894119869

4119888119894V119896 119869






0

1

2

3

4

5

6

0 02 04 06 08 10

05

1

15

2

25

3

k

0 02 04 06 08 1

0 02 04 06 08 1 0 02 04 06 08 1

k

0 02 04 06 08 1

k

0 02 04 06 08 1

k

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

CL

0

05

1

15

2

25

CL

CL

0

1

2

3

4

5

CL

0

1

2

3

4

5

6

CLV

0

1

2

3

4

5

6

CLV

Ha = 0001

Re = 2

Ha = 05

Re = 2

Re = 2

Ha = 05

Re = 5

Re = 5

Ha = 0001

Re = 5

Ha Ha


119871versus 119896


lowast lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 0

lowast

lowast

lowast lowast

lowast

lowast


lowast lowast

lowast lowast

lowast lowast

lowast

lowast lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 02120591 = 0

lowast lowast

lowast


lowast lowast


lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = 0


lowast lowast


lowast


lowast

lowast lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 1


lowast lowast


lowast lowast

lowast lowast


lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 02120591 = 1

lowast lowast

lowast lowast


lowast

lowast

lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

23

Ha = 04120591 = 1


lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

Ha = 005120591 = minus1


lowast

lowast lowast

lowast

lowast

lowast lowast


lowast lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

1

2

2

3

3

Ha = 02120591 = minus1





lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = minus1


119877lt 0 and 119869

2lt 0 (2) 119862

119877lt 0 and 119869

2gt 0 (3) 119862

119877gt 0 and 119869

2lt 0 The



119898= 119896119888radic2




111= 0 and 120597[119896b

111]120597119896 = 0 give the neutral


V119888=

119889

1198891199051205791(119905) = 119888

111+ [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]

times 1198862(119905) + 119888

1311198862

1(119905) + 119888

1321198862

2(119905)

(25)





reads as

ℎ (119909 119905) = 1 + 2 [1198611cos1199091015840 + 119861

2cos (21199091015840 minus 120601)] (26)


1(119905) = 0 119886

2(119905) = 0 and 120601(119905) =

0 (1199111(119905) = 0 119911





gives 1198861(119905) = 0 and 119886

2

2(119905) = minus119887

211119887232


211119887232


2 is set From



120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119904= 03671 119896

119904= 07933 119896

119904= 04467 119896

119904= 08707

Ha = 02 119896119904= 02309 119896

119904= 06056 119896

119904= 03438 119896

119904= 07045

Ha = 04 119896119904= 00268 119896

119904= 04399 119896

119904= 02561 119896

119904= 05686

120591 = 0

Ha = 0 119896119904= 02733 119896

119904= 07132 119896

119904= 03734 119896

119904= 07992

Ha = 02 119896119904= 01536 119896

119904= 05517 119896

119904= 02974 119896

119904= 06588

Ha = 04 119896119904= 0 119896

119904= 04073 119896

119904= 02319 119896

119904= 05436

120591 = minus1

Ha = 0 119896119904= 01211 119896

119904= 06237 119896

119904= 02823 119896

119904= 07205

Ha = 02 119896119904= 0 119896

119904= 04861 119896

119904= 02301 119896

119904= 06049

Ha = 04 119896119904= 0 119896

119904= 03602 119896

119904= 01835 119896

119904= 05094


120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119888= 07342 119896

119888= 15866 119896

119888= 08933 119896

119888= 17415

Ha = 02 119896119888= 04618 119896

119888= 12112 119896

119888= 06875 119896

119888= 14092

Ha = 04 119896119888= 00536 119896

119888= 08799 119896

119888= 05121 119896

119888= 11371

120591 = 0

Ha = 0 119896119888= 05467 119896

119888= 14263 119896

119888= 07468 119896

119888= 15984

Ha = 02 119896119888= 03071 119896

119888= 11034 119896

119888= 05947 119896

119888= 13177

Ha = 04 119896119888= 0 119896

119888= 08145 119896

119888= 04638 119896

119888= 10872

120591 = minus1

Ha = 0 119896119888= 02421 119896

119888= 12475 119896

119888= 05646 119896

119888= 14411

Ha = 02 119896119888= 0 119896

119888= 09722 119896

119888= 04602 119896

119888= 12099

Ha = 04 119896119888= 0 119896

119888= 07205 119896

119888= 01834 119896

119888= 10187

(D4c) 119888121

cos120601(119905) minus 119887121

sin120601(119905) = (1198962)11986010158401015840

1119886 is obtained


120601 (119905) = 120601 (119905) = tanminus1 [(4119887121

119888121

plusmn 11989611986010158401015840

1119886

times 41198872

121+ 41198882

121minus 1198962119860101584010158402

1119886212

)

times (41198872

121minus 1198962119860101584010158402

11198862)minus1

]

(27)


((

(

1198891198861(119905)

119889119905

1198891198862(119905)

119889119905

119889120601 (119905)

119889119905

))

)

= (

1199011

0 0

0 1199012

0

0 1199013

1199014

)(

1198861(119905)

1198862(119905)

120601 (119905)

) + (

1199021

1199022

1199023

)

(28)

such that

1199011= 119887111

+ (119887121

cos120601 + 119888121

sin120601) 119886 + 119887132

1198862

1199012= 119887211

+ 3119887232

1198862

1199013= minus2119887

121sin120601 + 2119888

121cos120601 minus 2119896119860

10158401015840

1119886

1199014= minus (2119887

121cos120601 + 2119888

121sin120601) 119886

1199021= 0 119902

2= minus2119887

2321198863

1199023= 11989611986010158401015840

11198862+ (2119887121

cos120601 + 2119888121

sin120601) 120601119886

(29)


119894 The


1205821= 119887211

+ 31198862119887232

= minus2119887211

1205822= minus2119886 (119887

121cos120601 + 119888

121sin120601)

1205823= 119887111

+ 119886 (119887121

cos120601 + 119888121

sin120601) + 119887132

1198862

(30)


0

0005

001

0015

002

0025

003

0035

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

0

0005

001

0015

002

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

Nc

Nc

Ha = 005

Re = 5

Ha = 005

Re = 5

Ha = 02

Re = 5

Ha = 02

Re = 5


(3) 120591 = 1


1= 24120576119896






1remains







11990511

= 119887111

+ (119887121

cos120601lowast + 119888121


+ 3119887131

119886lowast2

1+ 119887132

119886lowast2

2

11990512

= (119887121

cos120601lowast + 119888121


119886lowast

1119886lowast

2

11990513

= (minus119887121

sin120601lowast + 119888121


2

11990521

= 2 (119887221



119886lowast

1119886lowast

2

11990522

= 119887211

+ 119887231

119886lowast2

1+ 3119887232

119886lowast2

2

11990523

= minus (119887221

sin120601lowast + 119888221



0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

1205821=minus2b

211

k

120591 = 1

120591 = 0

120591 = minus1

120591 = minus2

120591 = 2

Ha = 03

Re = 2


∘

11990531

= 211989611986010158401015840

1119886lowast

1minus2119886lowast

1

119886lowast2

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990532

= minus 211989611986010158401015840

1119886lowast

2+ 2 (119888121


sin120601lowast)

+119886lowast2

1

119886lowast22

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990533

= minus 2 (119888121

sin120601lowast + 119887121


minus (119887221



1

119886lowast2

(31)







1(119905) rarr 120572119898

1(119905)

and 1198862(119905) rarr 120572


1198891198981(119905)

119889119905= 119887111

1198981(119905) + 120572

2

times [119887121

cos120601 (119905) + 119888121

sin120601 (119905)]1198981(119905)1198982(119905)

+ 1205722119887131

1198983

1(119905) + O (120572

4)

(32a)

1198891198982(119905)

119889119905= 119887211

1198982(119905)

+ [119887221

cos120601 (119905) minus 119888221

sin120601 (119905)]1198982

1(119905)

+ 1205722119887231

1198982

1(119905)1198982(119905) + O (120572

4)

(32b)

119889120601 (119905)

119889119905= minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198982

1(119905)

1198982(119905)

+ 120572211989611986010158401015840

11198982

1(119905) + 2120572

2

times [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]1198982(119905) + O (120572

4)

(32c)


tan120601 (119905) =

(minus11989711198972plusmn 1198973(1198972

1+ 1198972

2minus 1198972

3)12

)

(11989721minus 11989723)

(33)

where 1198971= (1 + 2120572

2119887121

1198982

2(119905)119887221

1198982

1(119905)) 1198972= (119888221

119887221

) minus

(21205722119888121

1198982

2(119905)119887221

1198982

1(119905)) and 119897

3= 120572211989611986010158401015840

11198982(119905)119887221



221

119887221

) + O (1205722) (34)


1205722119887131

1198982

1(119905) = minus119887

111minus 120572212059311198982(119905) (35)


