Fully Discrete Local Discontinuous Galerkin Approximation ...downloads.hindawi.com/journals/amp/2017/4961797.pdf · Fully Discrete Local Discontinuous Galerkin Approximation for Time-Space

Research ArticleFully Discrete Local Discontinuous Galerkin Approximation forTime-Space Fractional SubdiffusionSuperdiffusion Equations

Meilan Qiu12 LiquanMei1 and Dewang Li2

1School of Mathematics and Statistics Xirsquoan Jiaotong University Xirsquoan 710049 China2Department of Mathematics Huizhou University Huizhou Guangdong 516007 China

Correspondence should be addressed to Liquan Mei lqmeimailxjtueducn and Dewang Li ldwldw1976126com

Received 16 September 2016 Revised 24 December 2016 Accepted 19 January 2017 Published 16 March 2017

Academic Editor Giampaolo Cristadoro

Copyright copy 2017 Meilan Qiu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We focus on developing the finite difference (ie backward Euler difference or second-order central difference)local discontinuousGalerkin finite element mixed method to construct and analyze a kind of efficient accurate flexible numerical schemesfor approximately solving time-space fractional subdiffusionsuperdiffusion equations Discretizing the time Caputo fractionalderivative by using the backward Euler difference for the derivative parameter (0 lt 120572 lt 1) or second-order central differencemethod for (1 lt 120572 lt 2) combined with local discontinuous Galerkinmethod to approximate the spatial derivative which is definedby a fractional Laplacian operator two high-accuracy fully discrete local discontinuous Galerkin (LDG) schemes of the time-space fractional subdiffusionsuperdiffusion equations are proposed respectively Through the mathematical induction methodwe show the concrete analysis for the stability and the convergence under the 1198712 norm of the LDG schemes Several numericalexperiments are presented to validate the proposed model and demonstrate the convergence rate of numerical schemes Thenumerical experiment results show that the fully discrete local discontinuous Galerkin (LDG) methods are efficient and powerfulfor solving fractional partial differential equations

1 Introduction

During the last few decades fractional calculus emerges as anatural description for a broad range of nonclassical phenom-ena in the applied sciences and engineering For exampleanomalous diffusion is one of the most important conceptsinmodern physics [1ndash3] and is presented in extremely diverseengineering fields such as vibration and acoustic dissipationin soft matter [4] polymer dynamics [5] turbulence flow[6] glassy and porous media [7] charges transport in amor-phous semiconductor [8] chemistry and biochemistry [9]contaminant transport in groundwater flow [10] and chaoticdynamics of classical conservative systems [11]

To date many solution techniques for fractional differ-ential equations have exploited the properties of the Fourierand Laplace transforms of the operators to confirm a classicalsolution Typically Gorenflo et al in [12] used the methodof Laplace transform to obtain the scale-invariant solutionof the time fractional diffusion-wave equation according

to the Wright function Starting from the Fourier-Laplacerepresentation Mainardi et al in [13] derived the explicitexpression of the Green function (ie the fundamentalsolution) for the Cauchy problem with respect to its scalingand similarity properties Since fractional order differen-tial operators possess their own properties (i) they arenonlocal operators and (ii) the adjoint operator itself offractional differential operator is a nonnegative operatorTherefore most fractional differential equations do not haveexact solutions Thus some potent and accurate numericalmethods for these equations are developed For examplea finite difference method was applied to construct thenumerical approximation for the time fractional diffusionequation and introduce an implicit difference scheme forthe space-time fractional diffusion equation was studied in[14ndash17] Ervin and Roop in [18] considered the stationaryfractional advection dispersion equation by using the stan-dard finite element method Zhao et al adopted the similarmethod to discuss the numerical solutions of the multiterm

HindawiAdvances in Mathematical PhysicsVolume 2017 Article ID 4961797 20 pageshttpsdoiorg10115520174961797

2 Advances in Mathematical Physics

time-space Riesz fractional advection-diffusion equations intheir paper [19] and Deng in [20] presented the finite elementapproximation for the space-time fractional Fokker-Planckequation Lin and Xu in [21] and Li and Xu in [22] proposeda spectral method for time or space fractional diffusionequations based on a weak formulation and the detailederror analysis was carried out A new matrix approach tosolve fractional partial differential equations numericallyhas been developed by Podlubny in [23] Li et al in [24]used the fractional difference method in time and the finiteelement method in space for the time-space fractional ordersubdiffusion and superdiffusion equations in which theypresented the error estimates for the full discrete schemesAnd for the subdiffusion linear problem they obtain theapproximation of order Δ1199052minus120572 + Δ119905minus120572ℎ2 in theoretical for thesuperdiffusion linear problem they got the approximation oforder Δ1199052 + Δ1199051minus120572ℎ2 However their numerical experimentscannot be consistent with the corresponding theoreticalresults very well particularly for the superdiffusion prob-lems

Moreover high order accuracy numerical methods forsolving fractional differential equations are still very limitedDiscontinuous Galerkin method is famous for extreme flex-ibility and high-accuracy properties [25] By applying thismethod differential equations can be solved locally and accu-rately while we only need to assemble and solve a large linearsystem Deng and Hesthaven in [26] studied the convergencerate analysis of local discontinuous Galerkin method (LDG)for solving spatial fractional diffusion equation which wasthe first attempt to solve fractional derivative equations byusing the LDG method at the same time the authors inthis paper speculated that the LDG method has significantpotential in solving fractional calculus problems After thatlots of researchers use the LDG method to analyze the timediscretization of fractional differential equations [27 28]Recently Xu and Hesthaven [29] obtained a consistent andhigh-accuracy numerical scheme for fractional convection-diffusion equationswith a space fractional Laplacian operatorof order120572 (1 lt 120572 lt 2) through exploiting LDGmethodTheyintroduced a new technique by rewriting the fractional Lap-lacian operator as a composite of first-order derivativesand a fractional integral then they converted the frac-tional convection-diffusion equation into a system of loworder equations Optimal order of convergence 119874(ℎ119896+1)for the space fractional diffusion problem and suboptimalorder of convergence 119874(ℎ119896+12) for the general spatial frac-tional convection-diffusion problem are established respec-tively

In this paper motivated by the research work of [24]we intend to construct and analyze the numerical schemesfor solving the time-space fractional subdiffusion equation asfollows120597120572119906 (119909 119905)120597119905120572 = 120598 (minus (minus)1205732) 119906 (119909 119905)

119909 isin Ω 119905 isin (0 119879] 119906 (119909 0) = 1199060 (119909) 119909 isin Ω(1)

and the time-space fractional superdiffusion equation asfollows120597120572119906 (119909 119905)120597119905120572 = 120598 (minus (minus)1205732) 119906 (119909 119905)

119909 isin Ω 119905 isin (0 119879] 119906 (119909 0) = 120601 (119909) 119909 isin Ω119906119905 (119909 0) = 120595 (119909) 119909 isin Ω(2)

where Ω = (119886 119887) 119879 gt 0 1 lt 120573 lt 2 denotes the ordernumber of space fractional derivatives 0 lt 120572 lt 1 describesthe fractional subdiffusion equation (1) 1 lt 120572 lt 2 describesthe fractional superdiffusion equation (2) 120598 is a real positiveparameter and 1199060 120601(119909) 120595(119909) isin 1198671

0 (Ω) are the given smoothfunctions

For the sake of simplicity we just consider the periodicboundary conditions in this paper Notice that this assump-tion is not essential our method can be directly extended tothe problems with aperiodic boundary condition

The time fractional derivative in (1) uses the Caputofractional partial derivative of order 120572 defined as [30 31]120597120572119906 (119909 119905)120597119905120572

= 1Γ (1 minus 120572) int119905

0

120597119906 (119909 119904)120597119904 119889119904(119905 minus 119904)120572 0 lt 120572 lt 1120597119906 (119909 119905)120597119905 120572 = 1(3)

and the time fractional Caputo derivative in (2) defined as[24]120597120572119906 (119909 119905)120597119905120572

= 1Γ (2 minus 120572) int119905

0(119905 minus 119904)1minus120572 1205972119906 (119909 119904)1205971199042 119889119904 1 lt 120572 lt 21205972119906 (119909 119905)1205971199052 120572 = 2

(4)

where Γ(sdot) is the Gamma functionThe space fractional terms in (1) and (2) are defined

through the fractional Laplacian operator (minus(minus )1205732)119906(119909 119905)1 lt 120573 lt 2 which is also known as the Riesz fractionalderivative (120597120573120597|119909|120573)119906(119909 119905)under the assumption that119906(119909 119905)1199061015840(119909 119905) 119906(119899minus1)(119909 119905) vanishes at the end point of an infinitedomain (ie 119909 = ∓infin) [32]

The spatial fractional differential operator is also animportant tool to describe the anomalous diffusion phe-nomenon when 1 lt 120573 lt 2 it represents a Levy 120573-stable flight[33] when 120573 rarr 2 it models a Brownian diffusion process

The rest of this paper is organized as follows In thenext section we review the appropriate functional spacesand some important definitions and properties of fractionalderivatives By using the technique which is proposed in[29] that is through converting the space fractional operatorof order 120573 into a composite of first-order derivative and

Advances in Mathematical Physics 3

a fractional integral of order 2 minus 120573 the fully discrete LDGschemes for the time-space fractional subdiffusion equationand the time-space fractional superdiffusion equation areconstructed respectively in Section 3 Sections 4 and 5provide the detailed stability and 1198712 error analysis for thetwo fully discrete schemes respectively Several examples arepresented to test our theoretical results in Section 6 Section 7contains the concluding remarks

2 Definitions and Auxiliary Results

In this section we present a few definitions and recall someproperties of fractional calculus [34 35]

Definition 1 For function 119906(119909 119905) given in the domain Ω =(119886 119887) (119886 119887 can be finite or infinite) the expressions

119877119886 119868120572119909119906 (119909 119905) = 1Γ (120572) int119909

119886(119909 minus 120585)120572minus1 119906 (120585 119905) 119889120585 119909 gt 119886

119877119909119868120572119887 119906 (119909 119905) = 1Γ (120572) int119887

119909(120585 minus 119909)120572minus1 119906 (120585 119905) 119889120585 119909 lt 119887 (5)

are called left- and right-side Riemann-Liouville fractionalintegrals of order 120572 (0 le 119899 minus 1 lt 120572 lt 119899 119899 being an integer)respectively Γ(sdot) is a Gamma function

Definition 2 The left and right of Riemann-Liouville frac-tional derivatives of order 120572 (119899 minus 1 lt 120572 lt 119899) in the domainΩ = (119886 119887) (119886 119887 can be finite or infinite) can be derived byRiemann-Liouville fractional integrals as follows

119877119886119863120572

119909119906 (119909 119905) = 1Γ (119899 minus 120572) 119889119899119889119909119899int119909

119886(119909 minus 120585)119899minus120572minus1 119906 (120585 119905) 119889120585

119909 gt 119886 (6)

119877119909119863120572

119887119906 (119909 119905) = (minus1)119899Γ (119899 minus 120572) 119889119899119889119909119899int119887


119909 lt 119887 (7)

where Γ(sdot) is a Gamma function When 120572 is an integer 119863120572 =119889120572 = 119889119909120572

Definition 3 (see [34]) The fractional Riesz derivatives oforder 120572 (119899 minus 1 lt 120572 le 119899) in the domain Ω = (119886 119887) (119886 119887 can befinite or infinite) are defined as follows

120597120572120597 |119909|120572 119906 (119909 119905) = minus119862120572 (119886119863120572119909 + 119909119863120572

119887 ) 119906 (119909 119905) (8)

where 119862120572 = 12 cos(1205871205722) 120572 = 1 119886119863120572119909 119909119863120572

119887 denote theleft and the right Riemann-Liouville fractional derivatives oforder 120572 respectively

Definition 4 For a function 119906(119909 119905) defined on the infinitedomain (minusinfin lt 119909 lt infin) 119905 isin (0 119879] the following equalityholds [34]minus (minus)1205722 119906 (119909 119905)

= minus 12 cos (1205871205722) (minusinfin119863120572119909 + 119909119863120572

infin) 119906 (119909 119905)= 120597120572120597 |119909|120572 119906 (119909 119905)

(9)

Remark 5 Thedefinition of the fractional Laplacian operatoruses the Fourier transform on an infinite domain with anatural extension to include finite domainswhen the function119906(119909 119905) is subject to homogeneous Dirichlet boundary condi-tions In this paper the developments and analysis are basedon this definition

Lemma6 Suppose that (120597119895120597119909)119906(119909 119905) = 0 for119909 rarr ∓infinforall0 le119895 le 119899 (119899 minus 1 lt 120572 lt 119899) and then the relationship betweenRiemann-Liouville fractional integrals and Riemann-Liouvillefractional derivatives satisfies [30]

119877119886119863120572

119909119906 (119909 119905) = 119863119899 (119886119868119899minus120572119909 119906 (119909 119905)) = 119886119868119899minus120572119909 (119863119899119906 (119909 119905)) 119877119909119863120572

119887119906 (119909 119905) = (minus119863)119899 (119909119868119899minus120572119887 119906 (119909 119905))= 119909119868119899minus120572119887 ((minus119863)119899 119906 (119909 119905)) (10)

In order to carry out the analysis of the fully discreteLDG scheme we need the following properties of fractionalintegrals and derivatives and some lemmas which werecollected from [29]

Lemma 7 (linearity) One has

119886119868120572119909 (120582119891 (119909) + 120583119892 (119909)) = 120582119886119868120572119909119891 (119909) + 120583119886119868120572119909119892 (119909) 119886119863120572

119909 (120582119891 (119909) + 120583119892 (119909)) = 120582119886119863120572119909119891 (119909) + 120583119886119863120572

119909119892 (119909) (11)

Lemma 8 (semigroup property)

119886119868120572+120573119909 119891 (119909) = 119886119868120572119909 (119886119868120573119909119891 (119909)) = 119886119868120573119909 (119886119868120572119909119891 (119909)) (12)

By equalities (10) for any continuous function119906(119909 119905)withlim119909rarrplusmninfin119906119895(119909 119905) = 0 (0 le 119895 le 2) and 1 lt 120573 lt 2 we canobtain the following result [29]

minus (minus)1205732 119906 (119909 119905) = minus 12 cos (1205871205732)sdot 12059721205971199092

(minusinfin1198682minus120573119909 119906 (119909 119905) + 1199091198682minus120573infin 119906 (119909 119905))= minus 12 cos (1205871205732)sdot 120597120597119909 (minusinfin1198682minus120573119909

120597119906 (119909 119905)120597119909 + 1199091198682minus120573infin

120597119906 (119909 119905)120597119909 )= 120597120597119909 (minusminusinfin1198682minus120573119909 + 1199091198682minus120573infin2 cos (1205871205732) 120597119906 (119909 119905)120597119909 )

(13)


In the above equation if 120573 lt 0 the fractional Laplacianoperator becomes the fractional integral operator Similar to[29] since 1 lt 120573 lt 2 then minus1 lt 120573 minus 2 lt 0 and we define

(120573minus2)2 = minusminusinfin1198682minus120573119909 + 1199091198682minus120573infin2 cos (1205871205732) (14)

Lemma 9 When 1 lt 120573 lt 2 we can rewrite the spacefractional Laplacian operator in a composite form of first-orderderivative and a fractional integral of order 2minus120573 as follows [29]minus (minus)1205732 119906 (119909 119905) = 120597120597119909 ((120573minus2)2

120597119906 (119909 119905)120597119909 ) (15)

Definition 10 We define the left- and right-fractional spacenorms as [18]119906119869120572119871 (119877) = (|119906|2119869120572119871 (119877) + 11990621198712(119877))12

119906119869120572119877(119877) = (|119906|2119869120572119877(119877) + 11990621198712(119877))12 (16)

where |119906|119869120572(119877) = 1198631205721199061198712(119877) is the seminorm of fractionalspace and 119869120572119871 (119877) 119869120572119877(119877) refer to the closure of 119862infin

0 (119877) withrespect to sdot 119869120572119871 (119877) and sdot 119869120572119877(119877) respectivelyLemma 11 (adjoint property) One has(minusinfin119868120572119909119906 119906)119877 = (119906 119909119868120572infin119906)119877 (17)Lemma 12 (see [18]) One has(minusinfin119868120572119909119906 119909119868120572infin119906)119877 = cos (120572120587) |119906|2119869minus120572119871 (119877)= cos (120572120587) |119906|2119869minus120572119877 (119877) (18)

Lemma 13 (see [19]) Let 0 lt 120579 lt 2 120579 = 1 Then for any119908 V isin 11986712057920 (0 119883) then(1198770 119863120579

119909119908 V)1198712(0119883)

= (1198770 1198631205792119909 119908 119877

119905 1198631205792119883 V)

1198712(0119883) (19)

From the above definitions and Lemma 8 together withthe properties of Riemann-Liouville fractional integrals thefollowing lemmas hold

Lemma 14 For any 0 lt 119904 lt 1 the fractional integral satisfiesthe following property(minus119904119906 119906)119877 = |119906|2119869minus120572119871 (119877) = |119906|2119869minus120572119877 (119877) (20)

Proof Since 0 lt 119904 lt 1 by Lemmas 9 11 12 and 13 we obtain

(minus119904119906 119906)119877 = (minusinfin1198682119904119909 119906 + 1199091198682119904infin1199062 cos (119904120587) 119906)119877= 12 cos (119904120587) [(minusinfin1198682119904119909 119906 119906)

119877+ (1199091198682119904infin119906 119906)

119877]

= 12 cos (119904120587) [(minusinfin119868119904119909119906 119909119868119904infin119906)119877 + (119909119868119904infin119906 minusinfin119868119904119909119906)119877]= 12 cos (119904120587) 2 cos (119904120587) |119906|2119869minus119904119871 (119877)= 12 cos (119904120587) 2 cos (119904120587) |119906|2119869minus119904119877 (119877)

(21)

Theorem 15 (see [26]) If minus1205722 lt minus1205721 lt 0 then 119869minus12057211198710 (Ω) (or119869minus12057211198770 (Ω) or 119869minus12057211198780 (Ω)) is embedded into 119869minus12057221198710 (Ω) (or 119869minus12057221198770 (Ω) or119869minus12057221198780 (Ω)) and 1198712(Ω) is embedded into both of them

Lemma 16 (fractional Poincare-Friedrichs inequality [18])One has1199061198712(Ω) le 119862 |119906|1198691205831198710(Ω) 119906 isin 1198691205831198710 (Ω) 120583 isin 119877

1199061198712(Ω) le 119862 |119906|1198691205831198770(Ω) 119906 isin 1198691205831198770 (Ω) 120583 isin 119877 (22)

Lemma 17 Suppose the fractional integral is defined on [0 119887]and let 119892(119910) = 119891(119887 minus 119910) Then

119909119868120572119887 119891 (119909) =0119868120572119910 119892 (119910) (23)

Proof One has

119909119868120572119887 119891 (119909) = int119887

119909(119905 minus 119909)120572minus1 119891 (119905) 119889119905

119905=119887minus120585= minus int0

119887minus119909(119887 minus 120585 minus 119909)120572minus1 119891 (119887 minus 120585) 119889120585