+ 120572212059311198982(119905)]119887131

lt 0 where1205931= (119887121

cos120601lowast + 119888121



1205724119887231

12059311198982

2(119905) + 120572

2(119887111

119887231

minus 119887211

119887131

+ 12059311205932)

times 1198982(119905) + 120593

2119887111

= 0

(36)

where 1205932

= 119887221



111119887231

minus 119887211

119887131

+ 12059311205932)2minus

4120572412059311205932119887111

119887231

gt 0 Since 119887111


1198982(119905) =

119887211

119887131

minus 12059311205932

1205722119887231

1205931

=3119896211986211198692

1205722 (119861101584010158401minus 4119896211986210158401015840

1) 1205931

(37)

When 1198692= 0 119898




08

10

12

minus120587k x 120587k

t = 15 t = 145

h(xt)

(a)

08

10

12

minus120587k x 120587k

t = 2995

t = 2985

h(xt)

(b)

08

10

12

minus120587k x 120587k

t = 2t = 0

h(xt)

(c)





119869 = (

21205722119887131

119898lowast2

112057221205931119898lowast

1minus1205722 sin120601lowast (119887

121+119888121

119887221

119888221

)119898lowast

1119898lowast

2

2 (1205932+ 1205722119887231

119898lowast

2)119898lowast

11205722119887211

119898lowast2

10

2120572211989611986010158401015840

1119898lowast

1minus21205722(119887121

+119888121

119887221

119888221


2+ 1205932

119898lowast2

1

119898lowast2

)

) (38)



(119909 119905) 119909 isin (minus120587119896119898

120587119896119898


119898119909)



0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03


0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)



0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 1

0

001

002

003

minus001

minus002

minus003

120591 = 1

120591 = 0

120591 = minus1

CR

k

0 1

0

00025

0005

minus00025

minus0005

120591 = 1

120591 = 0

120591 = minus1

CR

02 04 06 08k

Ha = 0001

Re = 2

Ha = 03

Re = 2

(a)

0 02 04 06 08 1

0

001

002

003

004

005

0 02 04 06 08 1

0

0005

001

0015

CR

minus001

minus0005

minus001

minus0015

120591 = 1 120591 = 1

120591 = 0

120591 = 0

120591 = minus1

120591 = minus1

CR

kk

Ha = 0001

Re = 2

Ha = 03

Re = 2

(b)


∘


120599 =1


= minus119888119894+ 11989621198691119903plusmn1

2radic4 (1198882119894+ 1198962V2) minus

81198941198694119888119894V119896

1198692

= 1206031plusmn 1206032

(23)

where

tr (M) = 2119888119894+ 211989621198691119903minus 41198692

1003816100381610038161003816Γ11990410038161003816100381610038162

= minus2119888119894+ 211989621198691119903

det (M) = 1198692

11198964minus (V2 + 2119888

1198941198691) 1198962minus2119894119888119894V1198694

1198692

119896

(24)


seen from (23) that 1205991= 1206031+ 1206032= 11989621198691119903

lt 0 and 1205992=

1206031minus 1206032= minus2119888

119894+ 11989621198691119903






2rarr infin



hand if (1198882119894+ 1198962V2) minus (8119894119869

4119888119894V119896 119869






0

1

2

3

4

5

6

0 02 04 06 08 10

05

1

15

2

25

3

k

0 02 04 06 08 1

0 02 04 06 08 1 0 02 04 06 08 1

k

0 02 04 06 08 1

k

0 02 04 06 08 1

k

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

CL

0

05

1

15

2

25

CL

CL

0

1

2

3

4

5

CL

0

1

2

3

4

5

6

CLV

0

1

2

3

4

5

6

CLV

Ha = 0001

Re = 2

Ha = 05

Re = 2

Re = 2

Ha = 05

Re = 5

Re = 5

Ha = 0001

Re = 5

Ha Ha


119871versus 119896


lowast lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 0

lowast

lowast

lowast lowast

lowast

lowast


lowast lowast

lowast lowast

lowast lowast

lowast

lowast lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 02120591 = 0

lowast lowast

lowast


lowast lowast


lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = 0


lowast lowast


lowast


lowast

lowast lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 1


lowast lowast


lowast lowast

lowast lowast


lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 02120591 = 1

lowast lowast

lowast lowast


lowast

lowast

lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

23

Ha = 04120591 = 1


lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

Ha = 005120591 = minus1


lowast

lowast lowast

lowast

lowast

lowast lowast


lowast lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

1

2

2

3

3

Ha = 02120591 = minus1





lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = minus1


119877lt 0 and 119869

2lt 0 (2) 119862

119877lt 0 and 119869

2gt 0 (3) 119862

119877gt 0 and 119869

2lt 0 The



119898= 119896119888radic2




111= 0 and 120597[119896b

111]120597119896 = 0 give the neutral


V119888=

119889

1198891199051205791(119905) = 119888

111+ [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]

times 1198862(119905) + 119888

1311198862

1(119905) + 119888

1321198862

2(119905)

(25)





reads as

ℎ (119909 119905) = 1 + 2 [1198611cos1199091015840 + 119861

2cos (21199091015840 minus 120601)] (26)


1(119905) = 0 119886

2(119905) = 0 and 120601(119905) =

0 (1199111(119905) = 0 119911





gives 1198861(119905) = 0 and 119886

2

2(119905) = minus119887

211119887232


211119887232


2 is set From



120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119904= 03671 119896

119904= 07933 119896

119904= 04467 119896

119904= 08707

Ha = 02 119896119904= 02309 119896

119904= 06056 119896

119904= 03438 119896

119904= 07045

Ha = 04 119896119904= 00268 119896

119904= 04399 119896

119904= 02561 119896

119904= 05686

120591 = 0

Ha = 0 119896119904= 02733 119896

119904= 07132 119896

119904= 03734 119896

119904= 07992

Ha = 02 119896119904= 01536 119896

119904= 05517 119896

119904= 02974 119896

119904= 06588

Ha = 04 119896119904= 0 119896

119904= 04073 119896

119904= 02319 119896

119904= 05436

120591 = minus1

Ha = 0 119896119904= 01211 119896

119904= 06237 119896

119904= 02823 119896

119904= 07205

Ha = 02 119896119904= 0 119896

119904= 04861 119896

119904= 02301 119896

119904= 06049

Ha = 04 119896119904= 0 119896

119904= 03602 119896

119904= 01835 119896

119904= 05094


120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119888= 07342 119896

119888= 15866 119896

119888= 08933 119896

119888= 17415

Ha = 02 119896119888= 04618 119896

119888= 12112 119896

119888= 06875 119896

119888= 14092

Ha = 04 119896119888= 00536 119896

119888= 08799 119896

119888= 05121 119896

119888= 11371

120591 = 0

Ha = 0 119896119888= 05467 119896

119888= 14263 119896

119888= 07468 119896

119888= 15984

Ha = 02 119896119888= 03071 119896

119888= 11034 119896

119888= 05947 119896

119888= 13177

Ha = 04 119896119888= 0 119896

119888= 08145 119896

119888= 04638 119896

119888= 10872

120591 = minus1

Ha = 0 119896119888= 02421 119896

119888= 12475 119896

119888= 05646 119896

119888= 14411

Ha = 02 119896119888= 0 119896

119888= 09722 119896

119888= 04602 119896

119888= 12099

Ha = 04 119896119888= 0 119896

119888= 07205 119896

119888= 01834 119896

119888= 10187

(D4c) 119888121

cos120601(119905) minus 119887121

sin120601(119905) = (1198962)11986010158401015840

1119886 is obtained


120601 (119905) = 120601 (119905) = tanminus1 [(4119887121

119888121

plusmn 11989611986010158401015840

1119886

times 41198872

121+ 41198882

121minus 1198962119860101584010158402

1119886212

)

times (41198872

121minus 1198962119860101584010158402

11198862)minus1

]

(27)


((

(

1198891198861(119905)

119889119905

1198891198862(119905)

119889119905

119889120601 (119905)

119889119905

))

)

= (

1199011

0 0

0 1199012

0

0 1199013

1199014

)(

1198861(119905)

1198862(119905)

120601 (119905)

) + (

1199021

1199022

1199023

)

(28)

such that

1199011= 119887111

+ (119887121

cos120601 + 119888121

sin120601) 119886 + 119887132

1198862

1199012= 119887211

+ 3119887232

1198862

1199013= minus2119887

121sin120601 + 2119888

121cos120601 minus 2119896119860

10158401015840

1119886

1199014= minus (2119887

121cos120601 + 2119888

121sin120601) 119886

1199021= 0 119902

2= minus2119887

2321198863

1199023= 11989611986010158401015840

11198862+ (2119887121

cos120601 + 2119888121

sin120601) 120601119886

(29)


119894 The


1205821= 119887211

+ 31198862119887232

= minus2119887211

1205822= minus2119886 (119887

121cos120601 + 119888

121sin120601)

1205823= 119887111

+ 119886 (119887121

cos120601 + 119888121

sin120601) + 119887132

1198862

(30)


0

0005

001

0015

002

0025

003

0035

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

0

0005

001

0015

002

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

Nc

Nc

Ha = 005

Re = 5

Ha = 005

Re = 5

Ha = 02

Re = 5

Ha = 02

Re = 5


(3) 120591 = 1


1= 24120576119896






1remains







11990511

= 119887111

+ (119887121

cos120601lowast + 119888121


+ 3119887131

119886lowast2

1+ 119887132

119886lowast2

2

11990512

= (119887121

cos120601lowast + 119888121


119886lowast

1119886lowast

2

11990513

= (minus119887121

sin120601lowast + 119888121


2

11990521

= 2 (119887221



119886lowast

1119886lowast

2

11990522

= 119887211

+ 119887231

119886lowast2

1+ 3119887232

119886lowast2

2

11990523

= minus (119887221

sin120601lowast + 119888221



0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

1205821=minus2b

211

k

120591 = 1

120591 = 0

120591 = minus1

120591 = minus2

120591 = 2

Ha = 03

Re = 2


∘

11990531

= 211989611986010158401015840

1119886lowast

1minus2119886lowast

1

119886lowast2

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990532

= minus 211989611986010158401015840

1119886lowast

2+ 2 (119888121


sin120601lowast)

+119886lowast2

1

119886lowast22

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990533

= minus 2 (119888121

sin120601lowast + 119887121


minus (119887221



1

119886lowast2

(31)







1(119905) rarr 120572119898

1(119905)

and 1198862(119905) rarr 120572


1198891198981(119905)

119889119905= 119887111

1198981(119905) + 120572

2

times [119887121

cos120601 (119905) + 119888121

sin120601 (119905)]1198981(119905)1198982(119905)

+ 1205722119887131

1198983

1(119905) + O (120572

4)

(32a)

1198891198982(119905)

119889119905= 119887211

1198982(119905)

+ [119887221

cos120601 (119905) minus 119888221

sin120601 (119905)]1198982

1(119905)

+ 1205722119887231

1198982

1(119905)1198982(119905) + O (120572

4)

(32b)

119889120601 (119905)

119889119905= minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198982

1(119905)

1198982(119905)

+ 120572211989611986010158401015840

11198982

1(119905) + 2120572

2

times [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]1198982(119905) + O (120572

4)

(32c)


tan120601 (119905) =

(minus11989711198972plusmn 1198973(1198972

1+ 1198972

2minus 1198972

3)12

)

(11989721minus 11989723)

(33)

where 1198971= (1 + 2120572

2119887121

1198982

2(119905)119887221

1198982

1(119905)) 1198972= (119888221

119887221

) minus

(21205722119888121

1198982

2(119905)119887221

1198982

1(119905)) and 119897

3= 120572211989611986010158401015840

11198982(119905)119887221



221

119887221

) + O (1205722) (34)


1205722119887131

1198982

1(119905) = minus119887

111minus 120572212059311198982(119905) (35)


+ 120572212059311198982(119905)]119887131

lt 0 where1205931= (119887121

cos120601lowast + 119888121



1205724119887231

12059311198982

2(119905) + 120572

2(119887111

119887231

minus 119887211

119887131

+ 12059311205932)

times 1198982(119905) + 120593

2119887111

= 0

(36)

where 1205932

= 119887221



111119887231

minus 119887211

119887131

+ 12059311205932)2minus

4120572412059311205932119887111

119887231

gt 0 Since 119887111


1198982(119905) =

119887211

119887131

minus 12059311205932

1205722119887231

1205931

=3119896211986211198692

1205722 (119861101584010158401minus 4119896211986210158401015840

1) 1205931

(37)

When 1198692= 0 119898




08

10

12

minus120587k x 120587k

t = 15 t = 145

h(xt)

(a)

08

10

12

minus120587k x 120587k

t = 2995

t = 2985

h(xt)

(b)

08

10

12

minus120587k x 120587k

t = 2t = 0

h(xt)

(c)





119869 = (

21205722119887131

119898lowast2

112057221205931119898lowast

1minus1205722 sin120601lowast (119887

121+119888121

119887221

119888221

)119898lowast

1119898lowast

2

2 (1205932+ 1205722119887231

119898lowast

2)119898lowast

11205722119887211

119898lowast2

10

2120572211989611986010158401015840

1119898lowast

1minus21205722(119887121

+119888121

119887221

119888221


2+ 1205932

119898lowast2

1

119898lowast2

)

) (38)



(119909 119905) 119909 isin (minus120587119896119898

120587119896119898


119898119909)



0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03


0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)



0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

1

2

3

4

5

6

0 02 04 06 08 10

05

1

15

2

25

3

k

0 02 04 06 08 1

0 02 04 06 08 1 0 02 04 06 08 1

k

0 02 04 06 08 1

k

0 02 04 06 08 1

k

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

120591 = 0

120591 = 1

120591 = minus1

CL

0

05

1

15

2

25

CL

CL

0

1

2

3

4

5

CL

0

1

2

3

4

5

6

CLV

0

1

2

3

4

5

6

CLV

Ha = 0001

Re = 2

Ha = 05

Re = 2

Re = 2

Ha = 05

Re = 5

Re = 5

Ha = 0001

Re = 5

Ha Ha


119871versus 119896


lowast lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 0

lowast

lowast

lowast lowast

lowast

lowast


lowast lowast

lowast lowast

lowast lowast

lowast

lowast lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 02120591 = 0

lowast lowast

lowast


lowast lowast


lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = 0


lowast lowast


lowast


lowast

lowast lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 1


lowast lowast


lowast lowast

lowast lowast


lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 02120591 = 1

lowast lowast

lowast lowast


lowast

lowast

lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

23

Ha = 04120591 = 1


lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

Ha = 005120591 = minus1


lowast

lowast lowast

lowast

lowast

lowast lowast


lowast lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

1

2

2

3

3

Ha = 02120591 = minus1





lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = minus1


119877lt 0 and 119869

2lt 0 (2) 119862

119877lt 0 and 119869

2gt 0 (3) 119862

119877gt 0 and 119869

2lt 0 The



119898= 119896119888radic2




111= 0 and 120597[119896b

111]120597119896 = 0 give the neutral


V119888=

119889

1198891199051205791(119905) = 119888

111+ [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]

times 1198862(119905) + 119888

1311198862

1(119905) + 119888

1321198862

2(119905)

(25)





reads as

ℎ (119909 119905) = 1 + 2 [1198611cos1199091015840 + 119861

2cos (21199091015840 minus 120601)] (26)


1(119905) = 0 119886

2(119905) = 0 and 120601(119905) =

0 (1199111(119905) = 0 119911





gives 1198861(119905) = 0 and 119886

2

2(119905) = minus119887

211119887232


211119887232


2 is set From



120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119904= 03671 119896

119904= 07933 119896

119904= 04467 119896

119904= 08707

Ha = 02 119896119904= 02309 119896

119904= 06056 119896

119904= 03438 119896

119904= 07045

Ha = 04 119896119904= 00268 119896

119904= 04399 119896

119904= 02561 119896

119904= 05686

120591 = 0

Ha = 0 119896119904= 02733 119896

119904= 07132 119896

119904= 03734 119896

119904= 07992

Ha = 02 119896119904= 01536 119896

119904= 05517 119896

119904= 02974 119896

119904= 06588

Ha = 04 119896119904= 0 119896

119904= 04073 119896

119904= 02319 119896

119904= 05436

120591 = minus1

Ha = 0 119896119904= 01211 119896

119904= 06237 119896

119904= 02823 119896

119904= 07205

Ha = 02 119896119904= 0 119896

119904= 04861 119896

119904= 02301 119896

119904= 06049

Ha = 04 119896119904= 0 119896

119904= 03602 119896

119904= 01835 119896

119904= 05094


120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119888= 07342 119896

119888= 15866 119896

119888= 08933 119896

119888= 17415

Ha = 02 119896119888= 04618 119896

119888= 12112 119896

119888= 06875 119896

119888= 14092

Ha = 04 119896119888= 00536 119896

119888= 08799 119896

119888= 05121 119896

119888= 11371

120591 = 0

Ha = 0 119896119888= 05467 119896

119888= 14263 119896

119888= 07468 119896

119888= 15984

Ha = 02 119896119888= 03071 119896

119888= 11034 119896

119888= 05947 119896

119888= 13177

Ha = 04 119896119888= 0 119896

119888= 08145 119896

119888= 04638 119896

119888= 10872

120591 = minus1

Ha = 0 119896119888= 02421 119896

119888= 12475 119896

119888= 05646 119896

119888= 14411

Ha = 02 119896119888= 0 119896

119888= 09722 119896

119888= 04602 119896

119888= 12099

Ha = 04 119896119888= 0 119896

119888= 07205 119896

119888= 01834 119896

119888= 10187

(D4c) 119888121

cos120601(119905) minus 119887121

sin120601(119905) = (1198962)11986010158401015840

1119886 is obtained


120601 (119905) = 120601 (119905) = tanminus1 [(4119887121

119888121

plusmn 11989611986010158401015840

1119886

times 41198872

121+ 41198882

121minus 1198962119860101584010158402

1119886212

)

times (41198872

121minus 1198962119860101584010158402

11198862)minus1

]