= int119887minus119909

0(119887 minus 119909 minus 120585)120572minus1 119891 (119887 minus 120585) 119889120585

119910=119887minus119909= int119910

0(119910 minus 120585)120572minus1 119891 (119887 minus 120585) 119889120585

119891(119887minus120585)=119892(120585)= int119910

0(119910 minus 120585)120572minus1 119892 (120585) 119889120585

=0119868120572119910 119892 (119910)

(24)

Lemma 18 (see [29]) Suppose 119906(119909) is a smooth functiondefined on Ω sub 119877 Ωℎ is discretization of the domain withinterval width ℎ and 119906ℎ(119909) is an approximation of u in 119875119896

ℎ Forall 119895 119906ℎ(119909) isin 119868119895 is a polynomial of degree up to order 119896 and(119906 V)119868119895 = (119906ℎ V)119868119895 forallV isin 119875119896 119896 is the degree of the polynomialThen for minus1 lt 120572 le 0 we obtain10038171003817100381710038171003817()1205732 119906 (119909) minus ()1205732 119906ℎ (119909)100381710038171003817100381710038171198712(Ω)

le 119862ℎ119896+1 (25)

where 119862 is a constant independent of ℎAssume the following mesh to cover the computational

domain Ω = [119886 119887] consisting of cells 119868119895 = [119909119895minus(12) 119909119895+(12)]for 119895 = 1 119873 where 119886 = 11990912 lt 11990932 lt sdot sdot sdot lt 119909119873+(12) = 119887denoting the cell lengths 119909119895 = 119909119895+(12) minus 119909119895minus(12) 1 le 119895 le 119873ℎ = max1le119895le119873 119909119895 Define 119881119896

ℎ as the space of polynomials ofthe degree up to 119896 in each cell 119868119895 that is119881119896

ℎ = V V isin 119875119896 (119868119895) 119909 isin 119868119895 119895 = 1 2 119873 (26)

To prepare for the analysis of the error estimates we needtwo projections in one dimension [119886 119887] denoted by Q thatis for each 119895int

119868119895

(Q120596 (119909) minus 120596 (119909)) 120592 (119909) 119889119909 = 0 forall120592 isin 119875119896 (119868119895) (27)


and special projectionPplusmn that is for each 119895int119868119895

(P+120596 (119909) minus 120596 (119909)) 120592 (119909) 119889119909 = 0forall120592 isin 119875119896minus1 (119868119895) P

+120596 (119909+119895minus(12)) = 120596 (119909119895minus(12))

int119868119895

(Pminus120596 (119909) minus 120596 (119909)) 120592 (119909) 119889119909 = 0forall120592 isin 119875119896minus1 (119868119895) P

minus120596 (119909minus119895+(12)) = 120596 (119909119895+(12))

(28)

The two projections satisfy the following approximationinequality [36]10038171003817100381710038171205961198901003817100381710038171003817 + ℎ 10038171003817100381710038171205961198901003817100381710038171003817infin + ℎ12 10038171003817100381710038171205961198901003817100381710038171003817120591ℎ le 119862ℎ119896+1 (29)

where 120596119890 = P120596(119909) minus 120596(119909) or 120596119890 = Pplusmn120596(119909) minus 120596(119909) Thepositive constant 119862 solely depending on 120596 is independent ofℎ 120591ℎ denotes the set of boundary points of all elements 119868119895

In the present paper we use 119862 to denote a positiveconstant whichmay have a different value in each occurrenceThe usual notation of norms in Sobolev spaces will be usedLet the scalar inner product on 1198712(119863) be denoted by (sdot sdot)119863and the associated norm by sdot 119863 If 119863 = Ω we drop 119863

3 Numerical Schemes

In this section we present the fully discrete local discontinu-ous Galerkin (LDG) schemes for solving the time-space frac-tional subdiffusionsuperdiffusion problems respectively

Let 120591 = 119879119872 be the timemesh size119872 is a positive integerlet 119905119899 = 119899120591 119899 = 0 1 119872 be mesh point and let 119905119899minus(12) =(119905119899 + 119905119899minus1)2 119899 = 0 1 119872 be mid-mesh points Here andbelow an unmarked norm sdot refers to the usual 1198712 norm

31 Subdiffusion Case In the rest of this section we willfollow the time discrete method of [24] to show the timediscretization for the time fractional derivative 120597120572119906(119909 119905)120597119905120572of subdiffusion equation

For the case of subdiffusion (0 lt 120572 lt 1) equation(1) we adopt the backward Euler method to numericallyapproximate the time fractional derivative 120597120572119906(119909 119905)120597119905120572 at 119905119899as follows [21 27]

120597120572119906 (119909 119905119899)120597119905120572 = 1Γ (1 minus 120572) 119899minus1sum119894=0

int119905119894+1

119905119894

120597119906 (119909 119904)120597119904 119889119904(119905119899 minus 119904)120572= 1Γ (1 minus 120572) 119899minus1sum

119894=0

119906 (119909 119905119894+1) minus 119906 (119909 119905119894)120591sdot int(119894+1)120591

119894120591

119889119904(119905119899 minus 119904)120572 + 119903119899 (119909) = 1205911minus120572Γ (2 minus 120572)sdot 119899minus1sum119894=0

119906 (119909 119905119899minus119894) minus 119906 (119909 119905119899minus119894minus1)120591 [(119894 + 1)1minus120572 minus 1198941minus120572]

+ 119903119899 (119909) = 1205911minus120572Γ (2 minus 120572) 119899minus1sum119894=0

119887119894 119906 (119909 119905119899minus119894) minus 119906 (119909 119905119899minus119894minus1)120591+ 119903119899 (119909) (30)

Here 119887119894 = (119894 + 1)1minus120572 minus 1198941minus120572 satisfy the following properties1 = 1198870 gt 1198871 gt 1198872 gt sdot sdot sdot gt 119887119899 gt 0119887119899 997888rarr 0 (119899 997888rarr infin) 119899sum119894=1

(119887119894minus1 minus 119887119894) + 119887119899 = 1 (31)

119903119899(119909) is the truncation error and satisfies the followinginequality [21] 1003817100381710038171003817119903119899 (119909)1003817100381710038171003817 le 119862119906Γ (2 minus 120572) 1205912minus120572 (32)

where 119862119906 is a constant only depending on 119906 We rewrite (1) asa first-order system120597120572119906 (119909 119905)120597119905120572 minus radic120598 120597120597119909 119902 = 0

119902 minus (120573minus2)2119901 = 0119901 minus radic120598 120597120597119909 119906 (119909 119905) = 0

(33)

And with (30) we give the weak form of system (33) at 119905119899 asfollows

intΩ

119906 (119909 119905119899) V 119889119909 + 1205720radic120598 (intΩ

119902 (119909 119905119899) V119909119889119909minus 119873sum

119895=1

(119902 (119909 119905119899) Vminus119895+(12) minus 119902 (119909 119905119899) V+119895minus(12)))= 119899minus1sum

119894=1

(119887119894minus1 minus 119887119894) intΩ

119906 (119909 119905119899minus119894) V 119889119909+ 119887119899minus1 int

Ω119906 (119909 1199050) V 119889119909 minus 1205720 int

Ω119903119899 (119909) V 119889119909

intΩ

119902 (119909 119905119899) 119908 119889119909 minus intΩ

(120573minus2)2119901 (119909 119905119899) 119908 119889119909 = 0intΩ

119901 (119909 119905119899) 119911 119889119909 + radic120598 (intΩ

119906 (119909 119905119899) 119911119909119889119909minus 119873sum

119895=1

(119906 (119909 119905119899) 119911minus119895+(12) minus 119906 (119909 119905119899) 119911+

119895minus(12))) = 0intΩ

119906 (119909 1199050) V 119889119909 minus intΩ

1199060 (119909) V 119889119909 = 0

(34)

where 1205720 = 120591120572Γ(2 minus 120572)


Let 119906119899ℎ 119901119899

ℎ 119902119899ℎ isin 119881119896ℎ be the approximation of 119906(sdot 119905119899) 119901(sdot119905119899) 119902(sdot 119905119899) respectively We define a fully discrete local

discontinuous Galerkin (LDG) scheme as follows Find119906119899ℎ 119901119899

ℎ 119902119899ℎ isin 119881119896ℎ such that for all V 119908 119911 isin 119881119896

ℎ

intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1

(119902119899ℎVminus119895+(12) minus 119902119899ℎV+119895minus(12))) = 119899minus1sum119894=1

(119887119894minus1 minus 119887119894)sdot int

Ω119906119899minus119894ℎ V 119889119909 + 119887119899minus1 int

Ω1199060ℎV 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(35)

where 1205720 = 120591120572Γ(2 minus 120572)The ldquohatrdquo terms in (35) in the cell boundary terms from

integration by parts are the so-called ldquonumerical fluxesrdquo [37]which are single valued functions defined on the edges andshould be designed based on different guiding principles fordifferent PDEs to ensure the stability Here we take the simplechoices such that 119906119899

ℎ = 119906119899ℎminus119902119899ℎ = 119902119899ℎ+ (36)

We remark that the choice for the fluxes (36) is not unique Infact the crucial part is taking 119906119899

ℎ and 119902119899ℎ from opposite sidesWe know the truncation error is 119903119899(119909) from (30)

32 Superdiffusion Case Similar to the subdiffusion casewe adopt the center difference scheme to estimate the timefractional derivative 120597120572119906(119909 119905)120597119905120572 of superdiffusion case (2)at 119905119899 as follows [24]120597120572119906 (119909 119905119899)120597119905120572 = 1Γ (2 minus 120572) int119905119899

0

1205972119906 (119909 119904)1205971199042 119889119904(119905119899 minus 119904)120572minus1= 1Γ (2 minus 120572) 119899minus1sum

119894=0

int119905119894+1

119905119894


119894=0

119906 (119909 119905119894+1) minus 2119906 (119909 119905119894) + 119906 (119909 119905119894minus1)1205912sdot int(119894+1)120591

119894120591(119905119899 minus 119904)1minus120572 119889119904 + 119877119899 (119909) = 1205912minus120572Γ (3 minus 120572)

sdot 119899minus1sum119894=0

119906 (119909 119905119899minus119894) minus 2119906 (119909 119905119899minus119894minus1) + 119906 (119909 119905119899minus119894minus2)1205912sdot [(119894 + 1)2minus120572 minus 1198942minus120572] + 119877119899 (119909) = 1205912minus120572Γ (3 minus 120572)sdot 119899minus1sum119894=0

]119894119906 (119909 119905119899minus119894) minus 119906 (119909 119905119899minus119894minus1)120591 + 119877119899 (119909)

(37)

where ]119894 = (119894 + 1)2minus120572 minus 1198942minus120572 119894 = 0 1 119899 minus 1 satisfy thefollowing properties

1 = ]0 gt ]1 gt ]2 gt sdot sdot sdot gt ]119899 gt 0]119899 997888rarr 0 (119899 997888rarr infin)

119899sum119894=1

(]119894minus1 minus ]119894) + ]119899 = 1 (38)

119877119899(119909) is the truncation error that is

119877119899 (119909) = 1Γ (2 minus 120572) 119899minus1sum119894=0

int119905119894+1

119905119894

(119905119899 minus 119904)1minus120572 [ 1205972119906 (119909 119904)1205971199042minus 12059721199061205971199052 (119909 119905119894) minus 120591212 12059741199061205971199054 (119909 119905119894) + O (1205914)] 119889119904= 1Γ (2 minus 120572) 119899minus1sum

119894=0

int119905119894+1

119905119894

(119905119899 minus 119904)1minus120572 [(119904 minus 119905119894) 12059731199061205971199053 (119909 119905119894)+ O (1205912)] 119889119904

(39)

which satisfies the following estimate

119877119899 (119909)le 119862120591Γ (2 minus 120572) max

1199050le119905le119905119899

100381610038161003816100381610038161003816100381610038161003816 12059731199061205971199053 (119909 119905119894)100381610038161003816100381610038161003816100381610038161003816 119899minus1sum119894=0

int119905119894+1

119905119894

(119905119899 minus 119904)1minus120572 119889119904+ O (1205912) le 1198621199061198793minus120572Γ (2 minus 120572) 120591 + O (1205912)

(40)

where 119862119906 is the constant which only depends on 119906Analogous to the subdiffusion equation we also can

rewrite (2) as the first-order system (33) From the timediscretization equation (37) we obtain the weak formulationof (2) at 119905119899

intΩ

119906 (119909 119905119899) V 119889119909 + 1205721radic120598 (intΩ

119902 (119909 119905119899) V119909119889119909minus 119873sum

119895=1

(119902 (119909 119905119899) Vminus119895+(12) minus 119902 (119909 119905119899) V+119895minus(12)))


= 119899minus2sum119894=1

(minus]119894minus1 + 2]119894 minus ]119894+1) intΩ

119906 (119909 119905119899minus119894minus1) V 119889119909+ (2]0 minus ]1) int

Ω119906 (119909 119905119899minus1) V 119889119909 + (2]119899minus1 minus ]119899minus2)

sdot intΩ

119906 (119909 1199050) V 119889119909 minus ]119899minus1 intΩ

119906 (119909 119905minus1) V 119889119909minus 1205721 int

Ω119877119899 (119909) V 119889119909

intΩ

119902 (119909 119905119899) 119908 119889119909 minus intΩ

(120573minus2)2119901 (119909 119905119899) 119908 119889119909 = 0intΩ

119901 (119909 119905119899) 119911 119889119909 + radic120598 (intΩ

119906 (119909 119905119899) 119911119909119889119909minus 119873sum

119895=1

(119906 (119909 119905119899) 119911minus119895+(12) minus 119906 (119909 119905119899) 119911+

119895minus(12))) = 0intΩ

119906 (119909 1199050) V 119889119909 minus intΩ

120601 (119909) V 119889119909 = 0intΩ

119906119905 (119909 1199050) V 119889119909 minus intΩ

120595 (119909) V 119889119909 = 0(41)

where 1205721 = 120591120572Γ(3 minus 120572)Let 119906119899

ℎ 119901119899ℎ 119902119899ℎ isin 119881119896

ℎ be the approximation of 119906(sdot 119905119899) 119901(sdot119905119899) 119902(sdot 119905119899) respectively We define a fully discrete localdiscontinuous Galerkin (LDG) scheme as follows find119906119899ℎ 119901119899


ℎ

intΩ

119906119899ℎV 119889119909 + 1205721radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1

(119902119899ℎVminus119895+(12) minus 119902119899ℎV+119895minus(12)))= 119899minus2sum

119894=1


119906119899minus119894minus1ℎ V 119889119909 + (2]0

minus ]1) intΩ

119906119899minus1ℎ V 119889119909 + (2]119899minus1 minus ]119899minus2) int

Ω1199060ℎV 119889119909

minus ]119899minus1 intΩ

119906minus1ℎ V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0

intΩ

1199060ℎV 119889119909 minus int

Ω120601 (119909) V 119889119909 = 0

intΩ

(1199060ℎ)

119905V 119889119909 minus int

Ω120595 (119909) V 119889119909 = 0

(42)

Remarks 1 Originally we assume that 119906 has compact supportand restrict the problem to the bounded domain Ω As aconsequence we impose homogeneous Dirichlet boundaryconditions for 119906 notin Ω

First we define the fluxes at the boundary as 119873+(12) =119906(119887 119905) 119902119873+(12) = 119902minus119873+(12) + (120583ℎ)[119906119873+(12)] for the rightboundary and 12 = 119906(119886 119905) 11990212 = 119902minus12 + (120583ℎ)[11990612] forthe left boundary [29] where 120583 is a positive constant and [sdot]is the jump term operator of function 119906

Next we define the boundary term operatorL as follows

L (V 119908 119911) = radic120598119906 (119886 119905) 119911+12 minus radic120598120583ℎ 119906 (119887 119905) Vminus119873+(12)minus radic120598119906 (119887 119905) 119911minus

119873+(12) = 0 (43)

4 Stability

In this subsection we present the stability analysis for thefully discrete LDG schemes (35) and (42) correspondingly

Theorem 19 For periodic or compactly supported boundaryconditions the fully discrete LDG scheme (35) is uncondition-ally stable and the numerical solution 119906119899

ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 (44)

Using the fluxes (36) and the fluxes at the boundariescombining equality (43) after a simple rearrangement ofterms the fully LDG scheme (35) becomes

intΩ

119906119899ℎV 119889119909 + 1205720radic120598 int

Ω119902119899ℎV119909119889119909 + radic120598 int

Ω119906119899ℎ119911119909119889119909

minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 + int

Ω119902119899ℎ119908 119889119909 + int

Ω119901119899ℎ119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(119902119899ℎ)+119895+(12) [V]119895+(12)+ radic120598119873minus1sum

119895=1

(119906119899ℎ)minus119895+(12) [119911]119895+(12)

+ 1205720radic120598 ((119902119899ℎ)+12 V+12 minus (119902119899ℎ)minus119873+(12) Vminus119873+(12))

+ radic120598120583ℎ ((119906119899ℎ)minus119873+(12))2

= 119899minus1sum119894=1

(119887119894minus1 minus 119887119894) intΩ

119906119899minus119894ℎ V 119889119909 + 119887119899minus1 int

Ω1199060ℎV 119889119909

(45)

Now we use the mathematical induction to proofTheorem 19


Proof First consider 119899 = 1 the LDG scheme (35) is

intΩ

1199061ℎV 119889119909 + 1205720radic120598 int

Ω1199021ℎV119909119889119909 + radic120598 int

Ω1199061ℎz119909119889119909

minus intΩ

(120573minus2)21199011ℎ119908 119889119909 + int

Ω1199021ℎ119908 119889119909 + int

Ω1199011ℎ119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1

(1199061ℎ)minus

119895+(12)[119911]119895+(12)

+ 1205720radic120598 (1199021ℎ)+12

V+12 minus 1205720radic120598 (1199021ℎ)minus119873+(12)

Vminus119873+(12)

+ radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎV 119889119909

(46)

Taking the test functions V = 1199061ℎ 119908 = minus12057201199011

ℎ 119911 = 12057201199021ℎ we getintΩ

1199061ℎ1199061

ℎ119889119909 + 1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909

+ 1205720radic120598 intΩ

1199061ℎ (1199021ℎ)

119909119889119909 + 1205720 int

Ω(120573minus2)21199011

ℎ1199011ℎ119889119909

minus 1205720 intΩ

1199021ℎ1199011ℎ119889119909 + 1205720 int

Ω1199011ℎ1199021ℎ119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

+ 1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+

12minus 1205720radic120598 (1199021ℎ)minus119873+(12)

(1199061ℎ)minus

119873+(12)

+ 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎ1199061

ℎ119889119909

(47)

Considering the integration by parts formulation(1199021ℎ (1199061ℎ)119909)119868119895 + (1199061

ℎ (1199021ℎ)119909)119868119895 = 1199061ℎ1199021ℎ|119909minus119895+(12)

119909+119895minus(12)

we obtain the inter-face condition

1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909 + 1205720radic120598 int

Ω1199061ℎ (1199021ℎ)

119909119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= 1205720radic120598 119873sum119895=1

1199061ℎ1199021ℎ10038161003816100381610038161003816119909minus119895+(12)119909+

119895minus(12)

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= minus1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+

12+ 1205720radic120598 (1199021ℎ)minus119873+(12)

(1199061ℎ)minus

119873+(12)

(48)

Thus we obtain

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

= intΩ

1199060ℎ1199061

ℎ119889119909 (49)

From the fact that

intΩ

1199060ℎ1199061

ℎ119889119909 le 12 100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (50)

we have

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

le 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (51)

Combining the boundary condition and Lemma 14 we get

100381710038171003817100381710038171199061ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (52)

Now we assume that the following inequality holds

1003817100381710038171003817119906119898ℎ

1003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 119898 = 1 2 119875 (53)

We need to prove 119906119875+1ℎ le 1199060

ℎ Let 119899 = 119875 + 1 and take thetest functions V = 119906119875+1

ℎ 119908 = minus1205720119901119875+1ℎ 119911 = 1205720119902119875+1ℎ in scheme

(35) we obtain

10038171003817100381710038171003817119906119875+1ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)2119901119875+1ℎ 119901119875+1

ℎ )+ 1205720radic120598120583ℎ ((119906119875+1

ℎ )minus119873+(12)

)2


= 119875sum119894=1

(119887119894minus1 minus 119887119894) intΩ

119906119875+1minus119894ℎ 119906119875+1

ℎ 119889119909+ 119887119875 int

Ω1199060ℎ119906119875+1

ℎ 119889119909 le 119875sum119894=1

(119887119894minus1 minus 119887119894) 10038171003817100381710038171003817119906119875+1minus119894ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 119875sum119894=1

(119887119894minus1 minus 119887119894) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817= ( 119875sum

119894=1

(119887119894minus1 minus 119887119894) + 119887119875) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172+ 12 10038171003817100381710038171003817119906119875+1

ℎ

100381710038171003817100381710038172 (54)

Calling Lemma 14 again and considering the boundary con-dition we have 10038171003817100381710038171003817119906119875+1

ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (55)

Theorem 20 For a sufficiently small step size 0 lt 120591 lt 119879 thefully discrete LDG scheme (42) is unconditional stable and thenumerical solution 119906119899

ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 119862 (1003817100381710038171003817120601 (119909)1003817100381710038171003817 + 120591 1003817100381710038171003817120595 (119909)1003817100381710038171003817) 119899 = 0 1 119873 (56)

Proof Similar to the subdiffusion case using the inequality119906minus1 le 119862(120601(119909)+120591120595(119909)) and adopting the samemethodsand techniques asTheorem 19 we can obtain the conclusionsof Theorem 20 For brevity we ignore the details in proofprocess here

5 Error Estimates

In this section we discuss the error estimates for the fullydiscrete LDG schemes (35) (42) respectively

Theorem 21 Let 119906(119909 119905119899) be the exact solution of the problem(1) which is sufficiently smooth with bounded derivatives and119906119899ℎ be the numerical solution of the fully discrete LDG scheme

(35) then the following error estimates holdWhen 0 lt 120572 lt 11003817100381710038171003817119906 (119909 119905119899) minus 119906119899

ℎ1003817100381710038171003817

le 1198621199061198791205721 minus 120572 (120591minus120572ℎ119896+1 + (120591)2minus120572 + (120591)minus1205722 ℎ119896+1 + ℎ119896+1) (57)

When 120572 rarr 11003817100381710038171003817119906 (119909 119905119899) minus 119906119899ℎ1003817100381710038171003817le 119862119906119879 (120591minus1ℎ119896+1 + 120591 + (120591)minus12 ℎ119896+1 + ℎ119896+1) (58)

where 119862119906 is a constant only depending on 119906

Proof Denote119890119899119906 = 119906 (119909 119905119899) minus 119906119899ℎ= P

minus119890119899119906 minus (Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899)) 119890119899119901 = 119901 (119909 119905119899) minus 119901119899ℎ = Q119890119899119901 minus (Q119901 (119909 119905119899) minus 119901 (119909 119905119899)) 119890119899119902 = 119902 (119909 119905119899) minus 119902119899ℎ= P

+119890119899119902 minus (P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) (59)

Subtracting equation (35) from equation (34) with the fluxeson boundaries we obtain the error equation

intΩ

119890119899119906V 119889119909 + 1205720radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1

((119890119899119902)+ Vminus119895+(12) minus (119890119899119902)+ V+119895minus(12)))minus 119899minus1sum

119894=1

(119887119894minus1 minus 119887119894) intΩ

119890119899minus119894119906 V 119889119909 minus 119887119899minus1 intΩ

1198900119906V 119889119909+ 1205720 int

Ω119903119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1

((119890119899119906)minus 119911minus119895+(12) minus (119890119899119906)minus 119911+

119895minus(12))) = 0

(60)

Using equations (59) and (43) after a simple rearrange-ment of terms the error equation (60) can be rewritten as

intΩP

minus119890119899119906V 119889119909 + 1205720radic120598 intΩP

+119890119899119902V119909119889119909+ radic120598 int

ΩP

minus119890119899119906119911119909119889119909 minus intΩ

(120573minus2)2Q119890119899119901119908 119889119909+ int

ΩP

+119890119899119902119908 119889119909 + intΩQ119890119899119901119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(P+119890119899119902)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1

(Pminus119890119899119906)minus119895+(12) [119911]119895+(12)+ 1205720radic120598 (P+119890119899119902)+

12V+12 minus 1205720radic120598 (P+119890119899119902)minus

119873+(12)

sdot Vminus119873+(12) + radic120598120583ℎ (Pminus119890119899119906)minus119873+(12) Vminus119873+(12)


= 119899minus1sum119894=1

(119887119894minus1 minus 119887119894) intΩP

minus119890119899minus119894119906 V 119889119909 + 119887119899minus1 intΩP

minus1198900119906V 119889119909minus 1205720 int

Ω119903119899 (119909) V 119889119909

+ intΩ

(Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899)) V 119889119909+ 1205720radic120598 int

Ω(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) V119909119889119909

+ radic120598 intΩ

(Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899)) 119911119909119889119909minus int

Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899)) 119908 119889119909

+ intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) 119908 119889119909+ int

Ω(Q119901 (119909 119905119899) minus 119901 (119909 119905119899)) 119911 119889119909 minus 119899minus1sum

119894=1


Ω(Pminus119906 (119909 119905119899minus119894) minus 119906 (119909 119905119899minus119894)) V 119889119909

minus 119887119899minus1 intΩ

(Pminus119906 (119909 1199050) minus 119906 (119909 1199050)) V 119889119909+ 1205720radic120598119873minus1sum

119895=1

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899))+119895+(12) [V]119895+(12)+ radic120598119873minus1sum

119895=1

(Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899))minus119895+(12) [119911]119895+(12)+ 1205720radic120598 (P+119902 (119909 119905119899) minus 119902 (119909 119905119899))+12 V+12minus 1205720radic120598 (P+119902 (119909 119905119899) minus 119902 (119909 119905119899))minus119873+(12) V

minus119873+(12)

+ radic120598120583ℎ (Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899))minus119873+(12) Vminus119873+(12)

(61)

Taking the test functions V = Pminus119890119899119906 119908 = minus1205720Q119890119899119901 119911 =1205720P+119890119899119902 in (61) using the properties (27)-(28) then the

following equality holds1003817100381710038171003817Pminus11989011989911990610038171003817100381710038172 + 1205720 ((120573minus2)2Q119890119899119901Q119890119899119901)+ 1205720radic120598120583ℎ (Pminus119890119899119906)2119873+(12) = 119899minus1sum

119894=1


ΩP

minus119890119899minus119894119906 Pminus119890119899119906119889119909 + 119887119899minus1 int

ΩP

minus1198900119906Pminus119890119899119906119889119909minus 1205720 int

Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ

(Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899))Pminus119890119899119906119889119909

+ 1205720radic120598 intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) (Pminus119890119899119906)119909 119889119909+ 1205720radic120598 int

Ω(Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899))

sdot (P+119890119899119902)119909

119889119909+ 1205720 int

Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899))Q119890119899119901119889119909

minus 1205720 intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899))Q119890119899119901119889119909+ 1205720 int

Ω(Q119901 (119909 119905119899) minus 119901 (119909 119905119899))P+119890119899119902119889119909

minus 119899minus1sum119894=1

(119887119894minus1 minus 119887119894) intΩ

(Pminus119906 (119909 119905119899minus119894) minus 119906 (119909 119905119899minus119894))sdot Pminus119890119899119906119889119909 minus 119887119899minus1 int

Ω(Pminus119906 (119909 1199050) minus 119906 (119909 1199050))

sdot Pminus119890119899119906119889119909+ 1205720radic120598119873minus1sum

119895=1

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899))+119895+(12)sdot [Pminus119890119899119906]119895+(12)+ 1205720radic120598119873minus1sum

119895=1

(Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899))minus119895+(12)sdot [P+119890119899119902]

119895+(12)+ 1205720radic120598 (P+119902 (119909 119905119899) minus 119902 (119909 119905119899))+12sdot (Pminus119890119899119906)+12 minus 1205720radic120598 (P+119902 (119909 119905119899)

minus 119902 (119909 119905119899))minus119873+(12) (Pminus119890119899119906)minus119873+(12)

+ radic120598120583ℎ (Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899))minus119873+(12)

sdot ((Pminus119890119899119906)minus)119873+(12)

(62)

That is1003817100381710038171003817Pminus11989011989911990610038171003817100381710038172 + 1205720 ((120573minus2)2Q119890119899119901Q119890119899119901)+ 1205720radic120598120583ℎ (Pminus119890119899119906)2119873+(12) = 119899minus1sum

i=1(119887119894minus1 minus 119887119894)

sdot intΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ



+ 1205720 ((120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899))minus (P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) Q119890119899119901)minus 1205720radic120598 (P+119902 (119909 119905119899) minus 119902 (119909 119905119899)minus)

119873+(12)

sdot (Pminus119890119899119906)minus119873+(12) minus 119899minus1sum119894=1


Ω(Pminus119906 (119909 119905119899minus119894) minus 119906 (119909 119905119899minus119894))Pminus119890119899119906119889119909


(Pminus119906 (119909 1199050) minus 119906 (119909 1199050))Pminus119890119899119906119889119909(63)

We proveTheorem 21 by mathematical inductionFirst we consider the case when 119899 = 1 (63) becomes10038171003817100381710038171003817Pminus1198901119906100381710038171003817100381710038172 + 1205720 ((120573minus2)2Q1198901119901Q1198901119901)

+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

= intΩP


Ω1199031 (119909)Pminus1198901119906119889119909

+ intΩ

(Pminus119906 (119909 1199051) minus 119906 (119909 1199051))Pminus1198901119906119889119909+ 1205720 ((120573minus2)2 (Q119901 (119909 1199051) minus 119901 (119909 1199051))minus (P+119902 (119909 1199051) minus 119902 (119909 1199051)) Q1198901119901)minus 1205720radic120598 (P+119902 (119909 1199051) minus 119902 (119909 1199051)minus)

119873+(12)sdot (Pminus1198901119906)minus119873+(12)

minus intΩ


(64)

Notice the fact that Pminus1198900119906 le 119862ℎ119896+1 119886119887 le 1205981198862 + (14120598)1198872Together with property (29) and Lemma 18 we can get10038171003817100381710038171003817(120573minus2)2 (Q119901 (119909 1199051) minus 119901 (119909 1199051))

minus (P+119902 (119909 1199051) minus 119902 (119909 1199051))10038171003817100381710038171003817= 10038171003817100381710038171003817(120573minus2)2 (Q119901 (119909 1199051) minus 119901 (119909 1199051))minus (120573minus2)2 (P+119901 (119909 1199051) minus 119901 (119909 1199051))10038171003817100381710038171003817 le 119862ℎ119896+1

(65)

Combining this with Definition 4 and (59) using Youngrsquosinequality we obtain10038171003817100381710038171003817Pminus1198901119906100381710038171003817100381710038172 + 1205720 ((120573minus2)2Q1198901119901Q1198901119901)

+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

le (10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

100381710038171003817100381710038171199031 (119909)10038171003817100381710038171003817

+ 1003817100381710038171003817Pminus119906 (119909 1199051) minus 119906 (119909 1199051)1003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 1199050) minus 119906 (119909 1199050)1003817100381710038171003817) 10038171003817100381710038171003817Pminus119890111990610038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q1198901119901100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus1198901119906)2

119873+(12)

(66)

Using Youngrsquos inequality again in (66) we get

10038171003817100381710038171003817Pminus1198901119906100381710038171003817100381710038172 + 1205720 ((120573minus2)2Q1198901119901Q1198901119901)le 119862 (ℎ119896+1 + 1205912)2 + 120598 10038171003817100381710038171003817Pminus1198901119906100381710038171003817100381710038172 + 119862ℎ2119896+2120591120572

+ 1205720120598119862 10038171003817100381710038171003817Q1198901119901100381710038171003817100381710038172 (67)

If we choose 120598 le min1 1119862 and recall the fractionalPoincare-Friedrichs Lemma 16 we have10038171003817100381710038171003817Pminus119890111990610038171003817100381710038171003817 le 119887minus10 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) (68)

where 119862 is a constant depending on 119906 119879 120572Next we suppose that the following inequality holds

1003817100381710038171003817Pminus119890119898119906 1003817100381710038171003817 le 119887minus1119898minus1119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) 119898 = 1 2 119897 (69)

When 119899 = 119897 + 1 from (63) and Youngrsquos inequality we canobtain10038171003817100381710038171003817Pminus119890119897+1119906

100381710038171003817100381710038172 + 1205720 ((120573minus2)2Q119890119897+1119901 Q119890119897+1119901 )+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)= 119897sum

119894=1


ΩP

minus119890119897+1minus119894119906 Pminus119890119897+1119906 119889119909 + 119887119897 int

ΩP

minus1198900119906Pminus119890119897+1119906 119889119909minus 1205720 int

Ω119903119897+1 (119909)Pminus119890119897+1119906 119889119909

+ intΩ

(Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1))Pminus119890119897+1119906 119889119909minus 1205720radic120598 (P+119902 (119909 119905119897+1) minus 119902 (119909 119905119897+1)minus)

119873+(12)sdot (Pminus119890119897+1119906 )minus119873+(12)+ 1205720 ((120573minus2)2 (Q119901 (119909 119905119897+1) minus 119901 (119909 119905119897+1))

minus (P+119902 (119909 119905119897+1) minus 119902 (119909 119905119897+1)) Q119890119897+1119901 )minus 119897sum

119894=1

(119887119894minus1 minus 119887119894)


sdot intΩ

(Pminus119906 (119909 119905119897+1minus119894) minus 119906 (119909 119905119897+1minus119894))Pminus119890119897+1119906 119889119909minus 119887119897 int

Ω(Pminus119906 (119909 1199050) minus 119906 (119909 1199050))Pminus119890119897+1119906 119889119909

le ( 119897sum119894=1

(119887119894minus1 minus 119887119894) 10038171003817100381710038171003817Pminus119890119897+1minus119894119906

10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817+ 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817 + 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ 119897sum

119894=1

(119887119894minus1 minus 119887119894) 1003817100381710038171003817Pminus119906 (119909 119905119897+1minus119894) minus 119906 (119909 119905119897+1minus119894)1003817100381710038171003817+ 119887119897 1003817100381710038171003817Pminus119906 (119909 1199050) minus 119906 (119909 1199050)1003817100381710038171003817) times 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1

(119887119894minus1 minus 119887119894) 119887minus1119897minus119894119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)+ 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ ( 119897sum

119894=1

(119887119894minus1 minus 119887119894) + 119887119897) 119862ℎ119896+1) times 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1

(119887119894minus1 minus 119887119894) 119887minus1119897minus119894119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)+ 119862 (ℎ119896+1 + 1205912)) 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(70)

Since 119887minus1119894minus1 lt 119887minus1119894 1205911205722ℎ119896+1 gt 0 (71)

then 10038171003817100381710038171003817Pminus119890119897+1119906


119873+(12)

le ( 119897sum119894=1

(119887119894minus1 minus 119887119894) 119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)+ 119887119897119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)) times 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le (( 119897sum119894=1

(119887119894minus1 minus 119887119894) + 119887119897)sdot 119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)) 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

(72)

Using Definition (7) into above inequalities and combiningYoungrsquos inequalities we have10038171003817100381710038171003817Pminus119890119897+1119906

100381710038171003817100381710038172 + 1205720 ((120573minus2)2Q119890119897+1119901 Q119890119897+1119901 )le (119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1))2 + 120598 10038171003817100381710038171003817Pminus119890119897+1119906

100381710038171003817100381710038172+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 (73)

Analogously if we choose 120598 small enough and recalling thefractional Poincare-Friedrichs Lemma 16 again we obtain10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 le 119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) (74)

where 119862 is a constant related to 119906 119879 120572According to the principle of mathematical induction we

get 1003817100381710038171003817Pminus1198901198991199061003817100381710038171003817 le 119887minus1119899minus1119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) (75)

By [21 27] we know that 119899minus120572119887minus1119899minus1 is monotonly increasingand tends to 1(1 minus 120572) 119899 rarr infin thus1003817100381710038171003817Pminus1198901198991199061003817100381710038171003817 le 119887minus1119899minus1119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)

le 119899120572119899minus120572119887minus1119899minus1119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le ( 119879120591 )120572 11 minus 120572 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le 1198621199061198791205721 minus 120572 (120591minus120572ℎ119896+1 + 1205912minus120572 + 120591minus1205722ℎ119896+1)

(76)



When 120572 rarr 1 1(1 minus 120572) rarr infin it is meaningless for (76)so as for the case of 120572 rarr 1 we should give out the estimateagain

We prove the following estimate by using the mathemat-ical induction1003817100381710038171003817Pminus1198901198991199061003817100381710038171003817 le 119899119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) (77)

Obviously when 119899 = 1 (77) is workable clearlyWhen 119899 = 119897 + 1 by (63) we obtain10038171003817100381710038171003817Pminus119890119897+1119906


119873+(12)le ( 119897sum

119894=1

(119887119894minus1 minus 119887119894)sdot 10038171003817100381710038171003817Pminus119890119897+1minus119894119906

10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817 + 119897sum

119894=1

(119887119894minus1 minus 119887119894)sdot 1003817100381710038171003817Pminus119906 (119909 119905119897+1minus119894) minus 119906 (119909 119905119897+1minus119894)1003817100381710038171003817 + 119887119897 1003817100381710038171003817Pminus119906 (119909 1199050)minus 119906 (119909 1199050)1003817100381710038171003817) times 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

le ( 119897sum119894=1

(119887119894minus1 minus 119887119894) (119897 + 1 minus 119894)sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) + 119862 (ℎ119896+1 + 1205912))sdot 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( sum119897119894=1 (119887119894minus1 minus 119887119894) (119897 + 1 minus 119894) + 1119897 + 1 (119897 + 1) 119862 (ℎ119896+1

+ 1205912 + 1205911205722ℎ119896+1)) 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(78)

Using Youngrsquos inequality into the last inequality above andchoosing 120598 le min1 1119862 then combining the property of 119887119894and Lemma 16 (Poincare-Friedrichs inequality) we obtain10038171003817100381710038171003817Pminus119890119897+1119906

100381710038171003817100381710038172 le ( sum119897119894=1 (119887119894minus1 minus 119887119894) (119897 + 1 minus 119894) + 1119897 + 1 (119897 + 1)

sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1))2 + 119862ℎ2119896+2120591120572