(27)


((

(

1198891198861(119905)

119889119905

1198891198862(119905)

119889119905

119889120601 (119905)

119889119905

))

)

= (

1199011

0 0

0 1199012

0

0 1199013

1199014

)(

1198861(119905)

1198862(119905)

120601 (119905)

) + (

1199021

1199022

1199023

)

(28)

such that

1199011= 119887111

+ (119887121

cos120601 + 119888121

sin120601) 119886 + 119887132

1198862

1199012= 119887211

+ 3119887232

1198862

1199013= minus2119887

121sin120601 + 2119888

121cos120601 minus 2119896119860

10158401015840

1119886

1199014= minus (2119887

121cos120601 + 2119888

121sin120601) 119886

1199021= 0 119902

2= minus2119887

2321198863

1199023= 11989611986010158401015840

11198862+ (2119887121

cos120601 + 2119888121

sin120601) 120601119886

(29)


119894 The


1205821= 119887211

+ 31198862119887232

= minus2119887211

1205822= minus2119886 (119887

121cos120601 + 119888

121sin120601)

1205823= 119887111

+ 119886 (119887121

cos120601 + 119888121

sin120601) + 119887132

1198862

(30)


0

0005

001

0015

002

0025

003

0035

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

0

0005

001

0015

002

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

Nc

Nc

Ha = 005

Re = 5

Ha = 005

Re = 5

Ha = 02

Re = 5

Ha = 02

Re = 5


(3) 120591 = 1


1= 24120576119896






1remains







11990511

= 119887111

+ (119887121

cos120601lowast + 119888121


+ 3119887131

119886lowast2

1+ 119887132

119886lowast2

2

11990512

= (119887121

cos120601lowast + 119888121


119886lowast

1119886lowast

2

11990513

= (minus119887121

sin120601lowast + 119888121


2

11990521

= 2 (119887221



119886lowast

1119886lowast

2

11990522

= 119887211

+ 119887231

119886lowast2

1+ 3119887232

119886lowast2

2

11990523

= minus (119887221

sin120601lowast + 119888221



0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

1205821=minus2b

211

k

120591 = 1

120591 = 0

120591 = minus1

120591 = minus2

120591 = 2

Ha = 03

Re = 2


∘

11990531

= 211989611986010158401015840

1119886lowast

1minus2119886lowast

1

119886lowast2

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990532

= minus 211989611986010158401015840

1119886lowast

2+ 2 (119888121


sin120601lowast)

+119886lowast2

1

119886lowast22

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990533

= minus 2 (119888121

sin120601lowast + 119887121


minus (119887221



1

119886lowast2

(31)







1(119905) rarr 120572119898

1(119905)

and 1198862(119905) rarr 120572


1198891198981(119905)

119889119905= 119887111

1198981(119905) + 120572

2

times [119887121

cos120601 (119905) + 119888121

sin120601 (119905)]1198981(119905)1198982(119905)

+ 1205722119887131

1198983

1(119905) + O (120572

4)

(32a)

1198891198982(119905)

119889119905= 119887211

1198982(119905)

+ [119887221

cos120601 (119905) minus 119888221

sin120601 (119905)]1198982

1(119905)

+ 1205722119887231

1198982

1(119905)1198982(119905) + O (120572

4)

(32b)

119889120601 (119905)

119889119905= minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198982

1(119905)

1198982(119905)

+ 120572211989611986010158401015840

11198982

1(119905) + 2120572

2

times [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]1198982(119905) + O (120572

4)

(32c)


tan120601 (119905) =

(minus11989711198972plusmn 1198973(1198972

1+ 1198972

2minus 1198972

3)12

)

(11989721minus 11989723)

(33)

where 1198971= (1 + 2120572

2119887121

1198982

2(119905)119887221

1198982

1(119905)) 1198972= (119888221

119887221

) minus

(21205722119888121

1198982

2(119905)119887221

1198982

1(119905)) and 119897

3= 120572211989611986010158401015840

11198982(119905)119887221



221

119887221

) + O (1205722) (34)


1205722119887131

1198982

1(119905) = minus119887

111minus 120572212059311198982(119905) (35)


+ 120572212059311198982(119905)]119887131

lt 0 where1205931= (119887121

cos120601lowast + 119888121



1205724119887231

12059311198982

2(119905) + 120572

2(119887111

119887231

minus 119887211

119887131

+ 12059311205932)

times 1198982(119905) + 120593

2119887111

= 0

(36)

where 1205932

= 119887221



111119887231

minus 119887211

119887131

+ 12059311205932)2minus

4120572412059311205932119887111

119887231

gt 0 Since 119887111


1198982(119905) =

119887211

119887131

minus 12059311205932

1205722119887231

1205931

=3119896211986211198692

1205722 (119861101584010158401minus 4119896211986210158401015840

1) 1205931

(37)

When 1198692= 0 119898




08

10

12

minus120587k x 120587k

t = 15 t = 145

h(xt)

(a)

08

10

12

minus120587k x 120587k

t = 2995

t = 2985

h(xt)

(b)

08

10

12

minus120587k x 120587k

t = 2t = 0

h(xt)

(c)





119869 = (

21205722119887131

119898lowast2

112057221205931119898lowast

1minus1205722 sin120601lowast (119887

121+119888121

119887221

119888221

)119898lowast

1119898lowast

2

2 (1205932+ 1205722119887231

119898lowast

2)119898lowast

11205722119887211

119898lowast2

10

2120572211989611986010158401015840

1119898lowast

1minus21205722(119887121

+119888121

119887221

119888221


2+ 1205932

119898lowast2

1

119898lowast2

)

) (38)



(119909 119905) 119909 isin (minus120587119896119898

120587119896119898


119898119909)



0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03


0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)



0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










lowast lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast lowast

lowast

lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 0

lowast

lowast

lowast lowast

lowast

lowast


lowast lowast

lowast lowast

lowast lowast

lowast

lowast lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 02120591 = 0

lowast lowast

lowast


lowast lowast


lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = 0


lowast lowast


lowast


lowast

lowast lowast

lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 005120591 = 1


lowast lowast


lowast lowast

lowast lowast


lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2

3

Ha = 02120591 = 1

lowast lowast

lowast lowast


lowast

lowast

lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

23

Ha = 04120591 = 1


lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast lowast

lowast

lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

Ha = 005120591 = minus1


lowast

lowast lowast

lowast

lowast

lowast lowast


lowast lowast

0 2 4 6 8 100

02

04

06

08

1

Re

k

1

1

2

2

3

3

Ha = 02120591 = minus1





lowast

lowast lowast


0 2 4 6 8 100

02

04

06

08

1

Re

k

1

2 3

Ha = 04120591 = minus1


119877lt 0 and 119869

2lt 0 (2) 119862

119877lt 0 and 119869

2gt 0 (3) 119862

119877gt 0 and 119869

2lt 0 The



119898= 119896119888radic2




111= 0 and 120597[119896b

111]120597119896 = 0 give the neutral


V119888=

119889

1198891199051205791(119905) = 119888

111+ [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]

times 1198862(119905) + 119888

1311198862

1(119905) + 119888

1321198862

2(119905)

(25)





reads as

ℎ (119909 119905) = 1 + 2 [1198611cos1199091015840 + 119861

2cos (21199091015840 minus 120601)] (26)


1(119905) = 0 119886

2(119905) = 0 and 120601(119905) =

0 (1199111(119905) = 0 119911





gives 1198861(119905) = 0 and 119886

2

2(119905) = minus119887

211119887232


211119887232


2 is set From



120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119904= 03671 119896

119904= 07933 119896

119904= 04467 119896

119904= 08707

Ha = 02 119896119904= 02309 119896

119904= 06056 119896

119904= 03438 119896

119904= 07045

Ha = 04 119896119904= 00268 119896

119904= 04399 119896

119904= 02561 119896

119904= 05686

120591 = 0

Ha = 0 119896119904= 02733 119896

119904= 07132 119896

119904= 03734 119896

119904= 07992

Ha = 02 119896119904= 01536 119896

119904= 05517 119896

119904= 02974 119896

119904= 06588

Ha = 04 119896119904= 0 119896

119904= 04073 119896

119904= 02319 119896

119904= 05436

120591 = minus1

Ha = 0 119896119904= 01211 119896

119904= 06237 119896

119904= 02823 119896

119904= 07205

Ha = 02 119896119904= 0 119896

119904= 04861 119896

119904= 02301 119896

119904= 06049

Ha = 04 119896119904= 0 119896

119904= 03602 119896

119904= 01835 119896

119904= 05094


120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119888= 07342 119896

119888= 15866 119896

119888= 08933 119896

119888= 17415

Ha = 02 119896119888= 04618 119896

119888= 12112 119896

119888= 06875 119896

119888= 14092

Ha = 04 119896119888= 00536 119896

119888= 08799 119896

119888= 05121 119896

119888= 11371

120591 = 0

Ha = 0 119896119888= 05467 119896

119888= 14263 119896

119888= 07468 119896

119888= 15984

Ha = 02 119896119888= 03071 119896

119888= 11034 119896

119888= 05947 119896

119888= 13177

Ha = 04 119896119888= 0 119896

119888= 08145 119896

119888= 04638 119896

119888= 10872

120591 = minus1

Ha = 0 119896119888= 02421 119896

119888= 12475 119896

119888= 05646 119896

119888= 14411

Ha = 02 119896119888= 0 119896

119888= 09722 119896

119888= 04602 119896

119888= 12099

Ha = 04 119896119888= 0 119896

119888= 07205 119896

119888= 01834 119896

119888= 10187

(D4c) 119888121

cos120601(119905) minus 119887121

sin120601(119905) = (1198962)11986010158401015840

1119886 is obtained


120601 (119905) = 120601 (119905) = tanminus1 [(4119887121

119888121

plusmn 11989611986010158401015840

1119886

times 41198872

121+ 41198882

121minus 1198962119860101584010158402

1119886212

)

times (41198872

121minus 1198962119860101584010158402

11198862)minus1

]