10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 le sum119897119894=1 (119887119894minus1 minus 119887119894) (119897 + 1 minus 119894) + 1119897 + 1 (119897 + 1)

sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le sum119897

119894=1 (119887119894minus1 minus 119887119894) 119897 + 1119897 + 1 (119897 + 1) 119862 (ℎ119896+1 + 1205912+ 1205911205722ℎ119896+1) le (119897 + 1) 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)

(79)

By mathematical induction we get inequality (77)Since 119899120591 le 119879 119899 = 0 1 119872 that is 119899 le 119879120591 hence

when 120572 rarr 1 inequality (77) becomes1003817100381710038171003817Pminus1198901198991199061003817100381710038171003817 le 119899119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le 119879120591 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 120591 + 120591minus12ℎ119896+1)

(80)

where 119862119906 is a constant only depending on 119906Combined with inequalities (76) and (80) and according

to the triangle inequality and the standard approximationproperty (29) coupling with the error associated with theprojection error we can prove the results of Theorem 21

Theorem 22 Let 119906(119909 119905119899) be the exact solution of problem (2)which is sufficiently smooth with bounded derivatives and 119906119899

ℎbe the numerical solution of the fully discrete LDG scheme (42)then the following error estimates hold

When 1 lt 120572 lt 21003817100381710038171003817119906 (119909 119905119899) minus 119906119899ℎ1003817100381710038171003817

le 119862119906119879120572minus12 minus 120572 (1205911minus120572ℎ119896+1 + 1205912 + (120591)1minus1205722 ℎ119896+1 + ℎ119896+1) (81)

when 120572 rarr 21003817100381710038171003817119906 (119909 119905119899) minus 119906119899ℎ1003817100381710038171003817 le 119862119906119879 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1) (82)

where 119862119906 is a constant only depending on 119906Proof Subtracting equation (42) from equation (41) andcombining the fluxes on boundaries we obtain the errorequation as follows

intΩ

119890119899119906V 119889119909 + 1205721radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1


119890119899minus119894minus1119906 V 119889119909 minus (2]0 minus ]1)


sdot intΩ

119890119899minus1119906 V 119889119909 minus (2]119899minus1 minus ]119899minus2) intΩ

1198900119906V 119889119909+ ]119899minus1 int

Ω119890minus1119906 V 119889119909 + 1205721 int

Ω119877119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0(83)

Similar to the error estimate of subdiffusion equation firstwe can prove the following error inequality10038171003817100381710038171003817119890minus1119906 10038171003817100381710038171003817 le 119862 (ℎ119896+1 + 1205912 + 120591ℎ119896+1) (84)

where 119862 is a constant related to 119906 119879 120572In fact

119890minus1119906 = 119906 (119909 119905minus1) minus 119906minus1ℎ = 119906 (119909 119905minus1) minus 119906minus1 + 119906minus1 minus 119906minus1

ℎ 119906 (119909 119905minus1) minus 119906minus1 = 119906 (119909 119905minus1) minus 119906 (119909 1199050) + 120591119906119905 (119909 1199050)

= 12059122 119906119905119905 (119909 1199050) + O (1205913) 10038171003817100381710038171003817119906minus1 minus 119906minus1ℎ

10038171003817100381710038171003817= 1003817100381710038171003817119906 (119909 1199050) minus 120591119906119905 (119909 1199050) minus (119906 (119909 1199050) minus 120591119906119905 (119909 1199050))ℎ1003817100381710038171003817= 10038171003817100381710038171003817119906 (119909 1199050) minus 1199060ℎ minus 120591 (119906119905 (119909 1199050) minus 1199060

119905ℎ)10038171003817100381710038171003817le 119862 (ℎ119896+1 + 120591ℎ119896+1) (85)

Substituting the final results of the last two equalities into thefirst equality we get the error inequality (84)

Next we adopt the same methods and techniques asTheorem 21 to obtain1003817100381710038171003817Pminus1198901198991199061003817100381710038171003817 le ]minus1119899minus1119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1) (86)

Now we consider the function

119891 (119909) = 1199091minus1205721199092minus120572 minus (119909 minus 1)2minus120572 119909 ge 1 (87)

Differentiating the function 119891(119909) with respect to 119909 wehave

1198911015840 (119909) = (1 minus 120572) 119909minus120572 [1199092minus120572 minus (119909 minus 1)2minus120572] minus (2 minus 120572) 1199091minus120572 [1199091minus120572 minus (119909 minus 1)1minus120572][1199092minus120572 minus (119909 minus 1)2minus120572]2 (88)

When 119909 gt 1 the denominator of equality (88) is strictlygreater than zero so we only need to consider the conditionof the molecular because(1 minus 120572) 119909minus120572 [1199092minus120572 minus (119909 minus 1)2minus120572] minus (2 minus 120572) 1199091minus120572 [1199091minus120572

minus (119909 minus 1)1minus120572] = (1 minus 120572) 1199092minus2120572 minus (1 minus 120572) 119909minus120572 (119909minus 1)2minus120572 minus (2 minus 120572) 1199092minus2120572 + (2 minus 120572) 1199091minus120572 (119909 minus 1)1minus120572= minus1199092minus2120572 + 119909minus120572 (119909 minus 1)2minus120572 minus (2 minus 120572) 119909minus120572 (119909 minus 1)2minus120572+ (2 minus 120572) 1199091minus120572 (119909 minus 1)1minus120572 = minus1199092minus2120572 + 119909minus120572 (119909minus 1)2minus120572 + (2 minus 120572) 119909minus120572 (119909 minus 1)1minus120572 [119909 minus (119909 minus 1)]= minus1199092minus2120572 + 119909minus120572 (119909 minus 1)2minus120572 + (2 minus 120572) 119909minus120572 (119909 minus 1)1minus120572= 119909minus120572 [(119909 minus 1)2minus120572 minus 1199092minus120572] + (2 minus 120572) 119909minus120572 (119909 minus 1)1minus120572= 119909minus120572 [(119909 minus 1)2minus120572 minus 1199092minus120572 + (119909 minus 1)1minus120572] + (1 minus 120572)sdot 119909minus120572 (119909 minus 1)1minus120572 = 1199091minus120572 [(119909 minus 1)1minus120572 minus 1199091minus120572] + (1minus 120572) 119909minus120572 (119909 minus 1)1minus120572 = 119909119909120572

[ 119909 minus 1(119909 minus 1)120572 minus 119909119909120572] + (1

minus 120572) 1119909120572

119909 minus 1(119909 minus 1)120572 = 11199092120572 (119909 minus 1)120572 [119909 (119909 minus 1) 119909120572

minus 1199092 (119909 minus 1)120572 + (1 minus 120572) (119909 minus 1) 119909120572] (89)

When 1 lt 120572 lt 2 and 119909 gt 1 always 1199092120572(119909 minus 1)120572 gt 0 so weonly need to estimate the positive or negative property of themolecular of the last equality above

119909 (119909 minus 1) 119909120572 minus 1199092 (119909 minus 1)120572 + (1 minus 120572) (119909 minus 1) 119909120572

= (119909 minus 1) 119909120572 (119909 + 1 minus 120572) minus 1199092 (119909 minus 1)120572= (119909 minus 1) 119909120572 [(119909 + 1 minus 120572) minus 1199092minus120572 (119909 minus 1)120572minus1] (90)

For 119909 gt 1 (119909 minus 1)119909120572 gt 0 always holdsFor any given 119909 we define the function as follows

119892 (120572) = (119909 + 1 minus 120572) minus 1199092minus120572 (119909 minus 1)120572minus1 (91)

Differentiating the function 119892(120572) with respect to 120572 we obtain1198921015840 (120572) = minus1 minus 1199092minus120572 (119909 minus 1)120572minus1 (ln119909 minus ln(119909minus1)) (92)


Let 1198921015840(120572) = 0 there is only one zero root of the derivedfunction 1198921015840(120572) on interval (1 2)120572 = log((1minus119909)119909

2ln119909(119909minus1))(119909minus1)119909 (93)

Since 119892(1) = 119892(2) = 0 for any given 119909 the function 119892(120572)with respect to 120572 can only increase first and then decrease ordecrease first and then increase on interval (1 2)

Now if we can prove that the value of any point oninterval (1 2) is positive then the function 119892(120572) must beincreasing first and then decreasing For example we choosethe parameter 120572 = 32 and estimate the positive or negativeproperty of the following equality 119892(32) = (119909 minus (12)) minus11990912(119909 minus 1)12 Since (119909 minus (12))11990912(119909 minus 1)12 = [(119909 minus(12))2119909(119909 minus 1)]12 gt 1 we get 119892(32) gt 0 hence thefunction 119892(120572) with respect to 120572 can only increase first andthen decrease on interval (1 2)That is for any given 119909 when1 lt 120572 lt 2 we have 119892(120572) gt 0 When 119909 = 1 (119909 minus 1)2119909120572 minus 1199092(119909 minus1)120572 + (1 minus 120572)(119909 minus 1)119909120572 = 0

From the above discussion we obtain 1198911015840(119909) ge 0 119909 isin[1 +infin) that is the function 119891(119909) is a monotone increasingfunction on interval [1 +infin)

Considering the function limit lim119909rarrinfin119891(119909) byHospitalrsquosrule we have

lim119909rarrinfin

119891 (119909) = lim119909rarrinfin

1199091minus1205721199092minus120572 minus (119909 minus 1)2minus120572= lim119909rarrinfin

(1 minus 120572) 119909minus120572(2 minus 120572) [1199091minus120572 minus (119909 minus 1)1minus120572]= lim

119909rarrinfin

1 minus 1205722 minus 120572 119909minus11 minus (119909 (119909 minus 1))120572minus1= lim

119909rarrinfin

12 minus 120572 119909minus2(119909 minus 1)minus2 (119909 (119909 minus 1))120572minus2= lim119909rarrinfin

12 minus 120572 ( 119909119909 minus 1 )minus2 ( 119909119909 minus 1 )2minus120572

= lim119909rarrinfin

12 minus 120572 ( 119909119909 minus 1 )minus120572


12 minus 120572 (1 minus 1119909 )120572 = 12 minus 120572

(94)

Hence 119891 (119909) le 12 minus 120572 119909 ge 1 (95)

Therefore the sequence 1198991minus120572]minus1119899minus1 monotone increasing tendsto 1(2 minus 120572) 119899 rarr infin We obtain1003817100381710038171003817Pminus1198901198991199061003817100381710038171003817 le ]minus1119899minus1119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)

le 119899120572minus11198991minus120572]minus1119899minus1119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le ( 119879120591 )120572minus1 12 minus 120572 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119862119906119879120572minus12 minus 120572 (1205911minus120572ℎ119896+1 + 1205912 + (120591)1minus1205722 ℎ119896+1)

(96)


When 120572 rarr 2 1(2 minus 120572) rarr infin estimate (96) is mean-ingless so for the case of 120572 rarr 2 we need to estimate again

Analogous to the subdiffusion case we can prove thefollowing estimate still using the mathematical induction1003817100381710038171003817Pminus1198901198991199061003817100381710038171003817 le 119899119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1) (97)

Because 119899120591 le 119879 119899 = 0 1 119872 that is 119899 le 119879120591 as a resultwhen 120572 rarr 2 the inequality (97) becomes1003817100381710038171003817Pminus1198901198991199061003817100381710038171003817 le 119899119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)

le 119879120591 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1)

(98)

where 119862119906 is a constant only depending on 119906Combining inequality (96) with inequality (98) accord-

ing to the triangle inequality and standard approximationTheorem (29) together with the property of projectionoperator we obtain the results of Theorem 22

6 Numerical Examples

In this section we present some numerical examples todemonstrate the effectiveness of the finite differenceLDGapproximation for solving the time-space fractional subdif-fusionsuperdiffusion equations We check the convergencebehavior of numerical solutions with respect to the time stepsize 120591 and the space step size ℎ By choosing the appropriatetime step size we avoid the contamination of the temporalerror compared with the spatial error

61 Implementation As examples we consider the time-space fractional subdiffusionsuperdiffusion equations asfollows120597120572119906 (119909 119905)120597119905120572 minus 120598 (minus (minus)1205732) 119906 (119909 119905) = 119891 (119909 119905)

in 119909 isin Ω 119905 isin (0 119879] 119906 (119909 0) = 1199060 (119909) in 119909 isin Ω

(99)

where Ω = [119886 119887] 119879 gt 0 0 lt 120572 lt 1 for the subdiffusion case(1) or 1 lt 120572 lt 2 for the superdiffusion case (2) and 1 lt 120573 lt 2120598 gt 0

For brevity we only derive the linear system obtainedfrom the fully discrete LDG scheme (35) For scheme (42)we can obtain the corresponding linear system similarly Weconsider the case of discontinuous piecewise polynomialsbasis function sequences 120601119894(119909)119896119894=1 in space 119881ℎ

119896 By thedefinition of the space 119881ℎ

119896 we know that for all V isin 119881ℎ119896

V(119909) = sum119873119894=1 V119894120601119894(119909) holds where V119894 = V(119909119894)


By the LDG scheme (35) we obtain the fully discretescheme of (99) as follows

intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)

where 1205720 = 120591120572Γ(2 minus 120572)In terms of the basis 120601119894(119909)119873119894=1 we denote approxi-

mate solution functions by 119906119899ℎ = sum119873

119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873

119894=1 119901119895(119905)120601119895(119909) and 119902119899ℎ = sum119873119894=1 119902119895(119905)120601119895(119909) and let test func-

tions V = sum119873119894=1 V119894120601119894(119909) V119894 = V(119909119894) 119908 = sum119873

119894=1 119908119894120601119894(119909) 119908119894 =119908(119909119894) and 119911 = sum119873119894=1 119911119894120601119894(119909) 119911119894 = 119911(119909119894) Putting these expres-

sions into the LDG scheme (100) and taking the flux in (36)we get the linear system of the scheme in 119895th element at 119905119899 asfollows

( 119872 0 1205720radic120598 (119860 minus 11986711 + 11986712)0 119877 119872radic120598 (119860 minus 11986713 + 11986714) 119872 0 ) (119880119899119895119875119899119895119876119899119895

)= 119865119899V + 119899minus1sum

119894=1

(119887119894minus1 minus 119887119894) 119880119899minus119894V + 119887119899minus11198800V(101)

where 119872 = (120601119895119898(119909) 120601119895

119896(119909))119895=1119873 is the mass matrix 119898 119896 =0 1 denote the polynomials order and

119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)

119877 is the quadrature of the fractional integral operator(120573minus2)2119901119899ℎ and test function 119908 in 119895th element From [29] we

know that

(120573minus2)2119901119899ℎ = minus 1198861198682minus120573119909 119901119899

ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where

1198861198682minus120573119909 119901119899ℎ (119909 119905) = 1Γ (2 minus 120573) int119909

119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)

and 119865119899 = (1198911 1198912 119891119899)119879is a vector valued function 119891119895(119905119899) =119891(119909119895 119905119899) Then by solving the linear system (101) we canobtain the solution 119880119899

62 Numerical Results

Example 1 Consider the fractional time-space subdiffusionequation (99) in domain [0 1] times [0 05] which satisfies thefollowing initial boundary conditions119906 (119909 0) = 1199092 (1 minus 119909)2 119909 isin [0 1] (105)

In order to obtain the exact solution 119906(119909 119905) = (2119905 minus 1)21199092(1 minus119909)2 we choose the source term as

119891 (119909 119905) = 1199092 (1 minus 119909)2 ( 81199052minus120572Γ (3 minus 120572) minus 41199051minus120572Γ (2 minus 120572)+ 119905minus120572Γ (1 minus 120572) ) + (2119905 minus 1)22 cos (120587 (120573 + 1) 2) [ 241199093minus120573Γ (4 minus 120573)minus 121199092minus120573Γ (3 minus 120573) + 21199091minus120573Γ (2 minus 120573) + 24 (1 minus 119909)3minus120573Γ (4 minus 120573)minus 12 (1 minus 119909)2minus120573Γ (3 minus 120573) + 2 (1 minus 119909)1minus120573Γ (2 minus 120573) ]

(106)

Choosing the fluxes in (36) and adopting the linear piecewisepolynomials (99) is solved numerically We investigate thenumerical solutions with respect to different fractional orderparameters 120572 and 120573 we select the appropriate time step size120591 such that the time discretization errors are negligible ascompared with the spatial errorsThen choose space step sizesequence as ℎ = 12119894 (119894 = 2 6) Tables 1 and 2 listout the 1198712 errors and the corresponding convergence orderrespectively fixing the value of the one parameter 120573 (or 120572)with the values change of the other one parameter120572 (or120573) Ascan be seen from the data in the two tables our schemes canachieve the optimal convergenceO(ℎ119896+1) in spatial directionthis matches the theoretical results of Theorem 21

The numerical example results listed in Tables 3 and 4show that when we choose the appropriate space step sizeℎ and the time step size sequence 120591 = 12119894 (119894 = 2 6)fixing 120573 (or 120572) and changing 120572 (or 120573) the temporal accuracy


Table 1 Example 1 The 1198712 errors and convergence rate in space direction 120573 = 16 119879 = 05ℎ 120572 = 01 Rate 120572 = 05 rate 120572 = 09 Rate14 33592 times 10minus2 mdash 27826 times 10minus2 mdash 14821 times 10minus2 mdash18 91511 times 10minus3 18761 69632 times 10minus3 19986 37341 times 10minus3 19888116 23499 times 10minus3 19613 17411 times 10minus3 19997 93353 times 10minus4 20000132 59437 times 10minus4 19832 43333 times 10minus4 20065 23241 times 10minus4 20060164 14875 times 10minus4 19984 10748 times 10minus4 20113 57422 times 10minus5 20170


Table 3 Example 1 The 1198712 errors and convergence rate in time direction 120573 = 16 119879 = 05120591 120572 = 02 Rate 120572 = 06 Rate 120572 = 0999 Rate14 48064 times 10minus3 mdash 55078 times 10minus3 mdash 54127 times 10minus3 mdash18 14486 times 10minus3 17302 21182 times 10minus3 13786 28977 times 10minus3 09014116 42962 times 10minus4 17536 80864 times 10minus4 13893 15246 times 10minus3 09265132 12320 times 10minus4 18020 30576 times 10minus4 14031 76910 times 10minus4 09872164 35291 times 10minus5 18037 11501 times 10minus4 14106 38372 times 10minus4 10031


Table 5 Example 2 The 1198712 errors and convergence rate in space direction 120573 = 16 119879 = 1ℎ 120572 = 02 Rate 120572 = 06 Rate 120572 = 099 Rate14 31592 times 10minus2 mdash 19328 times 10minus2 mdash 21532 times 10minus2 mdash18 95228 times 10minus3 17301 48649 times 10minus3 19902 54460 times 10minus3 19832116 24728 times 10minus3 19452 12174 times 10minus3 19985 13693 times 10minus3 19917132 12521 times 10minus3 19818 30317 times 10minus4 20057 33469 times 10minus4 20326164 31377 times 10minus4 19966 74936 times 10minus5 20164 83644 times 10minus5 20005

of the time-space fractional subdiffusion equation under the1198712 norm can converge to O(1205912minus120572) for 0 lt 120572 lt 1 and closeto 1 when 120572 rarr 1 the results agree with the theoreticalcomputation results in Theorem 21

Example 2 Consider the time-space fractional subdiffusion(0 lt 120572 lt 1) equation (99) for 119905 isin [0 1] 119909 isin [0 1] we choosethe exact solution as 119906(119909 119905) = sin(2120587119905)1199092(1minus119909)2 and 120598 = 005

Similar to Example 1 we give out the 1198712 errors andconvergence rate with fixing 120572 (or 120573) and changing 120573 or120572 We also select the appropriate time step size 120591 which

guarantee that the time discretization errors are negligible ascomparedwith the spatial errors Choosing the space step sizesequence as ℎ = 12119894 (119894 = 2 6) we obtain the optimalconvergence rate in space direction (as shown in Tables 5 and6 this corresponds to the theoretical results as illustrated inTheorem 21) Tables 7 and 8 listed out the temporal accuracyhere we choose the appropriate space step size ℎ and the timestep size sequence as 120591 = 12119894 (119894 = 2 6) The data inTable 7 display that for 0 lt 120572 lt 1 the convergence order intime direction can be up to O(1205912minus120572) and close to 1 order for120572 rarr 1 which are consistent withTheorem 21





Example 3 In this example we consider the case for 1 lt120572 lt 2 that is (99) is a time-space fractional superdiffusionmodel We assume that the equation satisfies the followinginitial boundary conditions in the domain [0 1] times [0 05]119906 (119909 0) = 1199092 (1 minus 119909)2 119909 isin [0 1] 119906119905 (119909 0) = minus41199092 (1 minus 119909)2 119909 isin [0 1] (107)

Here we choose 120598 = 1 and the source term is

119891 (119909 119905) = 1199092 (1 minus 119909)2 ( 81199052minus120572Γ (3 minus 120572) + 119905minus120572Γ (1 minus 120572) )+ (2119905 minus 1)22 cos (120587 (120573 + 1) 2) [ 241199093minus120573Γ (4 minus 120573) minus 121199092minus120573Γ (3 minus 120573)+ 21199091minus120573Γ (2 minus 120573) + 24 (1 minus 119909)3minus120573Γ (4 minus 120573) minus 12 (1 minus 119909)2minus120573Γ (3 minus 120573)+ 2 (1 minus 119909)1minus120573Γ (2 minus 120573) ]

(108)

Then the time-space fractional superdiffusion obtains thesame exact solution as Example 1 119906(119909 119905) = (2119905minus1)21199092(1minus119909)2

Analogous to the case of fractional subdiffusion equationwe consider the errors under the 1198712 norm and convergenceorders of the LDG numerical solutions for the originalequation with different values of the fractional derivative

parameters120572 and120573 First we choose the appropriate time stepsize 120591 such that the time discretization errors are negligibleas compared with the spatial errors Then choose the spacestep size sequence as ℎ = 12119894 (119894 = 2 6) Tables 9 and10 list out the errors and the convergence rate when we fix 120573(or 120572) and change 120572 (or 120573) As can be seen from the table weobtain the optimal spatial convergence rate O(ℎ119896+1) whichagrees with the theoretical results proved inTheorem 22Thenumerical results in Tables 11 and 12 show that when ℎ =1198621015840120591 (1198621015840 is a positive constant) we choose the time step sizesequence 120591 = 12119894 (119894 = 2 sdot sdot sdot 6) fix120573 (or120572) and then change120572 (or 120573) the temporal accuracy of the time-space fractionalsuperdiffusion can reach O(1205913minus120572) for 1 lt 120572 lt 2 and be closeto 1 when 120572 rarr 2 The numerical results are consistent withthe theoretical results proved inTheorem 22

7 Conclusions

In this paper we proposed the fully discrete local discontin-uous Galerkin scheme for solving the time-space fractionalsubdiffusionsuperdiffusion equations respectivelyThroughcarefully choosing the numerical flux on boundary terms andmaking a detailed theoretical analysis we proved that ournumerical schemes are unconditionally stable with conver-gence under the 1198712 norm From the numerical experimentresults we can observe that a convergence rateO(1205912minus120572 + ℎ119896+1)is achieved for 0 lt 120572 lt 1 and for 1 lt 120572 lt 2 when the spacestep size ℎ and the time step size 120591 satisfy the relationshipℎ = 1198621015840120591 (1198621015840 is a constant) the convergence rate of our scheme




Table 11 Example 3 The 1198712 errors and convergence rate in time direction ℎ = 1198621015840120591 1198621015840 = 05 and 120573 = 16120591 120572 = 105 Rate 120572 = 16 Rate 120572 = 1999 Rate14 56293 times 10minus2 mdash 49876 times 10minus2 mdash 43574 times 10minus2 mdash18 20868 times 10minus2 14316 22149 times 10minus2 12309 33742 times 10minus2 03689116 72314 times 10minus3 15290 87045 times 10minus3 12876 23331 times 10minus2 05323132 23828 times 10minus3 16016 33831 times 10minus3 13634 14645 times 10minus2 06718164 71811 times 10minus4 17304 12871 times 10minus3 13942 88796 times 10minus3 07219


can approach O(1205913minus120572 + ℎ119896+1) The numerical results showthat the fully discrete local discontinuous Galerkin (LDG)approximation is effective and powerful for solving time-space fractional subdiffusionsuperdiffusion equations

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Theproject is supported byNSF of China (11371289 11426068and 11501441) and the talent project of Huizhou University ofChina (2015JB018)

References

[1] R Gorenflo FMainardi DMoretti G Pagnini and P ParadisildquoDiscrete random walk models for space-time fractional diffu-sionrdquo Chemical Physics vol 284 no 1-2 pp 521ndash541 2002

[2] X Li Fractional calculus fractal geometry and stochastic pro-cesses [PhD thesis] University of Western Ontario LondonCanada 2003

[3] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000

[4] T L Szabo and J Wu ldquoAmodel for longitudinal and shear wavepropagation in viscoelastic mediardquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2437ndash2446 2000

[5] F Amblard A C Maggs B Yurke A N Pargellis and SLeibler ldquoSubdiffusion and anomalous local viscoelasticity inactin networksrdquo Physical Review Letters vol 77 no 21 pp4470ndash4473 1996

[6] I M Sokolov J Klafter and A Blumen ldquoBallistic versusdiffusive pair dispersion in the Richardson regimerdquo PhysicalReview E vol 61 no 3 pp 2717ndash2722 2000

[7] R Hilfer Ed Applications of Fractional Calculus in PhysicsWorld Scientific River Edge NJ USA 2000

[8] H Scher and E W Montroll ldquoAnomalous transit-time disper-sion in amorphous solidsrdquo Physical Review B vol 12 no 6 pp2455ndash2477 1975


[9] S B Yuste L Acedo and K Lindenberg ldquoReaction front in an119860 + 119861 rarr 119862 reaction subdiffusion processrdquo Physical Review Evol 69 pp 1ndash10 2004

[10] D A Benson S W Wheatcraft and M M Meerschaert ldquoThefractional-order governing equation of Levy motionrdquo WaterResources Research vol 36 no 6 pp 1413ndash1423 2000

[11] G M Zaslavsky D Stevens and H Weitzner ldquoSelf-similartransport in incomplete chaosrdquo Physical Review E vol 48 no3 pp 1683ndash1694 1993

[12] R Gorenflo Y Luchko and F Mainardi ldquoWright functionsas scale-invariant solutions of the diffusion-wave equationrdquoJournal of Computational and Applied Mathematics vol 118 no1-2 pp 175ndash191 2000

[13] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus amp Applied Analysis vol 4 no 2 pp 153ndash192 2001

[14] S B Yuste and L Acedo ldquoAn explicit finite difference methodand a new von Neumann-type stability analysis for fractionaldiffusion equationsrdquo SIAM Journal on Numerical Analysis vol42 no 5 pp 1862ndash1874 2005

[15] S B Yuste ldquoWeighted average finite differencemethods for frac-tional diffusion equationsrdquo Journal of Computational Physicsvol 216 no 1 pp 264ndash274 2006

[16] T A Langlands and B I Henry ldquoThe accuracy and stabilityof an implicit solution method for the fractional diffusionequationrdquo Journal of Computational Physics vol 205 no 2 pp719ndash736 2005

[17] M Cui ldquoCompact finite difference method for the fractionaldiffusion equationrdquo Journal of Computational Physics vol 228no 20 pp 7792ndash7804 2009

[18] V J Ervin and J P Roop ldquoVariational formulation for thestationary fractional advection dispersion equationrdquoNumericalMethods for Partial Differential Equations vol 22 no 3 pp 558ndash576 2006

[19] J Zhao J Xiao and Y Xu ldquoA finite element method forthe multiterm time-space Riesz fractional advection-diffusionequations in finite domainrdquo Abstract and Applied Analysis vol2013 Article ID 868035 15 pages 2013

[20] W Deng ldquoFinite element method for the space and timefractional Fokker-Planck equationrdquo SIAM Journal onNumericalAnalysis vol 47 no 1 pp 204ndash226 2008

[21] Y Lin and C Xu ldquoFinite differencespectral approximationsfor the time-fractional diffusion equationrdquo Journal of Compu-tational Physics vol 225 no 2 pp 1533ndash1552 2007

[22] X Li and C Xu ldquoExistence and uniqueness of the weak solutionof the space-time fractional diffusion equation and a spectralmethod approximationrdquo Communications in ComputationalPhysics vol 8 no 5 pp 1016ndash1051 2010

[23] I Podlubny A Chechkin T Skovranek Y Chen and B MVinagre Jara ldquoMatrix approach to discrete fractional calculusII Partial fractional differential equationsrdquo Journal of Compu-tational Physics vol 228 no 8 pp 3137ndash3153 2009

[24] C Li Z Zhao and Y Chen ldquoNumerical approximation of non-linear fractional differential equations with subdiffusion andsuperdiffusionrdquo Computers amp Mathematics with Applicationsvol 62 no 3 pp 855ndash875 2011

[25] B Cockburn and C-W Shu ldquoThe local discontinuous Galerkinmethod for time-dependent convection-diffusion systemsrdquoSIAM Journal on Numerical Analysis vol 35 no 6 pp 2440ndash2463 1998

[26] W H Deng and J S Hesthaven ldquoLocal discontinuous Galerkinmethods for fractional diffusion equationsrdquoESAIMMathemat-ical Modelling and Numerical Analysis vol 47 no 6 pp 1845ndash1864 2013

[27] L Wei X Zhang and Y He ldquoAnalysis of a local discontinuousGalerkin method for time-fractional advection-diffusion equa-tionsrdquo International Journal of Numerical Methods for Heat ampFluid Flow vol 23 no 4 pp 634ndash648 2013

[28] Q Xu and Z Zheng ldquoDiscontinuous Galerkin method fortime fractional diffusion equationrdquo Journal of Information andComputational Science vol 10 no 11 pp 3253ndash3264 2013

[29] QW Xu and J S Hesthaven ldquoDiscontinuous Galerkin methodfor fractional convection-diffusion equationsrdquo SIAM Journal onNumerical Analysis vol 52 no 1 pp 405ndash423 2014

[30] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[31] A A Kilbas H M Srivastava and J TrujilloTheory and Appli-cations of Fractional Differential Equations vol 204 of North-HollandMathematics Studies Elsevier Science BV AmsterdamThe Netherlands 2006

[32] G M Zaslavsky ldquoChaos fractional kinetics and anomaloustransportrdquo Physics Reports vol 371 no 6 pp 461ndash580 2002

[33] J Klafter B SWhite andM Levandowsky ldquoMicrozooplanktonfeeding behavior and the levy walkrdquo in Biological Motion vol89 of Lecture Notes in Biomathematics pp 281ndash296 SpringerBerlin Germany 1990

[34] Q Yang F Liu and I Turner ldquoNumerical methods for frac-tional partial differential equations with Riesz space fractionalderivativesrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol34 no 1 pp 200ndash218 2010

[35] S I Muslih and O P Agrawal ldquoRiesz fractional derivativesand fractional dimensional spacerdquo International Journal ofTheoretical Physics vol 49 no 2 pp 270ndash275 2010

[36] B Cockburn G Kanschat I Perugia and D Schotzau ldquoSuper-convergence of the local discontinuous Galerkin method forelliptic problems on Cartesian gridsrdquo SIAM Journal on Numer-ical Analysis vol 39 no 1 pp 264ndash285 2001

[37] J SHesthaven andTWarburtonNodalDiscontinuousGalerkinMethods Algorithms Analysis and Applications SpringerBerlin Germany 2008

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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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


time-space Riesz fractional advection-diffusion equations intheir paper [19] and Deng in [20] presented the finite elementapproximation for the space-time fractional Fokker-Planckequation Lin and Xu in [21] and Li and Xu in [22] proposeda spectral method for time or space fractional diffusionequations based on a weak formulation and the detailederror analysis was carried out A new matrix approach tosolve fractional partial differential equations numericallyhas been developed by Podlubny in [23] Li et al in [24]used the fractional difference method in time and the finiteelement method in space for the time-space fractional ordersubdiffusion and superdiffusion equations in which theypresented the error estimates for the full discrete schemesAnd for the subdiffusion linear problem they obtain theapproximation of order Δ1199052minus120572 + Δ119905minus120572ℎ2 in theoretical for thesuperdiffusion linear problem they got the approximation oforder Δ1199052 + Δ1199051minus120572ℎ2 However their numerical experimentscannot be consistent with the corresponding theoreticalresults very well particularly for the superdiffusion prob-lems

Moreover high order accuracy numerical methods forsolving fractional differential equations are still very limitedDiscontinuous Galerkin method is famous for extreme flex-ibility and high-accuracy properties [25] By applying thismethod differential equations can be solved locally and accu-rately while we only need to assemble and solve a large linearsystem Deng and Hesthaven in [26] studied the convergencerate analysis of local discontinuous Galerkin method (LDG)for solving spatial fractional diffusion equation which wasthe first attempt to solve fractional derivative equations byusing the LDG method at the same time the authors inthis paper speculated that the LDG method has significantpotential in solving fractional calculus problems After thatlots of researchers use the LDG method to analyze the timediscretization of fractional differential equations [27 28]Recently Xu and Hesthaven [29] obtained a consistent andhigh-accuracy numerical scheme for fractional convection-diffusion equationswith a space fractional Laplacian operatorof order120572 (1 lt 120572 lt 2) through exploiting LDGmethodTheyintroduced a new technique by rewriting the fractional Lap-lacian operator as a composite of first-order derivativesand a fractional integral then they converted the frac-tional convection-diffusion equation into a system of loworder equations Optimal order of convergence 119874(ℎ119896+1)for the space fractional diffusion problem and suboptimalorder of convergence 119874(ℎ119896+12) for the general spatial frac-tional convection-diffusion problem are established respec-tively

In this paper motivated by the research work of [24]we intend to construct and analyze the numerical schemesfor solving the time-space fractional subdiffusion equation asfollows120597120572119906 (119909 119905)120597119905120572 = 120598 (minus (minus)1205732) 119906 (119909 119905)

119909 isin Ω 119905 isin (0 119879] 119906 (119909 0) = 1199060 (119909) 119909 isin Ω(1)

and the time-space fractional superdiffusion equation asfollows120597120572119906 (119909 119905)120597119905120572 = 120598 (minus (minus)1205732) 119906 (119909 119905)

119909 isin Ω 119905 isin (0 119879] 119906 (119909 0) = 120601 (119909) 119909 isin Ω119906119905 (119909 0) = 120595 (119909) 119909 isin Ω(2)

where Ω = (119886 119887) 119879 gt 0 1 lt 120573 lt 2 denotes the ordernumber of space fractional derivatives 0 lt 120572 lt 1 describesthe fractional subdiffusion equation (1) 1 lt 120572 lt 2 describesthe fractional superdiffusion equation (2) 120598 is a real positiveparameter and 1199060 120601(119909) 120595(119909) isin 1198671

0 (Ω) are the given smoothfunctions

For the sake of simplicity we just consider the periodicboundary conditions in this paper Notice that this assump-tion is not essential our method can be directly extended tothe problems with aperiodic boundary condition

The time fractional derivative in (1) uses the Caputofractional partial derivative of order 120572 defined as [30 31]120597120572119906 (119909 119905)120597119905120572

= 1Γ (1 minus 120572) int119905

0

120597119906 (119909 119904)120597119904 119889119904(119905 minus 119904)120572 0 lt 120572 lt 1120597119906 (119909 119905)120597119905 120572 = 1(3)

and the time fractional Caputo derivative in (2) defined as[24]120597120572119906 (119909 119905)120597119905120572

= 1Γ (2 minus 120572) int119905

0(119905 minus 119904)1minus120572 1205972119906 (119909 119904)1205971199042 119889119904 1 lt 120572 lt 21205972119906 (119909 119905)1205971199052 120572 = 2

(4)

where Γ(sdot) is the Gamma functionThe space fractional terms in (1) and (2) are defined

through the fractional Laplacian operator (minus(minus )1205732)119906(119909 119905)1 lt 120573 lt 2 which is also known as the Riesz fractionalderivative (120597120573120597|119909|120573)119906(119909 119905)under the assumption that119906(119909 119905)1199061015840(119909 119905) 119906(119899minus1)(119909 119905) vanishes at the end point of an infinitedomain (ie 119909 = ∓infin) [32]

The spatial fractional differential operator is also animportant tool to describe the anomalous diffusion phe-nomenon when 1 lt 120573 lt 2 it represents a Levy 120573-stable flight[33] when 120573 rarr 2 it models a Brownian diffusion process

The rest of this paper is organized as follows In thenext section we review the appropriate functional spacesand some important definitions and properties of fractionalderivatives By using the technique which is proposed in[29] that is through converting the space fractional operatorof order 120573 into a composite of first-order derivative and






119877119886 119868120572119909119906 (119909 119905) = 1Γ (120572) int119909

119886(119909 minus 120585)120572minus1 119906 (120585 119905) 119889120585 119909 gt 119886

119877119909119868120572119887 119906 (119909 119905) = 1Γ (120572) int119887

119909(120585 minus 119909)120572minus1 119906 (120585 119905) 119889120585 119909 lt 119887 (5)



119877119886119863120572

119909119906 (119909 119905) = 1Γ (119899 minus 120572) 119889119899119889119909119899int119909


119909 gt 119886 (6)

119877119909119863120572

119887119906 (119909 119905) = (minus1)119899Γ (119899 minus 120572) 119889119899119889119909119899int119887


119909 lt 119887 (7)



120597120572120597 |119909|120572 119906 (119909 119905) = minus119862120572 (119886119863120572119909 + 119909119863120572

119887 ) 119906 (119909 119905) (8)

where 119862120572 = 12 cos(1205871205722) 120572 = 1 119886119863120572119909 119909119863120572




infin) 119906 (119909 119905)= 120597120572120597 |119909|120572 119906 (119909 119905)

(9)



119877119886119863120572

119909119906 (119909 119905) = 119863119899 (119886119868119899minus120572119909 119906 (119909 119905)) = 119886119868119899minus120572119909 (119863119899119906 (119909 119905)) 119877119909119863120572