(27)


((

(

1198891198861(119905)

119889119905

1198891198862(119905)

119889119905

119889120601 (119905)

119889119905

))

)

= (

1199011

0 0

0 1199012

0

0 1199013

1199014

)(

1198861(119905)

1198862(119905)

120601 (119905)

) + (

1199021

1199022

1199023

)

(28)

such that

1199011= 119887111

+ (119887121

cos120601 + 119888121

sin120601) 119886 + 119887132

1198862

1199012= 119887211

+ 3119887232

1198862

1199013= minus2119887

121sin120601 + 2119888

121cos120601 minus 2119896119860

10158401015840

1119886

1199014= minus (2119887

121cos120601 + 2119888

121sin120601) 119886

1199021= 0 119902

2= minus2119887

2321198863

1199023= 11989611986010158401015840

11198862+ (2119887121

cos120601 + 2119888121

sin120601) 120601119886

(29)


119894 The


1205821= 119887211

+ 31198862119887232

= minus2119887211

1205822= minus2119886 (119887

121cos120601 + 119888

121sin120601)

1205823= 119887111

+ 119886 (119887121

cos120601 + 119888121

sin120601) + 119887132

1198862

(30)


0

0005

001

0015

002

0025

003

0035

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

0

0005

001

0015

002

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

Nc

Nc

Ha = 005

Re = 5

Ha = 005

Re = 5

Ha = 02

Re = 5

Ha = 02

Re = 5


(3) 120591 = 1


1= 24120576119896






1remains







11990511

= 119887111

+ (119887121

cos120601lowast + 119888121


+ 3119887131

119886lowast2

1+ 119887132

119886lowast2

2

11990512

= (119887121

cos120601lowast + 119888121


119886lowast

1119886lowast

2

11990513

= (minus119887121

sin120601lowast + 119888121


2

11990521

= 2 (119887221



119886lowast

1119886lowast

2

11990522

= 119887211

+ 119887231

119886lowast2

1+ 3119887232

119886lowast2

2

11990523

= minus (119887221

sin120601lowast + 119888221



0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

1205821=minus2b

211

k

120591 = 1

120591 = 0

120591 = minus1

120591 = minus2

120591 = 2

Ha = 03

Re = 2


∘

11990531

= 211989611986010158401015840

1119886lowast

1minus2119886lowast

1

119886lowast2

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990532

= minus 211989611986010158401015840

1119886lowast

2+ 2 (119888121


sin120601lowast)

+119886lowast2

1

119886lowast22

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990533

= minus 2 (119888121

sin120601lowast + 119887121


minus (119887221



1

119886lowast2

(31)







1(119905) rarr 120572119898

1(119905)

and 1198862(119905) rarr 120572


1198891198981(119905)

119889119905= 119887111

1198981(119905) + 120572

2

times [119887121

cos120601 (119905) + 119888121

sin120601 (119905)]1198981(119905)1198982(119905)

+ 1205722119887131

1198983

1(119905) + O (120572

4)

(32a)

1198891198982(119905)

119889119905= 119887211

1198982(119905)

+ [119887221

cos120601 (119905) minus 119888221

sin120601 (119905)]1198982

1(119905)

+ 1205722119887231

1198982

1(119905)1198982(119905) + O (120572

4)

(32b)

119889120601 (119905)

119889119905= minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198982

1(119905)

1198982(119905)

+ 120572211989611986010158401015840

11198982

1(119905) + 2120572

2

times [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]1198982(119905) + O (120572

4)

(32c)


tan120601 (119905) =

(minus11989711198972plusmn 1198973(1198972

1+ 1198972

2minus 1198972

3)12

)

(11989721minus 11989723)

(33)

where 1198971= (1 + 2120572

2119887121

1198982

2(119905)119887221

1198982

1(119905)) 1198972= (119888221

119887221

) minus

(21205722119888121

1198982

2(119905)119887221

1198982

1(119905)) and 119897

3= 120572211989611986010158401015840

11198982(119905)119887221



221

119887221

) + O (1205722) (34)


1205722119887131

1198982

1(119905) = minus119887

111minus 120572212059311198982(119905) (35)


+ 120572212059311198982(119905)]119887131

lt 0 where1205931= (119887121

cos120601lowast + 119888121



1205724119887231

12059311198982

2(119905) + 120572

2(119887111

119887231

minus 119887211

119887131

+ 12059311205932)

times 1198982(119905) + 120593

2119887111

= 0

(36)

where 1205932

= 119887221



111119887231

minus 119887211

119887131

+ 12059311205932)2minus

4120572412059311205932119887111

119887231

gt 0 Since 119887111


1198982(119905) =

119887211

119887131

minus 12059311205932

1205722119887231

1205931

=3119896211986211198692

1205722 (119861101584010158401minus 4119896211986210158401015840

1) 1205931

(37)

When 1198692= 0 119898




08

10

12

minus120587k x 120587k

t = 15 t = 145

h(xt)

(a)

08

10

12

minus120587k x 120587k

t = 2995

t = 2985

h(xt)

(b)

08

10

12

minus120587k x 120587k

t = 2t = 0

h(xt)

(c)





119869 = (

21205722119887131

119898lowast2

112057221205931119898lowast

1minus1205722 sin120601lowast (119887

121+119888121

119887221

119888221

)119898lowast

1119898lowast

2

2 (1205932+ 1205722119887231

119898lowast

2)119898lowast

11205722119887211

119898lowast2

10

2120572211989611986010158401015840

1119898lowast

1minus21205722(119887121

+119888121

119887221

119888221


2+ 1205932

119898lowast2

1

119898lowast2

)

) (38)



(119909 119905) 119909 isin (minus120587119896119898

120587119896119898


119898119909)



0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03


0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)



0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119904= 03671 119896

119904= 07933 119896

119904= 04467 119896

119904= 08707

Ha = 02 119896119904= 02309 119896

119904= 06056 119896

119904= 03438 119896

119904= 07045

Ha = 04 119896119904= 00268 119896

119904= 04399 119896

119904= 02561 119896

119904= 05686

120591 = 0

Ha = 0 119896119904= 02733 119896

119904= 07132 119896

119904= 03734 119896

119904= 07992

Ha = 02 119896119904= 01536 119896

119904= 05517 119896

119904= 02974 119896

119904= 06588

Ha = 04 119896119904= 0 119896

119904= 04073 119896

119904= 02319 119896

119904= 05436

120591 = minus1

Ha = 0 119896119904= 01211 119896

119904= 06237 119896

119904= 02823 119896

119904= 07205

Ha = 02 119896119904= 0 119896

119904= 04861 119896

119904= 02301 119896

119904= 06049

Ha = 04 119896119904= 0 119896

119904= 03602 119896

119904= 01835 119896

119904= 05094


120573 = 55∘

120573 = 75∘

Re = 3 Re = 6 Re = 3 Re = 6

120591 = 1

Ha = 0 119896119888= 07342 119896

119888= 15866 119896

119888= 08933 119896

119888= 17415

Ha = 02 119896119888= 04618 119896

119888= 12112 119896

119888= 06875 119896

119888= 14092

Ha = 04 119896119888= 00536 119896

119888= 08799 119896

119888= 05121 119896

119888= 11371

120591 = 0

Ha = 0 119896119888= 05467 119896

119888= 14263 119896

119888= 07468 119896

119888= 15984

Ha = 02 119896119888= 03071 119896

119888= 11034 119896

119888= 05947 119896

119888= 13177

Ha = 04 119896119888= 0 119896

119888= 08145 119896

119888= 04638 119896

119888= 10872

120591 = minus1

Ha = 0 119896119888= 02421 119896

119888= 12475 119896

119888= 05646 119896

119888= 14411

Ha = 02 119896119888= 0 119896

119888= 09722 119896

119888= 04602 119896

119888= 12099

Ha = 04 119896119888= 0 119896

119888= 07205 119896

119888= 01834 119896

119888= 10187

(D4c) 119888121

cos120601(119905) minus 119887121

sin120601(119905) = (1198962)11986010158401015840

1119886 is obtained


120601 (119905) = 120601 (119905) = tanminus1 [(4119887121

119888121

plusmn 11989611986010158401015840

1119886

times 41198872

121+ 41198882

121minus 1198962119860101584010158402

1119886212

)

times (41198872

121minus 1198962119860101584010158402

11198862)minus1

]