119886119868120572119909 (120582119891 (119909) + 120583119892 (119909)) = 120582119886119868120572119909119891 (119909) + 120583119886119868120572119909119892 (119909) 119886119863120572

119909 (120582119891 (119909) + 120583119892 (119909)) = 120582119886119863120572119909119891 (119909) + 120583119886119863120572

119909119892 (119909) (11)


119886119868120572+120573119909 119891 (119909) = 119886119868120572119909 (119886119868120573119909119891 (119909)) = 119886119868120573119909 (119886119868120572119909119891 (119909)) (12)




120597119906 (119909 119905)120597119909 + 1199091198682minus120573infin


(13)





120597119906 (119909 119905)120597119909 ) (15)


119906119869120572119877(119877) = (|119906|2119869120572119877(119877) + 11990621198712(119877))12 (16)




119909119908 V)1198712(0119883)

= (1198770 1198631205792119909 119908 119877

119905 1198631205792119883 V)

1198712(0119883) (19)





119877+ (1199091198682119904infin119906 119906)

119877]


(21)



1199061198712(Ω) le 119862 |119906|1198691205831198770(Ω) 119906 isin 1198691205831198770 (Ω) 120583 isin 119877 (22)


119909119868120572119887 119891 (119909) =0119868120572119910 119892 (119910) (23)

Proof One has

119909119868120572119887 119891 (119909) = int119887

119909(119905 minus 119909)120572minus1 119891 (119905) 119889119905

119905=119887minus120585= minus int0


= int119887minus119909


119910=119887minus119909= int119910


119891(119887minus120585)=119892(120585)= int119910

0(119910 minus 120585)120572minus1 119892 (120585) 119889120585

=0119868120572119910 119892 (119910)

(24)



le 119862ℎ119896+1 (25)




ℎ = V V isin 119875119896 (119868119895) 119909 isin 119868119895 119895 = 1 2 119873 (26)


119868119895





+120596 (119909+119895minus(12)) = 120596 (119909119895minus(12))

int119868119895


minus120596 (119909minus119895+(12)) = 120596 (119909119895+(12))

(28)




3 Numerical Schemes





120597120572119906 (119909 119905119899)120597119905120572 = 1Γ (1 minus 120572) 119899minus1sum119894=0

int119905119894+1

119905119894


119894=0

119906 (119909 119905119894+1) minus 119906 (119909 119905119894)120591sdot int(119894+1)120591

119894120591






(119887119894minus1 minus 119887119894) + 119887119899 = 1 (31)




(33)


intΩ

119906 (119909 119905119899) V 119889119909 + 1205720radic120598 (intΩ

119902 (119909 119905119899) V119909119889119909minus 119873sum

119895=1


119894=1

(119887119894minus1 minus 119887119894) intΩ

119906 (119909 119905119899minus119894) V 119889119909+ 119887119899minus1 int

Ω119906 (119909 1199050) V 119889119909 minus 1205720 int

Ω119903119899 (119909) V 119889119909

intΩ

119902 (119909 119905119899) 119908 119889119909 minus intΩ

(120573minus2)2119901 (119909 119905119899) 119908 119889119909 = 0intΩ

119901 (119909 119905119899) 119911 119889119909 + radic120598 (intΩ

119906 (119909 119905119899) 119911119909119889119909minus 119873sum

119895=1

(119906 (119909 119905119899) 119911minus119895+(12) minus 119906 (119909 119905119899) 119911+

119895minus(12))) = 0intΩ

119906 (119909 1199050) V 119889119909 minus intΩ

1199060 (119909) V 119889119909 = 0

(34)

where 1205720 = 120591120572Γ(2 minus 120572)


Let 119906119899ℎ 119901119899




ℎ

intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(35)



ℎ = 119906119899ℎminus119902119899ℎ = 119902119899ℎ+ (36)




0


119894=0

int119905119894+1

119905119894


119894=0






(37)



119899sum119894=1

(]119894minus1 minus ]119894) + ]119899 = 1 (38)


119877119899 (119909) = 1Γ (2 minus 120572) 119899minus1sum119894=0

int119905119894+1

119905119894


119894=0

int119905119894+1

119905119894


(39)


119877119899 (119909)le 119862120591Γ (2 minus 120572) max

1199050le119905le119905119899

100381610038161003816100381610038161003816100381610038161003816 12059731199061205971199053 (119909 119905119894)100381610038161003816100381610038161003816100381610038161003816 119899minus1sum119894=0

int119905119894+1

119905119894


(40)



intΩ

119906 (119909 119905119899) V 119889119909 + 1205721radic120598 (intΩ

119902 (119909 119905119899) V119909119889119909minus 119873sum

119895=1



= 119899minus2sum119894=1




sdot intΩ


119906 (119909 119905minus1) V 119889119909minus 1205721 int

Ω119877119899 (119909) V 119889119909

intΩ

119902 (119909 119905119899) 119908 119889119909 minus intΩ

(120573minus2)2119901 (119909 119905119899) 119908 119889119909 = 0intΩ

119901 (119909 119905119899) 119911 119889119909 + radic120598 (intΩ

119906 (119909 119905119899) 119911119909119889119909minus 119873sum

119895=1

(119906 (119909 119905119899) 119911minus119895+(12) minus 119906 (119909 119905119899) 119911+

119895minus(12))) = 0intΩ

119906 (119909 1199050) V 119889119909 minus intΩ

120601 (119909) V 119889119909 = 0intΩ

119906119905 (119909 1199050) V 119889119909 minus intΩ

120595 (119909) V 119889119909 = 0(41)

where 1205721 = 120591120572Γ(3 minus 120572)Let 119906119899

ℎ 119901119899ℎ 119902119899ℎ isin 119881119896



ℎ

intΩ

119906119899ℎV 119889119909 + 1205721radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1


119894=1


119906119899minus119894minus1ℎ V 119889119909 + (2]0

minus ]1) intΩ


Ω1199060ℎV 119889119909


119906minus1ℎ V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0

intΩ

1199060ℎV 119889119909 minus int

Ω120601 (119909) V 119889119909 = 0

intΩ

(1199060ℎ)

119905V 119889119909 minus int

Ω120595 (119909) V 119889119909 = 0

(42)





119873+(12) = 0 (43)

4 Stability



ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 (44)


intΩ

119906119899ℎV 119889119909 + 1205720radic120598 int

Ω119902119899ℎV119909119889119909 + radic120598 int

Ω119906119899ℎ119911119909119889119909

minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 + int

Ω119902119899ℎ119908 119889119909 + int

Ω119901119899ℎ119911 119889119909

+ 1205720radic120598119873minus1sum119895=1


119895=1

(119906119899ℎ)minus119895+(12) [119911]119895+(12)


+ radic120598120583ℎ ((119906119899ℎ)minus119873+(12))2

= 119899minus1sum119894=1

(119887119894minus1 minus 119887119894) intΩ


Ω1199060ℎV 119889119909

(45)




intΩ

1199061ℎV 119889119909 + 1205720radic120598 int

Ω1199021ℎV119909119889119909 + radic120598 int

Ω1199061ℎz119909119889119909

minus intΩ

(120573minus2)21199011ℎ119908 119889119909 + int

Ω1199021ℎ119908 119889119909 + int

Ω1199011ℎ119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1

(1199061ℎ)minus

119895+(12)[119911]119895+(12)

+ 1205720radic120598 (1199021ℎ)+12


Vminus119873+(12)

+ radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎV 119889119909

(46)


ℎ 119911 = 12057201199021ℎ we getintΩ

1199061ℎ1199061

ℎ119889119909 + 1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909

+ 1205720radic120598 intΩ

1199061ℎ (1199021ℎ)

119909119889119909 + 1205720 int

Ω(120573minus2)21199011

ℎ1199011ℎ119889119909

minus 1205720 intΩ

1199021ℎ1199011ℎ119889119909 + 1205720 int

Ω1199011ℎ1199021ℎ119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

+ 1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+


(1199061ℎ)minus

119873+(12)

+ 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎ1199061

ℎ119889119909

(47)


ℎ (1199021ℎ)119909)119868119895 = 1199061ℎ1199021ℎ|119909minus119895+(12)

119909+119895minus(12)


1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909 + 1205720radic120598 int

Ω1199061ℎ (1199021ℎ)

119909119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= 1205720radic120598 119873sum119895=1

1199061ℎ1199021ℎ10038161003816100381610038161003816119909minus119895+(12)119909+

119895minus(12)

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= minus1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+

12+ 1205720radic120598 (1199021ℎ)minus119873+(12)

(1199061ℎ)minus

119873+(12)

(48)

Thus we obtain

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

= intΩ

1199060ℎ1199061

ℎ119889119909 (49)

From the fact that

intΩ

1199060ℎ1199061

ℎ119889119909 le 12 100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (50)

we have

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

le 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (51)


100381710038171003817100381710038171199061ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (52)


1003817100381710038171003817119906119898ℎ

1003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 119898 = 1 2 119875 (53)




(35) we obtain

10038171003817100381710038171003817119906119875+1ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)2119901119875+1ℎ 119901119875+1

ℎ )+ 1205720radic120598120583ℎ ((119906119875+1

ℎ )minus119873+(12)

)2


= 119875sum119894=1

(119887119894minus1 minus 119887119894) intΩ

119906119875+1minus119894ℎ 119906119875+1

ℎ 119889119909+ 119887119875 int

Ω1199060ℎ119906119875+1

ℎ 119889119909 le 119875sum119894=1


10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 119875sum119894=1

(119887119894minus1 minus 119887119894) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817= ( 119875sum

119894=1

(119887119894minus1 minus 119887119894) + 119887119875) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172+ 12 10038171003817100381710038171003817119906119875+1

ℎ

100381710038171003817100381710038172 (54)


ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (55)


ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 119862 (1003817100381710038171003817120601 (119909)1003817100381710038171003817 + 120591 1003817100381710038171003817120595 (119909)1003817100381710038171003817) 119899 = 0 1 119873 (56)


5 Error Estimates




ℎ1003817100381710038171003817






+119890119899119902 minus (P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) (59)


intΩ

119890119899119906V 119889119909 + 1205720radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1

(119887119894minus1 minus 119887119894) intΩ


1198900119906V 119889119909+ 1205720 int

Ω119903119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0

(60)


intΩP


+119890119899119902V119909119889119909+ radic120598 int

ΩP

minus119890119899119906119911119909119889119909 minus intΩ

(120573minus2)2Q119890119899119901119908 119889119909+ int

ΩP

+119890119899119902119908 119889119909 + intΩQ119890119899119901119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(P+119890119899119902)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1



119873+(12)



= 119899minus1sum119894=1



minus1198900119906V 119889119909minus 1205720 int

Ω119903119899 (119909) V 119889119909

+ intΩ


Ω(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) V119909119889119909

+ radic120598 intΩ


Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899)) 119908 119889119909

+ intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) 119908 119889119909+ int


119894=1





119895=1


119895=1


minus119873+(12)


(61)



119894=1


ΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ


+ 1205720radic120598 intΩ


Ω(Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899))

sdot (P+119890119899119902)119909

119889119909+ 1205720 int

Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899))Q119890119899119901119889119909

minus 1205720 intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899))Q119890119899119901119889119909+ 1205720 int

Ω(Q119901 (119909 119905119899) minus 119901 (119909 119905119899))P+119890119899119902119889119909


(119887119894minus1 minus 119887119894) intΩ


Ω(Pminus119906 (119909 1199050) minus 119906 (119909 1199050))


119895=1


119895=1






(62)


i=1(119887119894minus1 minus 119887119894)

sdot intΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ




119873+(12)







+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

= intΩP


Ω1199031 (119909)Pminus1198901119906119889119909

+ intΩ



minus intΩ


(64)



(65)


+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

le (10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

100381710038171003817100381710038171199031 (119909)10038171003817100381710038171003817


119873+(12)

(66)



+ 1205720120598119862 10038171003817100381710038171003817Q1198901119901100381710038171003817100381710038172 (67)






119873+(12)= 119897sum

119894=1


ΩP


ΩP


Ω119903119897+1 (119909)Pminus119890119897+1119906 119889119909

+ intΩ




119894=1

(119887119894minus1 minus 119887119894)


sdot intΩ



le ( 119897sum119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817+ 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817 + 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ ( 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(70)


then 10038171003817100381710038171003817Pminus119890119897+1119906


119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le (( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

(72)



100381710038171003817100381710038172+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 (73)


10038171003817100381710038171003817 le 119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) (74)





(76)







119873+(12)le ( 119897sum

119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817 + 119897sum

119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)


+ 1205912 + 1205911205722ℎ119896+1)) 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(78)



sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1))2 + 119862ℎ2119896+2120591120572

10038171003817100381710038171003817Pminus119890119897+1119906


sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le sum119897


(79)



(80)





When 1 lt 120572 lt 21003817100381710038171003817119906 (119909 119905119899) minus 119906119899ℎ1003817100381710038171003817




intΩ

119890119899119906V 119889119909 + 1205721radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1




sdot intΩ


1198900119906V 119889119909+ ]119899minus1 int

Ω119890minus1119906 V 119889119909 + 1205721 int

Ω119877119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0(83)







119905ℎ)10038171003817100381710038171003817le 119862 (ℎ119896+1 + 120591ℎ119896+1) (85)










minus 120572) 1119909120572











2ln119909(119909minus1))(119909minus1)119909 (93)





lim119909rarrinfin

119891 (119909) = lim119909rarrinfin



119909rarrinfin


119909rarrinfin







(94)

Hence 119891 (119909) le 12 minus 120572 119909 ge 1 (95)



(96)





le 119879120591 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1)

(98)







(99)








intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)



119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873






)= 119865119899V + 119899minus1sum

119894=1


where 119872 = (120601119895119898(119909) 120601119895


119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)


know that


ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where


119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)






(106)




















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119877119886 119868120572119909119906 (119909 119905) = 1Γ (120572) int119909

119886(119909 minus 120585)120572minus1 119906 (120585 119905) 119889120585 119909 gt 119886

119877119909119868120572119887 119906 (119909 119905) = 1Γ (120572) int119887

119909(120585 minus 119909)120572minus1 119906 (120585 119905) 119889120585 119909 lt 119887 (5)



119877119886119863120572

119909119906 (119909 119905) = 1Γ (119899 minus 120572) 119889119899119889119909119899int119909


119909 gt 119886 (6)

119877119909119863120572

119887119906 (119909 119905) = (minus1)119899Γ (119899 minus 120572) 119889119899119889119909119899int119887


119909 lt 119887 (7)



120597120572120597 |119909|120572 119906 (119909 119905) = minus119862120572 (119886119863120572119909 + 119909119863120572

119887 ) 119906 (119909 119905) (8)

where 119862120572 = 12 cos(1205871205722) 120572 = 1 119886119863120572119909 119909119863120572




infin) 119906 (119909 119905)= 120597120572120597 |119909|120572 119906 (119909 119905)

(9)



119877119886119863120572

119909119906 (119909 119905) = 119863119899 (119886119868119899minus120572119909 119906 (119909 119905)) = 119886119868119899minus120572119909 (119863119899119906 (119909 119905)) 119877119909119863120572




119886119868120572119909 (120582119891 (119909) + 120583119892 (119909)) = 120582119886119868120572119909119891 (119909) + 120583119886119868120572119909119892 (119909) 119886119863120572

119909 (120582119891 (119909) + 120583119892 (119909)) = 120582119886119863120572119909119891 (119909) + 120583119886119863120572

119909119892 (119909) (11)


119886119868120572+120573119909 119891 (119909) = 119886119868120572119909 (119886119868120573119909119891 (119909)) = 119886119868120573119909 (119886119868120572119909119891 (119909)) (12)




120597119906 (119909 119905)120597119909 + 1199091198682minus120573infin


(13)





120597119906 (119909 119905)120597119909 ) (15)


119906119869120572119877(119877) = (|119906|2119869120572119877(119877) + 11990621198712(119877))12 (16)




119909119908 V)1198712(0119883)

= (1198770 1198631205792119909 119908 119877

119905 1198631205792119883 V)

1198712(0119883) (19)





119877+ (1199091198682119904infin119906 119906)

119877]


(21)



1199061198712(Ω) le 119862 |119906|1198691205831198770(Ω) 119906 isin 1198691205831198770 (Ω) 120583 isin 119877 (22)


119909119868120572119887 119891 (119909) =0119868120572119910 119892 (119910) (23)

Proof One has

119909119868120572119887 119891 (119909) = int119887

119909(119905 minus 119909)120572minus1 119891 (119905) 119889119905

119905=119887minus120585= minus int0


= int119887minus119909


119910=119887minus119909= int119910


119891(119887minus120585)=119892(120585)= int119910

0(119910 minus 120585)120572minus1 119892 (120585) 119889120585

=0119868120572119910 119892 (119910)

(24)



le 119862ℎ119896+1 (25)




ℎ = V V isin 119875119896 (119868119895) 119909 isin 119868119895 119895 = 1 2 119873 (26)


119868119895





+120596 (119909+119895minus(12)) = 120596 (119909119895minus(12))

int119868119895


minus120596 (119909minus119895+(12)) = 120596 (119909119895+(12))

(28)




3 Numerical Schemes





120597120572119906 (119909 119905119899)120597119905120572 = 1Γ (1 minus 120572) 119899minus1sum119894=0

int119905119894+1

119905119894


119894=0

119906 (119909 119905119894+1) minus 119906 (119909 119905119894)120591sdot int(119894+1)120591

119894120591






(119887119894minus1 minus 119887119894) + 119887119899 = 1 (31)




(33)


intΩ

119906 (119909 119905119899) V 119889119909 + 1205720radic120598 (intΩ

119902 (119909 119905119899) V119909119889119909minus 119873sum

119895=1


119894=1

(119887119894minus1 minus 119887119894) intΩ

119906 (119909 119905119899minus119894) V 119889119909+ 119887119899minus1 int

Ω119906 (119909 1199050) V 119889119909 minus 1205720 int

Ω119903119899 (119909) V 119889119909

intΩ

119902 (119909 119905119899) 119908 119889119909 minus intΩ

(120573minus2)2119901 (119909 119905119899) 119908 119889119909 = 0intΩ

119901 (119909 119905119899) 119911 119889119909 + radic120598 (intΩ

119906 (119909 119905119899) 119911119909119889119909minus 119873sum

119895=1

(119906 (119909 119905119899) 119911minus119895+(12) minus 119906 (119909 119905119899) 119911+

119895minus(12))) = 0intΩ

119906 (119909 1199050) V 119889119909 minus intΩ

1199060 (119909) V 119889119909 = 0

(34)

where 1205720 = 120591120572Γ(2 minus 120572)


Let 119906119899ℎ 119901119899




ℎ

intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(35)



ℎ = 119906119899ℎminus119902119899ℎ = 119902119899ℎ+ (36)




0


119894=0

int119905119894+1

119905119894


119894=0






(37)



119899sum119894=1

(]119894minus1 minus ]119894) + ]119899 = 1 (38)


119877119899 (119909) = 1Γ (2 minus 120572) 119899minus1sum119894=0

int119905119894+1

119905119894


119894=0

int119905119894+1

119905119894


(39)


119877119899 (119909)le 119862120591Γ (2 minus 120572) max

1199050le119905le119905119899

100381610038161003816100381610038161003816100381610038161003816 12059731199061205971199053 (119909 119905119894)100381610038161003816100381610038161003816100381610038161003816 119899minus1sum119894=0

int119905119894+1

119905119894


(40)



intΩ

119906 (119909 119905119899) V 119889119909 + 1205721radic120598 (intΩ

119902 (119909 119905119899) V119909119889119909minus 119873sum

119895=1



= 119899minus2sum119894=1




sdot intΩ


119906 (119909 119905minus1) V 119889119909minus 1205721 int

Ω119877119899 (119909) V 119889119909

intΩ

119902 (119909 119905119899) 119908 119889119909 minus intΩ

(120573minus2)2119901 (119909 119905119899) 119908 119889119909 = 0intΩ

119901 (119909 119905119899) 119911 119889119909 + radic120598 (intΩ

119906 (119909 119905119899) 119911119909119889119909minus 119873sum

119895=1

(119906 (119909 119905119899) 119911minus119895+(12) minus 119906 (119909 119905119899) 119911+

119895minus(12))) = 0intΩ

119906 (119909 1199050) V 119889119909 minus intΩ

120601 (119909) V 119889119909 = 0intΩ

119906119905 (119909 1199050) V 119889119909 minus intΩ

120595 (119909) V 119889119909 = 0(41)

where 1205721 = 120591120572Γ(3 minus 120572)Let 119906119899

ℎ 119901119899ℎ 119902119899ℎ isin 119881119896



ℎ

intΩ

119906119899ℎV 119889119909 + 1205721radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1


119894=1


119906119899minus119894minus1ℎ V 119889119909 + (2]0

minus ]1) intΩ


Ω1199060ℎV 119889119909


119906minus1ℎ V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0

intΩ

1199060ℎV 119889119909 minus int

Ω120601 (119909) V 119889119909 = 0

intΩ

(1199060ℎ)

119905V 119889119909 minus int

Ω120595 (119909) V 119889119909 = 0

(42)





119873+(12) = 0 (43)

4 Stability



ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 (44)


intΩ

119906119899ℎV 119889119909 + 1205720radic120598 int

Ω119902119899ℎV119909119889119909 + radic120598 int

Ω119906119899ℎ119911119909119889119909

minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 + int

Ω119902119899ℎ119908 119889119909 + int

Ω119901119899ℎ119911 119889119909

+ 1205720radic120598119873minus1sum119895=1


119895=1

(119906119899ℎ)minus119895+(12) [119911]119895+(12)


+ radic120598120583ℎ ((119906119899ℎ)minus119873+(12))2

= 119899minus1sum119894=1

(119887119894minus1 minus 119887119894) intΩ


Ω1199060ℎV 119889119909

(45)




intΩ

1199061ℎV 119889119909 + 1205720radic120598 int

Ω1199021ℎV119909119889119909 + radic120598 int

Ω1199061ℎz119909119889119909

minus intΩ

(120573minus2)21199011ℎ119908 119889119909 + int

Ω1199021ℎ119908 119889119909 + int

Ω1199011ℎ119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1

(1199061ℎ)minus

119895+(12)[119911]119895+(12)

+ 1205720radic120598 (1199021ℎ)+12


Vminus119873+(12)

+ radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎV 119889119909

(46)


ℎ 119911 = 12057201199021ℎ we getintΩ

1199061ℎ1199061

ℎ119889119909 + 1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909

+ 1205720radic120598 intΩ

1199061ℎ (1199021ℎ)

119909119889119909 + 1205720 int

Ω(120573minus2)21199011

ℎ1199011ℎ119889119909

minus 1205720 intΩ

1199021ℎ1199011ℎ119889119909 + 1205720 int

Ω1199011ℎ1199021ℎ119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

+ 1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+


(1199061ℎ)minus

119873+(12)

+ 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎ1199061

ℎ119889119909

(47)


ℎ (1199021ℎ)119909)119868119895 = 1199061ℎ1199021ℎ|119909minus119895+(12)

119909+119895minus(12)


1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909 + 1205720radic120598 int

Ω1199061ℎ (1199021ℎ)

119909119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= 1205720radic120598 119873sum119895=1

1199061ℎ1199021ℎ10038161003816100381610038161003816119909minus119895+(12)119909+

119895minus(12)

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= minus1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+

12+ 1205720radic120598 (1199021ℎ)minus119873+(12)

(1199061ℎ)minus

119873+(12)

(48)

Thus we obtain

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

= intΩ

1199060ℎ1199061

ℎ119889119909 (49)

From the fact that

intΩ

1199060ℎ1199061

ℎ119889119909 le 12 100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (50)

we have

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

le 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (51)


100381710038171003817100381710038171199061ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (52)


1003817100381710038171003817119906119898ℎ

1003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 119898 = 1 2 119875 (53)




(35) we obtain

10038171003817100381710038171003817119906119875+1ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)2119901119875+1ℎ 119901119875+1

ℎ )+ 1205720radic120598120583ℎ ((119906119875+1

ℎ )minus119873+(12)

)2


= 119875sum119894=1

(119887119894minus1 minus 119887119894) intΩ

119906119875+1minus119894ℎ 119906119875+1

ℎ 119889119909+ 119887119875 int

Ω1199060ℎ119906119875+1

ℎ 119889119909 le 119875sum119894=1


10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 119875sum119894=1

(119887119894minus1 minus 119887119894) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817= ( 119875sum

119894=1

(119887119894minus1 minus 119887119894) + 119887119875) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172+ 12 10038171003817100381710038171003817119906119875+1

ℎ

100381710038171003817100381710038172 (54)


ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (55)


ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 119862 (1003817100381710038171003817120601 (119909)1003817100381710038171003817 + 120591 1003817100381710038171003817120595 (119909)1003817100381710038171003817) 119899 = 0 1 119873 (56)


5 Error Estimates




ℎ1003817100381710038171003817






+119890119899119902 minus (P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) (59)


intΩ

119890119899119906V 119889119909 + 1205720radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1

(119887119894minus1 minus 119887119894) intΩ


1198900119906V 119889119909+ 1205720 int

Ω119903119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0

(60)


intΩP


+119890119899119902V119909119889119909+ radic120598 int

ΩP

minus119890119899119906119911119909119889119909 minus intΩ

(120573minus2)2Q119890119899119901119908 119889119909+ int

ΩP

+119890119899119902119908 119889119909 + intΩQ119890119899119901119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(P+119890119899119902)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1



119873+(12)



= 119899minus1sum119894=1



minus1198900119906V 119889119909minus 1205720 int

Ω119903119899 (119909) V 119889119909

+ intΩ


Ω(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) V119909119889119909

+ radic120598 intΩ


Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899)) 119908 119889119909

+ intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) 119908 119889119909+ int


119894=1





119895=1


119895=1


minus119873+(12)


(61)



119894=1


ΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ


+ 1205720radic120598 intΩ


Ω(Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899))

sdot (P+119890119899119902)119909

119889119909+ 1205720 int

Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899))Q119890119899119901119889119909

minus 1205720 intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899))Q119890119899119901119889119909+ 1205720 int

Ω(Q119901 (119909 119905119899) minus 119901 (119909 119905119899))P+119890119899119902119889119909


(119887119894minus1 minus 119887119894) intΩ


Ω(Pminus119906 (119909 1199050) minus 119906 (119909 1199050))


119895=1


119895=1






(62)


i=1(119887119894minus1 minus 119887119894)

sdot intΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ




119873+(12)







+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

= intΩP


Ω1199031 (119909)Pminus1198901119906119889119909

+ intΩ



minus intΩ


(64)



(65)


+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

le (10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

100381710038171003817100381710038171199031 (119909)10038171003817100381710038171003817


119873+(12)

(66)



+ 1205720120598119862 10038171003817100381710038171003817Q1198901119901100381710038171003817100381710038172 (67)






119873+(12)= 119897sum

119894=1


ΩP


ΩP


Ω119903119897+1 (119909)Pminus119890119897+1119906 119889119909

+ intΩ




119894=1

(119887119894minus1 minus 119887119894)


sdot intΩ



le ( 119897sum119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817+ 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817 + 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ ( 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(70)


then 10038171003817100381710038171003817Pminus119890119897+1119906


119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le (( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

(72)



100381710038171003817100381710038172+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 (73)


10038171003817100381710038171003817 le 119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) (74)





(76)







119873+(12)le ( 119897sum

119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817 + 119897sum

119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)


+ 1205912 + 1205911205722ℎ119896+1)) 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(78)



sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1))2 + 119862ℎ2119896+2120591120572

10038171003817100381710038171003817Pminus119890119897+1119906


sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le sum119897


(79)



(80)





When 1 lt 120572 lt 21003817100381710038171003817119906 (119909 119905119899) minus 119906119899ℎ1003817100381710038171003817




intΩ

119890119899119906V 119889119909 + 1205721radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1




sdot intΩ


1198900119906V 119889119909+ ]119899minus1 int

Ω119890minus1119906 V 119889119909 + 1205721 int

Ω119877119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0(83)







119905ℎ)10038171003817100381710038171003817le 119862 (ℎ119896+1 + 120591ℎ119896+1) (85)










minus 120572) 1119909120572











2ln119909(119909minus1))(119909minus1)119909 (93)





lim119909rarrinfin

119891 (119909) = lim119909rarrinfin



119909rarrinfin


119909rarrinfin







(94)

Hence 119891 (119909) le 12 minus 120572 119909 ge 1 (95)



(96)





le 119879120591 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1)

(98)







(99)








intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)



119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873






)= 119865119899V + 119899minus1sum

119894=1


where 119872 = (120601119895119898(119909) 120601119895


119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)


know that


ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where


119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)






(106)




















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120597119906 (119909 119905)120597119909 ) (15)


119906119869120572119877(119877) = (|119906|2119869120572119877(119877) + 11990621198712(119877))12 (16)




119909119908 V)1198712(0119883)

= (1198770 1198631205792119909 119908 119877

119905 1198631205792119883 V)

1198712(0119883) (19)





119877+ (1199091198682119904infin119906 119906)

119877]


(21)



1199061198712(Ω) le 119862 |119906|1198691205831198770(Ω) 119906 isin 1198691205831198770 (Ω) 120583 isin 119877 (22)


119909119868120572119887 119891 (119909) =0119868120572119910 119892 (119910) (23)

Proof One has

119909119868120572119887 119891 (119909) = int119887

119909(119905 minus 119909)120572minus1 119891 (119905) 119889119905

119905=119887minus120585= minus int0


= int119887minus119909


119910=119887minus119909= int119910


119891(119887minus120585)=119892(120585)= int119910

0(119910 minus 120585)120572minus1 119892 (120585) 119889120585

=0119868120572119910 119892 (119910)

(24)



le 119862ℎ119896+1 (25)




ℎ = V V isin 119875119896 (119868119895) 119909 isin 119868119895 119895 = 1 2 119873 (26)


119868119895





+120596 (119909+119895minus(12)) = 120596 (119909119895minus(12))

int119868119895


minus120596 (119909minus119895+(12)) = 120596 (119909119895+(12))

(28)




3 Numerical Schemes





120597120572119906 (119909 119905119899)120597119905120572 = 1Γ (1 minus 120572) 119899minus1sum119894=0

int119905119894+1

119905119894


119894=0

119906 (119909 119905119894+1) minus 119906 (119909 119905119894)120591sdot int(119894+1)120591

119894120591






(119887119894minus1 minus 119887119894) + 119887119899 = 1 (31)




(33)


intΩ

119906 (119909 119905119899) V 119889119909 + 1205720radic120598 (intΩ

119902 (119909 119905119899) V119909119889119909minus 119873sum

119895=1


119894=1

(119887119894minus1 minus 119887119894) intΩ

119906 (119909 119905119899minus119894) V 119889119909+ 119887119899minus1 int

Ω119906 (119909 1199050) V 119889119909 minus 1205720 int

Ω119903119899 (119909) V 119889119909

intΩ

119902 (119909 119905119899) 119908 119889119909 minus intΩ

(120573minus2)2119901 (119909 119905119899) 119908 119889119909 = 0intΩ

119901 (119909 119905119899) 119911 119889119909 + radic120598 (intΩ

119906 (119909 119905119899) 119911119909119889119909minus 119873sum

119895=1

(119906 (119909 119905119899) 119911minus119895+(12) minus 119906 (119909 119905119899) 119911+

119895minus(12))) = 0intΩ

119906 (119909 1199050) V 119889119909 minus intΩ

1199060 (119909) V 119889119909 = 0

(34)

where 1205720 = 120591120572Γ(2 minus 120572)


Let 119906119899ℎ 119901119899




ℎ

intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(35)



ℎ = 119906119899ℎminus119902119899ℎ = 119902119899ℎ+ (36)




0


119894=0

int119905119894+1

119905119894


119894=0






(37)



119899sum119894=1

(]119894minus1 minus ]119894) + ]119899 = 1 (38)


119877119899 (119909) = 1Γ (2 minus 120572) 119899minus1sum119894=0

int119905119894+1

119905119894


119894=0

int119905119894+1

119905119894


(39)


119877119899 (119909)le 119862120591Γ (2 minus 120572) max

1199050le119905le119905119899

100381610038161003816100381610038161003816100381610038161003816 12059731199061205971199053 (119909 119905119894)100381610038161003816100381610038161003816100381610038161003816 119899minus1sum119894=0

int119905119894+1

119905119894


(40)



intΩ

119906 (119909 119905119899) V 119889119909 + 1205721radic120598 (intΩ

119902 (119909 119905119899) V119909119889119909minus 119873sum

119895=1



= 119899minus2sum119894=1




sdot intΩ


119906 (119909 119905minus1) V 119889119909minus 1205721 int

Ω119877119899 (119909) V 119889119909

intΩ

119902 (119909 119905119899) 119908 119889119909 minus intΩ

(120573minus2)2119901 (119909 119905119899) 119908 119889119909 = 0intΩ

119901 (119909 119905119899) 119911 119889119909 + radic120598 (intΩ

119906 (119909 119905119899) 119911119909119889119909minus 119873sum

119895=1

(119906 (119909 119905119899) 119911minus119895+(12) minus 119906 (119909 119905119899) 119911+

119895minus(12))) = 0intΩ

119906 (119909 1199050) V 119889119909 minus intΩ

120601 (119909) V 119889119909 = 0intΩ

119906119905 (119909 1199050) V 119889119909 minus intΩ

120595 (119909) V 119889119909 = 0(41)

where 1205721 = 120591120572Γ(3 minus 120572)Let 119906119899

ℎ 119901119899ℎ 119902119899ℎ isin 119881119896



ℎ

intΩ

119906119899ℎV 119889119909 + 1205721radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1


119894=1


119906119899minus119894minus1ℎ V 119889119909 + (2]0

minus ]1) intΩ


Ω1199060ℎV 119889119909


119906minus1ℎ V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0

intΩ

1199060ℎV 119889119909 minus int

Ω120601 (119909) V 119889119909 = 0

intΩ

(1199060ℎ)

119905V 119889119909 minus int

Ω120595 (119909) V 119889119909 = 0

(42)





119873+(12) = 0 (43)

4 Stability



ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 (44)


intΩ

119906119899ℎV 119889119909 + 1205720radic120598 int

Ω119902119899ℎV119909119889119909 + radic120598 int

Ω119906119899ℎ119911119909119889119909

minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 + int

Ω119902119899ℎ119908 119889119909 + int

Ω119901119899ℎ119911 119889119909

+ 1205720radic120598119873minus1sum119895=1


119895=1

(119906119899ℎ)minus119895+(12) [119911]119895+(12)


+ radic120598120583ℎ ((119906119899ℎ)minus119873+(12))2

= 119899minus1sum119894=1

(119887119894minus1 minus 119887119894) intΩ


Ω1199060ℎV 119889119909

(45)




intΩ

1199061ℎV 119889119909 + 1205720radic120598 int

Ω1199021ℎV119909119889119909 + radic120598 int

Ω1199061ℎz119909119889119909

minus intΩ

(120573minus2)21199011ℎ119908 119889119909 + int

Ω1199021ℎ119908 119889119909 + int

Ω1199011ℎ119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1

(1199061ℎ)minus

119895+(12)[119911]119895+(12)

+ 1205720radic120598 (1199021ℎ)+12


Vminus119873+(12)

+ radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎV 119889119909

(46)


ℎ 119911 = 12057201199021ℎ we getintΩ

1199061ℎ1199061

ℎ119889119909 + 1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909

+ 1205720radic120598 intΩ

1199061ℎ (1199021ℎ)

119909119889119909 + 1205720 int

Ω(120573minus2)21199011

ℎ1199011ℎ119889119909

minus 1205720 intΩ

1199021ℎ1199011ℎ119889119909 + 1205720 int

Ω1199011ℎ1199021ℎ119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

+ 1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+


(1199061ℎ)minus

119873+(12)

+ 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎ1199061

ℎ119889119909

(47)


ℎ (1199021ℎ)119909)119868119895 = 1199061ℎ1199021ℎ|119909minus119895+(12)

119909+119895minus(12)


1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909 + 1205720radic120598 int

Ω1199061ℎ (1199021ℎ)

119909119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= 1205720radic120598 119873sum119895=1

1199061ℎ1199021ℎ10038161003816100381610038161003816119909minus119895+(12)119909+

119895minus(12)

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= minus1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+

12+ 1205720radic120598 (1199021ℎ)minus119873+(12)

(1199061ℎ)minus

119873+(12)

(48)

Thus we obtain

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

= intΩ

1199060ℎ1199061

ℎ119889119909 (49)

From the fact that

intΩ

1199060ℎ1199061

ℎ119889119909 le 12 100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (50)

we have

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

le 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (51)


100381710038171003817100381710038171199061ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (52)


1003817100381710038171003817119906119898ℎ

1003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 119898 = 1 2 119875 (53)




(35) we obtain

10038171003817100381710038171003817119906119875+1ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)2119901119875+1ℎ 119901119875+1

ℎ )+ 1205720radic120598120583ℎ ((119906119875+1

ℎ )minus119873+(12)

)2


= 119875sum119894=1

(119887119894minus1 minus 119887119894) intΩ

119906119875+1minus119894ℎ 119906119875+1

ℎ 119889119909+ 119887119875 int

Ω1199060ℎ119906119875+1

ℎ 119889119909 le 119875sum119894=1


10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 119875sum119894=1

(119887119894minus1 minus 119887119894) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817= ( 119875sum

119894=1

(119887119894minus1 minus 119887119894) + 119887119875) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172+ 12 10038171003817100381710038171003817119906119875+1

ℎ

100381710038171003817100381710038172 (54)


ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (55)


ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 119862 (1003817100381710038171003817120601 (119909)1003817100381710038171003817 + 120591 1003817100381710038171003817120595 (119909)1003817100381710038171003817) 119899 = 0 1 119873 (56)


5 Error Estimates




ℎ1003817100381710038171003817






+119890119899119902 minus (P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) (59)


intΩ

119890119899119906V 119889119909 + 1205720radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1

(119887119894minus1 minus 119887119894) intΩ


1198900119906V 119889119909+ 1205720 int

Ω119903119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0

(60)


intΩP


+119890119899119902V119909119889119909+ radic120598 int

ΩP

minus119890119899119906119911119909119889119909 minus intΩ

(120573minus2)2Q119890119899119901119908 119889119909+ int

ΩP

+119890119899119902119908 119889119909 + intΩQ119890119899119901119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(P+119890119899119902)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1