(27)


((

(

1198891198861(119905)

119889119905

1198891198862(119905)

119889119905

119889120601 (119905)

119889119905

))

)

= (

1199011

0 0

0 1199012

0

0 1199013

1199014

)(

1198861(119905)

1198862(119905)

120601 (119905)

) + (

1199021

1199022

1199023

)

(28)

such that

1199011= 119887111

+ (119887121

cos120601 + 119888121

sin120601) 119886 + 119887132

1198862

1199012= 119887211

+ 3119887232

1198862

1199013= minus2119887

121sin120601 + 2119888

121cos120601 minus 2119896119860

10158401015840

1119886

1199014= minus (2119887

121cos120601 + 2119888

121sin120601) 119886

1199021= 0 119902

2= minus2119887

2321198863

1199023= 11989611986010158401015840

11198862+ (2119887121

cos120601 + 2119888121

sin120601) 120601119886

(29)


119894 The


1205821= 119887211

+ 31198862119887232

= minus2119887211

1205822= minus2119886 (119887

121cos120601 + 119888

121sin120601)

1205823= 119887111

+ 119886 (119887121

cos120601 + 119888121

sin120601) + 119887132

1198862

(30)


0

0005

001

0015

002

0025

003

0035

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

0

0005

001

0015

002

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

Nc

Nc

Ha = 005

Re = 5

Ha = 005

Re = 5

Ha = 02

Re = 5

Ha = 02

Re = 5


(3) 120591 = 1


1= 24120576119896






1remains







11990511

= 119887111

+ (119887121

cos120601lowast + 119888121


+ 3119887131

119886lowast2

1+ 119887132

119886lowast2

2

11990512

= (119887121

cos120601lowast + 119888121


119886lowast

1119886lowast

2

11990513

= (minus119887121

sin120601lowast + 119888121


2

11990521

= 2 (119887221



119886lowast

1119886lowast

2

11990522

= 119887211

+ 119887231

119886lowast2

1+ 3119887232

119886lowast2

2

11990523

= minus (119887221

sin120601lowast + 119888221



0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

1205821=minus2b

211

k

120591 = 1

120591 = 0

120591 = minus1

120591 = minus2

120591 = 2

Ha = 03

Re = 2


∘

11990531

= 211989611986010158401015840

1119886lowast

1minus2119886lowast

1

119886lowast2

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990532

= minus 211989611986010158401015840

1119886lowast

2+ 2 (119888121


sin120601lowast)

+119886lowast2

1

119886lowast22

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990533

= minus 2 (119888121

sin120601lowast + 119887121


minus (119887221



1

119886lowast2

(31)







1(119905) rarr 120572119898

1(119905)

and 1198862(119905) rarr 120572


1198891198981(119905)

119889119905= 119887111

1198981(119905) + 120572

2

times [119887121

cos120601 (119905) + 119888121

sin120601 (119905)]1198981(119905)1198982(119905)

+ 1205722119887131

1198983

1(119905) + O (120572

4)

(32a)

1198891198982(119905)

119889119905= 119887211

1198982(119905)

+ [119887221

cos120601 (119905) minus 119888221

sin120601 (119905)]1198982

1(119905)

+ 1205722119887231

1198982

1(119905)1198982(119905) + O (120572

4)

(32b)

119889120601 (119905)

119889119905= minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198982

1(119905)

1198982(119905)

+ 120572211989611986010158401015840

11198982

1(119905) + 2120572

2

times [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]1198982(119905) + O (120572

4)

(32c)


tan120601 (119905) =

(minus11989711198972plusmn 1198973(1198972

1+ 1198972

2minus 1198972

3)12

)

(11989721minus 11989723)

(33)

where 1198971= (1 + 2120572

2119887121

1198982

2(119905)119887221

1198982

1(119905)) 1198972= (119888221

119887221

) minus

(21205722119888121

1198982

2(119905)119887221

1198982

1(119905)) and 119897

3= 120572211989611986010158401015840

11198982(119905)119887221



221

119887221

) + O (1205722) (34)


1205722119887131

1198982

1(119905) = minus119887

111minus 120572212059311198982(119905) (35)


+ 120572212059311198982(119905)]119887131

lt 0 where1205931= (119887121

cos120601lowast + 119888121



1205724119887231

12059311198982

2(119905) + 120572

2(119887111

119887231

minus 119887211

119887131

+ 12059311205932)

times 1198982(119905) + 120593

2119887111

= 0

(36)

where 1205932

= 119887221



111119887231

minus 119887211

119887131

+ 12059311205932)2minus

4120572412059311205932119887111

119887231

gt 0 Since 119887111


1198982(119905) =

119887211

119887131

minus 12059311205932

1205722119887231

1205931

=3119896211986211198692

1205722 (119861101584010158401minus 4119896211986210158401015840

1) 1205931

(37)

When 1198692= 0 119898




08

10

12

minus120587k x 120587k

t = 15 t = 145

h(xt)

(a)

08

10

12

minus120587k x 120587k

t = 2995

t = 2985

h(xt)

(b)

08

10

12

minus120587k x 120587k

t = 2t = 0

h(xt)

(c)





119869 = (

21205722119887131

119898lowast2

112057221205931119898lowast

1minus1205722 sin120601lowast (119887

121+119888121

119887221

119888221

)119898lowast

1119898lowast

2

2 (1205932+ 1205722119887231

119898lowast

2)119898lowast

11205722119887211

119898lowast2

10

2120572211989611986010158401015840

1119898lowast

1minus21205722(119887121

+119888121

119887221

119888221


2+ 1205932

119898lowast2

1

119898lowast2

)

) (38)



(119909 119905) 119909 isin (minus120587119896119898

120587119896119898


119898119909)



0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03


0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)



0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

0005

001

0015

002

0025

003

0035

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

0

0005

001

0015

002

2

1

3

02 04 0806 1 12 14

k

120572Γ0

35

4

45

5

55

6

65

7

2

1

3

02 04 0806 1 12 14

k

Nc

Nc

Ha = 005

Re = 5

Ha = 005

Re = 5

Ha = 02

Re = 5

Ha = 02

Re = 5


(3) 120591 = 1


1= 24120576119896






1remains







11990511

= 119887111

+ (119887121

cos120601lowast + 119888121


+ 3119887131

119886lowast2

1+ 119887132

119886lowast2

2

11990512

= (119887121

cos120601lowast + 119888121


119886lowast

1119886lowast

2

11990513

= (minus119887121

sin120601lowast + 119888121


2

11990521

= 2 (119887221



119886lowast

1119886lowast

2

11990522

= 119887211

+ 119887231

119886lowast2

1+ 3119887232

119886lowast2

2

11990523

= minus (119887221

sin120601lowast + 119888221



0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

1205821=minus2b

211

k

120591 = 1

120591 = 0

120591 = minus1

120591 = minus2

120591 = 2

Ha = 03

Re = 2


∘

11990531

= 211989611986010158401015840

1119886lowast

1minus2119886lowast

1

119886lowast2

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990532

= minus 211989611986010158401015840

1119886lowast

2+ 2 (119888121


sin120601lowast)

+119886lowast2

1

119886lowast22

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990533

= minus 2 (119888121

sin120601lowast + 119887121


minus (119887221



1

119886lowast2

(31)







1(119905) rarr 120572119898

1(119905)

and 1198862(119905) rarr 120572


1198891198981(119905)

119889119905= 119887111

1198981(119905) + 120572

2

times [119887121

cos120601 (119905) + 119888121

sin120601 (119905)]1198981(119905)1198982(119905)

+ 1205722119887131

1198983

1(119905) + O (120572

4)

(32a)

1198891198982(119905)

119889119905= 119887211

1198982(119905)

+ [119887221

cos120601 (119905) minus 119888221

sin120601 (119905)]1198982

1(119905)

+ 1205722119887231

1198982

1(119905)1198982(119905) + O (120572

4)

(32b)

119889120601 (119905)

119889119905= minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198982

1(119905)

1198982(119905)

+ 120572211989611986010158401015840

11198982

1(119905) + 2120572

2

times [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]1198982(119905) + O (120572

4)

(32c)


tan120601 (119905) =

(minus11989711198972plusmn 1198973(1198972

1+ 1198972

2minus 1198972

3)12

)

(11989721minus 11989723)

(33)

where 1198971= (1 + 2120572

2119887121

1198982

2(119905)119887221

1198982

1(119905)) 1198972= (119888221

119887221

) minus

(21205722119888121

1198982

2(119905)119887221

1198982

1(119905)) and 119897

3= 120572211989611986010158401015840

11198982(119905)119887221



221

119887221

) + O (1205722) (34)


1205722119887131

1198982

1(119905) = minus119887

111minus 120572212059311198982(119905) (35)


+ 120572212059311198982(119905)]119887131

lt 0 where1205931= (119887121

cos120601lowast + 119888121



1205724119887231

12059311198982

2(119905) + 120572

2(119887111

119887231

minus 119887211

119887131

+ 12059311205932)

times 1198982(119905) + 120593

2119887111

= 0

(36)

where 1205932

= 119887221



111119887231

minus 119887211

119887131

+ 12059311205932)2minus

4120572412059311205932119887111

119887231

gt 0 Since 119887111


1198982(119905) =

119887211

119887131

minus 12059311205932

1205722119887231

1205931

=3119896211986211198692

1205722 (119861101584010158401minus 4119896211986210158401015840

1) 1205931

(37)

When 1198692= 0 119898




08

10

12

minus120587k x 120587k

t = 15 t = 145

h(xt)

(a)

08

10

12

minus120587k x 120587k

t = 2995

t = 2985

h(xt)

(b)

08

10

12

minus120587k x 120587k

t = 2t = 0

h(xt)

(c)





119869 = (

21205722119887131

119898lowast2

112057221205931119898lowast

1minus1205722 sin120601lowast (119887

121+119888121

119887221

119888221

)119898lowast

1119898lowast

2

2 (1205932+ 1205722119887231

119898lowast

2)119898lowast

11205722119887211

119898lowast2

10

2120572211989611986010158401015840

1119898lowast

1minus21205722(119887121

+119888121

119887221

119888221


2+ 1205932

119898lowast2

1

119898lowast2

)

) (38)



(119909 119905) 119909 isin (minus120587119896119898

120587119896119898


119898119909)



0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03


0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)



0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

1205821=minus2b

211

k

120591 = 1

120591 = 0

120591 = minus1

120591 = minus2

120591 = 2

Ha = 03

Re = 2


∘

11990531

= 211989611986010158401015840

1119886lowast

1minus2119886lowast

1

119886lowast2

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990532

= minus 211989611986010158401015840

1119886lowast

2+ 2 (119888121


sin120601lowast)

+119886lowast2

1

119886lowast22

(119887221

sin120601lowast + 119888221

cos120601lowast)

11990533

= minus 2 (119888121

sin120601lowast + 119887121


minus (119887221



1

119886lowast2

(31)







1(119905) rarr 120572119898

1(119905)

and 1198862(119905) rarr 120572


1198891198981(119905)

119889119905= 119887111

1198981(119905) + 120572

2

times [119887121

cos120601 (119905) + 119888121

sin120601 (119905)]1198981(119905)1198982(119905)

+ 1205722119887131

1198983

1(119905) + O (120572

4)

(32a)

1198891198982(119905)

119889119905= 119887211

1198982(119905)

+ [119887221

cos120601 (119905) minus 119888221

sin120601 (119905)]1198982

1(119905)

+ 1205722119887231

1198982

1(119905)1198982(119905) + O (120572

4)

(32b)

119889120601 (119905)

119889119905= minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198982

1(119905)

1198982(119905)

+ 120572211989611986010158401015840

11198982

1(119905) + 2120572

2

times [119888121

cos120601 (119905) minus 119887121

sin120601 (119905)]1198982(119905) + O (120572

4)

(32c)


tan120601 (119905) =

(minus11989711198972plusmn 1198973(1198972

1+ 1198972

2minus 1198972

3)12

)

(11989721minus 11989723)

(33)

where 1198971= (1 + 2120572

2119887121

1198982

2(119905)119887221

1198982

1(119905)) 1198972= (119888221

119887221

) minus

(21205722119888121

1198982

2(119905)119887221

1198982

1(119905)) and 119897

3= 120572211989611986010158401015840

11198982(119905)119887221



221

119887221

) + O (1205722) (34)


1205722119887131

1198982

1(119905) = minus119887

111minus 120572212059311198982(119905) (35)


+ 120572212059311198982(119905)]119887131

lt 0 where1205931= (119887121

cos120601lowast + 119888121



1205724119887231

12059311198982

2(119905) + 120572

2(119887111

119887231

minus 119887211

119887131

+ 12059311205932)

times 1198982(119905) + 120593

2119887111

= 0

(36)

where 1205932

= 119887221



111119887231

minus 119887211

119887131

+ 12059311205932)2minus

4120572412059311205932119887111

119887231

gt 0 Since 119887111


1198982(119905) =

119887211

119887131

minus 12059311205932

1205722119887231

1205931

=3119896211986211198692

1205722 (119861101584010158401minus 4119896211986210158401015840

1) 1205931

(37)

When 1198692= 0 119898




08

10

12

minus120587k x 120587k

t = 15 t = 145

h(xt)

(a)

08

10

12

minus120587k x 120587k

t = 2995

t = 2985

h(xt)

(b)

08

10

12

minus120587k x 120587k

t = 2t = 0

h(xt)

(c)





119869 = (

21205722119887131

119898lowast2

112057221205931119898lowast

1minus1205722 sin120601lowast (119887

121+119888121

119887221

119888221

)119898lowast

1119898lowast

2

2 (1205932+ 1205722119887231

119898lowast

2)119898lowast

11205722119887211

119898lowast2

10

2120572211989611986010158401015840

1119898lowast

1minus21205722(119887121

+119888121

119887221

119888221


2+ 1205932

119898lowast2

1

119898lowast2

)

) (38)



(119909 119905) 119909 isin (minus120587119896119898

120587119896119898


119898119909)



0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03


0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)



0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










08

10

12

minus120587k x 120587k

t = 15 t = 145

h(xt)

(a)

08

10

12

minus120587k x 120587k

t = 2995

t = 2985

h(xt)

(b)

08

10

12

minus120587k x 120587k

t = 2t = 0

h(xt)

(c)





119869 = (

21205722119887131

119898lowast2

112057221205931119898lowast

1minus1205722 sin120601lowast (119887

121+119888121

119887221

119888221

)119898lowast

1119898lowast

2

2 (1205932+ 1205722119887231

119898lowast

2)119898lowast

11205722119887211

119898lowast2

10

2120572211989611986010158401015840

1119898lowast

1minus21205722(119887121

+119888121

119887221

119888221


2+ 1205932

119898lowast2

1

119898lowast2

)

) (38)



(119909 119905) 119909 isin (minus120587119896119898

120587119896119898


119898119909)



0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03


0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)



0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

0minus120587k

x

120587k

106

104

102

1

098

096

094

h(xt)

Ha = 005Ha = 0

Ha = 02 Ha = 03


0 50 100 150 200 2501

105

11

115

120591 =1

120591 = 0

120591 = minus1

120591 = minus15

Hm

ax

t

(a)

0 100 200 300 400 500 600 7001

105

11

115

120591 = 0

120591 = 1

120591 = minus1 120591 = minus15

Hm

ax

t

(b)



0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800 1000 1200 1400

1

098

096

094

092

09

088

120591 = 0

120591 = 0

120591 = minus1120591 = minus1

Hm

in

t

1

105

11

115

120591 = 0120591 = minus1

120591 = 0 120591 = minus1

Hm

ax

0 200 400 600 800 1000 1200 1400

t


∘ 119878 = 5 and Ha = 03












ℎ

0119906(119909 119905)119889119910 for different







0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0995

1005

101

1

099

Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(a)

0998

0999

1

1001

1002

1003

1004

1005Time = 1299

Time = 1300

minus3120587k minus120587k 3120587k120587k

x

h(xt)

(b)










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus120587k 120587k

x

minus120587k 120587k

x

0minus05

0

05

1

15

2

25

3

35

4

1

1

1

1

1

2

2

2

2

2

3

3

33

3

u(xt)(xt)q(xt)

120591 = 1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 0

1

2

3

0minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = 1

minus05

0

05

1

15

2

25

3

35

4

u(xt)(xt)q(xt)

120591 = minus1

0

Ha = 0

Ha = 0

Ha = 0

Ha = 02

Ha = 02

Ha = 02







Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Appendices



119906119909+ V119910= 0

120576 (119906119905+ 119906119906119909+ V119906119910) = minus120576119901

119909+ 1205762119906119909119909

+ 119906119910119910

+ ReminusHa119906

1205762(V119905+ 119906V119909+ VV119910) = minus119901

119910+ 1205763V119909119909

+ 120576V119910119910

minus Re cot120573

(A1)


119906 = 0 V = 0 on 119910 = 0 (A2)

119901 + 2120576(1 minus 120576

2ℎ2

119909)

(1 + 1205762ℎ2119909)119906119909+

2120576ℎ119909

1 + 1205762ℎ2119909

(119906119910+ 1205762V119909)

= minus1205762Sℎ119909119909

(1 + 1205762ℎ2119909)32

on 119910 = ℎ

(A3)

(1 minus 1205762ℎ2

119909)

(1 + 1205762ℎ2119909)(119906119910+ 1205762V119909) minus

41205762ℎ119909

(1 + 1205762ℎ2119909)119906119909= 120591

on 119910 = ℎ

(A4)

V = ℎ119905+ 119906ℎ119909

on 119910 = ℎ (A5)



119867(120576 119909 1199091 119905 1199051 1199052) =

infin

sum

119894=1

120572119894119867119894(120576 119909 119909

1 119905 1199051 1199052) (B1)


(1198710+ 1205721198711+ 12057221198712) (120572119867

1+ 12057221198672+ 12057231198673) = minus120572

21198732minus 12057231198733

(B2)



2and


1198710=

120597

120597119905+ 1198601

120597

120597119909+ 1205761198611

1205972

1205971199092+ 1205761198621

1205974

1205971199094 (B3a)

1198711=

120597

1205971199051

+ 1198601

120597

1205971199091

+ 21205761198611

1205972

1205971199091205971199091

+ 41205761198621

1205974

12059711990931205971199091

(B3b)

1198712=

120597

1205971199052

+ 1205761198611

1205972

12059711990921

+ 61205761198621

1205974

120597119909212059711990921

(B3c)

1198732= 1198601015840

111986711198671119909

+ 1205761198611015840

111986711198671119909119909

+ 1205761198621015840

111986711198671119909119909119909119909

+ 1205761198611015840

11198672

1119909+ 1205761198621015840

111986711199091198671119909119909119909

(B3d)

1198733= 1198601015840

1(11986711198672119909

+ 119867111986711199091

+ 11986721198671119909)

+ 1205761198611015840

1(11986711198672119909119909

+ 2119867111986711199091199091

+ 11986721198671119909119909

)

+ 1205761198621015840

1(11986711198672119909119909119909119909

+ 4119867111986711199091199091199091199091

+ 11986721198671119909119909119909119909

)

+ 1205761198611015840

1(211986711199091198672119909

+ 2119867111990911986711199091

)

+ 1205761198621015840

1(1198672119909119909119909

1198671119909

+ 311986711199091199091199091

1198671119909

+1198671119909119909119909

1198672119909

+ 1198671119909119909119909

11986711199091

)

+11986010158401015840

1

21198672

11198671119909

+ 12057611986110158401015840

1

21198672

11198671119909119909

+ 12057611986210158401015840

1

21198672

11198671119909119909119909119909

+ 12057611986110158401015840

111986711198672

1119909+ 12057611986210158401015840

1119867111986711199091198671119909119909119909

(B3e)



120597120578

1205971199052

minus 120572minus2119862119877120578 + 120576 (119861

1minus 611989621198621)1205972120578

12059711990921

+ (1198692+ 1198941198694)100381610038161003816100381612057810038161003816100381610038162

120578 = 0

(B4)

where

119890119903=

2 (1198611015840

1minus 11989621198621015840

1)

(minus41198611+ 161198962119862

1) 119890

119894=

minus1198601015840

1

120576 (minus41198961198611+ 161198963119862

1)

1198692= 120576(7119896

41198901199031198621015840

1minus 11989621198901199031198611015840

1minus1198962

211986110158401015840

1+1198964

211986210158401015840

1) minus 119896119890

1198941198601015840

1

1198694= 120576 (7119896

41198901198941198621015840

1minus 11989621198901198941198611015840

1) + 119896119890

1199031198601015840

1+ 119896

11986010158401015840

1

2

(B5)


119894V120597120578

1205971199091

= 2119894119896120576 (1198611minus 211989621198621) 120572minus1 120597120578

1205971199091

(B6)




2) The


0(1199052)119890[minus119894119887(1199052)1199052]



namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











namely

119889Γ0

1198891199052

minus 119894Γ0

119889

1198891199052

[119887 (1199052) 1199052] minus 120572minus2119862119877Γ0+ (1198692+ 1198941198694) Γ3

0= 0

(B7)


119889 [Γ0(1199052)]

1198891199052

= (120572minus2119862119877minus 1198692Γ2

0) Γ0 (B8)

119889 [119887 (1199052) 1199052]

1198891199052

= 1198694Γ2

0 (B9)

If 1198692


119877and



0from (B8)


2)

120572Γ0= radic

119862119877

1198692

(B10)


119887 =1198621198771198694

12057221198692

(B11)



2= 1205722119905 This gives


119873119888= 119862119871119881

+ 1205722119887 = 119862

119871119881+ 119862119877

1198694

1198692

(B12)




120578 = Γ119904(1199052) + (ΔΓ

+(1199052) 119890119894119896119909

+ ΔΓminus(1199052) 119890minus119894119896119909

) 119890minus1198941198871199052 (C1)



1minus 211989621198621) lt 0 119888

119894=

120572minus2119862119877 and 119869

1119903= 120576(119861

1minus 611989621198621) lt 0 since 119861

1minus 11989621198621

=

O(1205722))

119889Γ+(1199052)

1198891199052

= (119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)

times Γ+(1199052) minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

Γminus(1199052)

(C2a)

119889Γminus(1199052)

1198891199052

= (119894P minus 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

)


10038161003816100381610038162

Γ+(1199052)

(C2b)


(

119889Γ+(1199052)

1198891199052

119889Γminus(1199052)

1198891199052

) = M(

Γ+(1199052)

Γminus(1199052)

) (C3)


11987211

= 119894P + 119896V + 119888119894+ 11989621198691119903minus 2 (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987222


10038161003816100381610038162

11987212

= minus (1198692+ 1198941198694)1003816100381610038161003816Γ119904

10038161003816100381610038162

11987221

= 11987212

(C4)

Setting Γ+(1199052) prop 119890

1205991199052 and Γminus(1199052) prop 119890




120599 =1





complex conjugate)

ℎ (119909 119905) = 1 +

2

sum

119899=1

119911119899(119905) 119890119899119894119896119909

+ 119911119899(119905) 119890minus119899119894119896119909 (D1)




2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2119894119896119909 terms

119889

1198891199051199111(119905) = 119886

1111199111(119905) + 119886

1211199111(119905) 1199112(119905)

+ 1199111(119905) (119886131

1003816100381610038161003816119911110038161003816100381610038162

+ 119886132

1003816100381610038161003816119911210038161003816100381610038162

)

(D2a)

119889

1198891199051199112(119905) = 119886

2111199112(119905) + 119886

2211199112

1(119905)

+ 1199112(119905) (119886231

1003816100381610038161003816119911110038161003816100381610038162

+ 119886232

1003816100381610038161003816119911210038161003816100381610038162

)

(D2b)


1| ≪ 1 and |119911

2| ≪ |119911


system are given by

119886111

= 120576 (11989621198611minus 11989641198621) minus 119894119896119860

1

119886211

= 120576 (411989621198611minus 16119896

41198621) minus 2119894119896119860

1

119886121

= 120576 (11989621198611015840

1minus 711989641198621015840

1) minus 119894119896119860

1015840

1

119886221

= 120576 (211989621198611015840

1minus 211989641198621015840

1) minus 119894119896119860

1015840

1

119886131

=120576

2(119896211986110158401015840

1minus 119896411986210158401015840

1) minus

119894

211989611986010158401015840

1

119886231

= 120576 (4119896211986110158401015840

1minus 16119896

411986210158401015840

1) minus 2119894119896119860

10158401015840

1

119886132

= 120576 (119896211986110158401015840

1minus 119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

119886232

= 120576 (2119896211986110158401015840

1minus 8119896411986210158401015840

1) minus 119894119896119860

10158401015840

1

(D3)


119899(119905) = 119886

119899(119905)119890119894120579119899(119905) in polar

form where 119886119899(119905) and 120579



1198891198861(119905)

119889119905= 1198911(1198861(119905) 1198862(119905) 120601 (119905))

= 119887111

1198861(119905) + [119887

121cos120601 (119905) + 119888

121sin120601 (119905)]

times 1198861(119905) 1198862(119905) + 119886

1(119905) [119887131

1198862

1(119905) + 119887

1321198862

2(119905)]

(D4a)

1198891198862(119905)

119889119905= 1198912(1198861(119905) 1198862(119905) 120601 (119905))

= 119887211

1198862(119905) + [119887

221cos120601 (119905) minus 119888

221sin120601 (119905)]

times 1198862

1(119905) + 119886

2(119905) [119887231

1198862

1(119905) + 119887

2321198862

2(119905)]

(D4b)

119889120601 (119905)

119889119905= 1198913(1198861(119905) 1198862(119905) 120601 (119905))

= 11989611986010158401015840

1(1198862

1minus 1198862

2) + 2 [119888

121cos120601 (119905) minus 119887

121sin120601 (119905)]

times 1198862(119905) minus [119887

221sin120601 (119905) + 119888

221cos120601 (119905)]

1198862

1(119905)

1198862(119905)

(D4c)



1(119905) minus 120579

2(119905)

The terms 119887119894119895119896

and 119888119894119895119896



119894 119896 = 1 2 and 119895 = 1 2 3

References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Air-Aided Shear on a Thin Film Subjected to a …downloads.hindawi.com/journals/isrn/2013/748613.pdf · 2017. 12. 4. · ISRN MathematicalPhysics Y B 0 h(x,t) g a