119873+(12)



= 119899minus1sum119894=1



minus1198900119906V 119889119909minus 1205720 int

Ω119903119899 (119909) V 119889119909

+ intΩ


Ω(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) V119909119889119909

+ radic120598 intΩ


Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899)) 119908 119889119909

+ intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) 119908 119889119909+ int


119894=1





119895=1


119895=1


minus119873+(12)


(61)



119894=1


ΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ


+ 1205720radic120598 intΩ


Ω(Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899))

sdot (P+119890119899119902)119909

119889119909+ 1205720 int

Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899))Q119890119899119901119889119909

minus 1205720 intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899))Q119890119899119901119889119909+ 1205720 int

Ω(Q119901 (119909 119905119899) minus 119901 (119909 119905119899))P+119890119899119902119889119909


(119887119894minus1 minus 119887119894) intΩ


Ω(Pminus119906 (119909 1199050) minus 119906 (119909 1199050))


119895=1


119895=1






(62)


i=1(119887119894minus1 minus 119887119894)

sdot intΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ




119873+(12)







+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

= intΩP


Ω1199031 (119909)Pminus1198901119906119889119909

+ intΩ



minus intΩ


(64)



(65)


+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

le (10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

100381710038171003817100381710038171199031 (119909)10038171003817100381710038171003817


119873+(12)

(66)



+ 1205720120598119862 10038171003817100381710038171003817Q1198901119901100381710038171003817100381710038172 (67)






119873+(12)= 119897sum

119894=1


ΩP


ΩP


Ω119903119897+1 (119909)Pminus119890119897+1119906 119889119909

+ intΩ




119894=1

(119887119894minus1 minus 119887119894)


sdot intΩ



le ( 119897sum119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817+ 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817 + 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ ( 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(70)


then 10038171003817100381710038171003817Pminus119890119897+1119906


119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le (( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

(72)



100381710038171003817100381710038172+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 (73)


10038171003817100381710038171003817 le 119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) (74)





(76)







119873+(12)le ( 119897sum

119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817 + 119897sum

119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)


+ 1205912 + 1205911205722ℎ119896+1)) 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(78)



sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1))2 + 119862ℎ2119896+2120591120572

10038171003817100381710038171003817Pminus119890119897+1119906


sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le sum119897


(79)



(80)





When 1 lt 120572 lt 21003817100381710038171003817119906 (119909 119905119899) minus 119906119899ℎ1003817100381710038171003817




intΩ

119890119899119906V 119889119909 + 1205721radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1




sdot intΩ


1198900119906V 119889119909+ ]119899minus1 int

Ω119890minus1119906 V 119889119909 + 1205721 int

Ω119877119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0(83)







119905ℎ)10038171003817100381710038171003817le 119862 (ℎ119896+1 + 120591ℎ119896+1) (85)










minus 120572) 1119909120572











2ln119909(119909minus1))(119909minus1)119909 (93)





lim119909rarrinfin

119891 (119909) = lim119909rarrinfin



119909rarrinfin


119909rarrinfin







(94)

Hence 119891 (119909) le 12 minus 120572 119909 ge 1 (95)



(96)





le 119879120591 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1)

(98)







(99)








intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)



119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873






)= 119865119899V + 119899minus1sum

119894=1


where 119872 = (120601119895119898(119909) 120601119895


119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)


know that


ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where


119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)






(106)




















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












+120596 (119909+119895minus(12)) = 120596 (119909119895minus(12))

int119868119895


minus120596 (119909minus119895+(12)) = 120596 (119909119895+(12))

(28)




3 Numerical Schemes





120597120572119906 (119909 119905119899)120597119905120572 = 1Γ (1 minus 120572) 119899minus1sum119894=0

int119905119894+1

119905119894


119894=0

119906 (119909 119905119894+1) minus 119906 (119909 119905119894)120591sdot int(119894+1)120591

119894120591






(119887119894minus1 minus 119887119894) + 119887119899 = 1 (31)




(33)


intΩ

119906 (119909 119905119899) V 119889119909 + 1205720radic120598 (intΩ

119902 (119909 119905119899) V119909119889119909minus 119873sum

119895=1


119894=1

(119887119894minus1 minus 119887119894) intΩ

119906 (119909 119905119899minus119894) V 119889119909+ 119887119899minus1 int

Ω119906 (119909 1199050) V 119889119909 minus 1205720 int

Ω119903119899 (119909) V 119889119909

intΩ

119902 (119909 119905119899) 119908 119889119909 minus intΩ

(120573minus2)2119901 (119909 119905119899) 119908 119889119909 = 0intΩ

119901 (119909 119905119899) 119911 119889119909 + radic120598 (intΩ

119906 (119909 119905119899) 119911119909119889119909minus 119873sum

119895=1

(119906 (119909 119905119899) 119911minus119895+(12) minus 119906 (119909 119905119899) 119911+

119895minus(12))) = 0intΩ

119906 (119909 1199050) V 119889119909 minus intΩ

1199060 (119909) V 119889119909 = 0

(34)

where 1205720 = 120591120572Γ(2 minus 120572)


Let 119906119899ℎ 119901119899




ℎ

intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(35)



ℎ = 119906119899ℎminus119902119899ℎ = 119902119899ℎ+ (36)




0


119894=0

int119905119894+1

119905119894


119894=0






(37)



119899sum119894=1

(]119894minus1 minus ]119894) + ]119899 = 1 (38)


119877119899 (119909) = 1Γ (2 minus 120572) 119899minus1sum119894=0

int119905119894+1

119905119894


119894=0

int119905119894+1

119905119894


(39)


119877119899 (119909)le 119862120591Γ (2 minus 120572) max

1199050le119905le119905119899

100381610038161003816100381610038161003816100381610038161003816 12059731199061205971199053 (119909 119905119894)100381610038161003816100381610038161003816100381610038161003816 119899minus1sum119894=0

int119905119894+1

119905119894


(40)



intΩ

119906 (119909 119905119899) V 119889119909 + 1205721radic120598 (intΩ

119902 (119909 119905119899) V119909119889119909minus 119873sum

119895=1



= 119899minus2sum119894=1




sdot intΩ


119906 (119909 119905minus1) V 119889119909minus 1205721 int

Ω119877119899 (119909) V 119889119909

intΩ

119902 (119909 119905119899) 119908 119889119909 minus intΩ

(120573minus2)2119901 (119909 119905119899) 119908 119889119909 = 0intΩ

119901 (119909 119905119899) 119911 119889119909 + radic120598 (intΩ

119906 (119909 119905119899) 119911119909119889119909minus 119873sum

119895=1

(119906 (119909 119905119899) 119911minus119895+(12) minus 119906 (119909 119905119899) 119911+

119895minus(12))) = 0intΩ

119906 (119909 1199050) V 119889119909 minus intΩ

120601 (119909) V 119889119909 = 0intΩ

119906119905 (119909 1199050) V 119889119909 minus intΩ

120595 (119909) V 119889119909 = 0(41)

where 1205721 = 120591120572Γ(3 minus 120572)Let 119906119899

ℎ 119901119899ℎ 119902119899ℎ isin 119881119896



ℎ

intΩ

119906119899ℎV 119889119909 + 1205721radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1


119894=1


119906119899minus119894minus1ℎ V 119889119909 + (2]0

minus ]1) intΩ


Ω1199060ℎV 119889119909


119906minus1ℎ V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0

intΩ

1199060ℎV 119889119909 minus int

Ω120601 (119909) V 119889119909 = 0

intΩ

(1199060ℎ)

119905V 119889119909 minus int

Ω120595 (119909) V 119889119909 = 0

(42)





119873+(12) = 0 (43)

4 Stability



ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 (44)


intΩ

119906119899ℎV 119889119909 + 1205720radic120598 int

Ω119902119899ℎV119909119889119909 + radic120598 int

Ω119906119899ℎ119911119909119889119909

minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 + int

Ω119902119899ℎ119908 119889119909 + int

Ω119901119899ℎ119911 119889119909

+ 1205720radic120598119873minus1sum119895=1


119895=1

(119906119899ℎ)minus119895+(12) [119911]119895+(12)


+ radic120598120583ℎ ((119906119899ℎ)minus119873+(12))2

= 119899minus1sum119894=1

(119887119894minus1 minus 119887119894) intΩ


Ω1199060ℎV 119889119909

(45)




intΩ

1199061ℎV 119889119909 + 1205720radic120598 int

Ω1199021ℎV119909119889119909 + radic120598 int

Ω1199061ℎz119909119889119909

minus intΩ

(120573minus2)21199011ℎ119908 119889119909 + int

Ω1199021ℎ119908 119889119909 + int

Ω1199011ℎ119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1

(1199061ℎ)minus

119895+(12)[119911]119895+(12)

+ 1205720radic120598 (1199021ℎ)+12


Vminus119873+(12)

+ radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎV 119889119909

(46)


ℎ 119911 = 12057201199021ℎ we getintΩ

1199061ℎ1199061

ℎ119889119909 + 1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909

+ 1205720radic120598 intΩ

1199061ℎ (1199021ℎ)

119909119889119909 + 1205720 int

Ω(120573minus2)21199011

ℎ1199011ℎ119889119909

minus 1205720 intΩ

1199021ℎ1199011ℎ119889119909 + 1205720 int

Ω1199011ℎ1199021ℎ119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

+ 1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+


(1199061ℎ)minus

119873+(12)

+ 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎ1199061

ℎ119889119909

(47)


ℎ (1199021ℎ)119909)119868119895 = 1199061ℎ1199021ℎ|119909minus119895+(12)

119909+119895minus(12)


1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909 + 1205720radic120598 int

Ω1199061ℎ (1199021ℎ)

119909119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= 1205720radic120598 119873sum119895=1

1199061ℎ1199021ℎ10038161003816100381610038161003816119909minus119895+(12)119909+

119895minus(12)

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= minus1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+

12+ 1205720radic120598 (1199021ℎ)minus119873+(12)

(1199061ℎ)minus

119873+(12)

(48)

Thus we obtain

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

= intΩ

1199060ℎ1199061

ℎ119889119909 (49)

From the fact that

intΩ

1199060ℎ1199061

ℎ119889119909 le 12 100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (50)

we have

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

le 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (51)


100381710038171003817100381710038171199061ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (52)


1003817100381710038171003817119906119898ℎ

1003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 119898 = 1 2 119875 (53)




(35) we obtain

10038171003817100381710038171003817119906119875+1ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)2119901119875+1ℎ 119901119875+1

ℎ )+ 1205720radic120598120583ℎ ((119906119875+1

ℎ )minus119873+(12)

)2


= 119875sum119894=1

(119887119894minus1 minus 119887119894) intΩ

119906119875+1minus119894ℎ 119906119875+1

ℎ 119889119909+ 119887119875 int

Ω1199060ℎ119906119875+1

ℎ 119889119909 le 119875sum119894=1


10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 119875sum119894=1

(119887119894minus1 minus 119887119894) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817= ( 119875sum

119894=1

(119887119894minus1 minus 119887119894) + 119887119875) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172+ 12 10038171003817100381710038171003817119906119875+1

ℎ

100381710038171003817100381710038172 (54)


ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (55)


ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 119862 (1003817100381710038171003817120601 (119909)1003817100381710038171003817 + 120591 1003817100381710038171003817120595 (119909)1003817100381710038171003817) 119899 = 0 1 119873 (56)


5 Error Estimates




ℎ1003817100381710038171003817






+119890119899119902 minus (P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) (59)


intΩ

119890119899119906V 119889119909 + 1205720radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1

(119887119894minus1 minus 119887119894) intΩ


1198900119906V 119889119909+ 1205720 int

Ω119903119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0

(60)


intΩP


+119890119899119902V119909119889119909+ radic120598 int

ΩP

minus119890119899119906119911119909119889119909 minus intΩ

(120573minus2)2Q119890119899119901119908 119889119909+ int

ΩP

+119890119899119902119908 119889119909 + intΩQ119890119899119901119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(P+119890119899119902)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1



119873+(12)



= 119899minus1sum119894=1



minus1198900119906V 119889119909minus 1205720 int

Ω119903119899 (119909) V 119889119909

+ intΩ


Ω(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) V119909119889119909

+ radic120598 intΩ


Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899)) 119908 119889119909

+ intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) 119908 119889119909+ int


119894=1





119895=1


119895=1


minus119873+(12)


(61)



119894=1


ΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ


+ 1205720radic120598 intΩ


Ω(Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899))

sdot (P+119890119899119902)119909

119889119909+ 1205720 int

Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899))Q119890119899119901119889119909

minus 1205720 intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899))Q119890119899119901119889119909+ 1205720 int

Ω(Q119901 (119909 119905119899) minus 119901 (119909 119905119899))P+119890119899119902119889119909


(119887119894minus1 minus 119887119894) intΩ


Ω(Pminus119906 (119909 1199050) minus 119906 (119909 1199050))


119895=1


119895=1






(62)


i=1(119887119894minus1 minus 119887119894)

sdot intΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ




119873+(12)







+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

= intΩP


Ω1199031 (119909)Pminus1198901119906119889119909

+ intΩ



minus intΩ


(64)



(65)


+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

le (10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

100381710038171003817100381710038171199031 (119909)10038171003817100381710038171003817


119873+(12)

(66)



+ 1205720120598119862 10038171003817100381710038171003817Q1198901119901100381710038171003817100381710038172 (67)






119873+(12)= 119897sum

119894=1


ΩP


ΩP


Ω119903119897+1 (119909)Pminus119890119897+1119906 119889119909

+ intΩ




119894=1

(119887119894minus1 minus 119887119894)


sdot intΩ



le ( 119897sum119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817+ 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817 + 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ ( 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(70)


then 10038171003817100381710038171003817Pminus119890119897+1119906


119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le (( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

(72)



100381710038171003817100381710038172+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 (73)


10038171003817100381710038171003817 le 119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) (74)





(76)







119873+(12)le ( 119897sum

119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817 + 119897sum

119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)


+ 1205912 + 1205911205722ℎ119896+1)) 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(78)



sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1))2 + 119862ℎ2119896+2120591120572

10038171003817100381710038171003817Pminus119890119897+1119906


sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le sum119897


(79)



(80)





When 1 lt 120572 lt 21003817100381710038171003817119906 (119909 119905119899) minus 119906119899ℎ1003817100381710038171003817




intΩ

119890119899119906V 119889119909 + 1205721radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1




sdot intΩ


1198900119906V 119889119909+ ]119899minus1 int

Ω119890minus1119906 V 119889119909 + 1205721 int

Ω119877119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0(83)







119905ℎ)10038171003817100381710038171003817le 119862 (ℎ119896+1 + 120591ℎ119896+1) (85)










minus 120572) 1119909120572











2ln119909(119909minus1))(119909minus1)119909 (93)





lim119909rarrinfin

119891 (119909) = lim119909rarrinfin



119909rarrinfin


119909rarrinfin







(94)

Hence 119891 (119909) le 12 minus 120572 119909 ge 1 (95)



(96)





le 119879120591 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1)

(98)







(99)








intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)



119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873






)= 119865119899V + 119899minus1sum

119894=1


where 119872 = (120601119895119898(119909) 120601119895


119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)


know that


ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where


119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)






(106)




















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Let 119906119899ℎ 119901119899




ℎ

intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(35)



ℎ = 119906119899ℎminus119902119899ℎ = 119902119899ℎ+ (36)




0


119894=0

int119905119894+1

119905119894


119894=0






(37)



119899sum119894=1

(]119894minus1 minus ]119894) + ]119899 = 1 (38)


119877119899 (119909) = 1Γ (2 minus 120572) 119899minus1sum119894=0

int119905119894+1

119905119894


119894=0

int119905119894+1

119905119894


(39)


119877119899 (119909)le 119862120591Γ (2 minus 120572) max

1199050le119905le119905119899

100381610038161003816100381610038161003816100381610038161003816 12059731199061205971199053 (119909 119905119894)100381610038161003816100381610038161003816100381610038161003816 119899minus1sum119894=0

int119905119894+1

119905119894


(40)



intΩ

119906 (119909 119905119899) V 119889119909 + 1205721radic120598 (intΩ

119902 (119909 119905119899) V119909119889119909minus 119873sum

119895=1



= 119899minus2sum119894=1




sdot intΩ


119906 (119909 119905minus1) V 119889119909minus 1205721 int

Ω119877119899 (119909) V 119889119909

intΩ

119902 (119909 119905119899) 119908 119889119909 minus intΩ

(120573minus2)2119901 (119909 119905119899) 119908 119889119909 = 0intΩ

119901 (119909 119905119899) 119911 119889119909 + radic120598 (intΩ

119906 (119909 119905119899) 119911119909119889119909minus 119873sum

119895=1

(119906 (119909 119905119899) 119911minus119895+(12) minus 119906 (119909 119905119899) 119911+

119895minus(12))) = 0intΩ

119906 (119909 1199050) V 119889119909 minus intΩ

120601 (119909) V 119889119909 = 0intΩ

119906119905 (119909 1199050) V 119889119909 minus intΩ

120595 (119909) V 119889119909 = 0(41)

where 1205721 = 120591120572Γ(3 minus 120572)Let 119906119899

ℎ 119901119899ℎ 119902119899ℎ isin 119881119896



ℎ

intΩ

119906119899ℎV 119889119909 + 1205721radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1


119894=1


119906119899minus119894minus1ℎ V 119889119909 + (2]0

minus ]1) intΩ


Ω1199060ℎV 119889119909


119906minus1ℎ V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0

intΩ

1199060ℎV 119889119909 minus int

Ω120601 (119909) V 119889119909 = 0

intΩ

(1199060ℎ)

119905V 119889119909 minus int

Ω120595 (119909) V 119889119909 = 0

(42)





119873+(12) = 0 (43)

4 Stability



ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 (44)


intΩ

119906119899ℎV 119889119909 + 1205720radic120598 int

Ω119902119899ℎV119909119889119909 + radic120598 int

Ω119906119899ℎ119911119909119889119909

minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 + int

Ω119902119899ℎ119908 119889119909 + int

Ω119901119899ℎ119911 119889119909

+ 1205720radic120598119873minus1sum119895=1


119895=1

(119906119899ℎ)minus119895+(12) [119911]119895+(12)


+ radic120598120583ℎ ((119906119899ℎ)minus119873+(12))2

= 119899minus1sum119894=1

(119887119894minus1 minus 119887119894) intΩ


Ω1199060ℎV 119889119909

(45)




intΩ

1199061ℎV 119889119909 + 1205720radic120598 int

Ω1199021ℎV119909119889119909 + radic120598 int

Ω1199061ℎz119909119889119909

minus intΩ

(120573minus2)21199011ℎ119908 119889119909 + int

Ω1199021ℎ119908 119889119909 + int

Ω1199011ℎ119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1

(1199061ℎ)minus

119895+(12)[119911]119895+(12)

+ 1205720radic120598 (1199021ℎ)+12


Vminus119873+(12)

+ radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎV 119889119909

(46)


ℎ 119911 = 12057201199021ℎ we getintΩ

1199061ℎ1199061

ℎ119889119909 + 1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909

+ 1205720radic120598 intΩ

1199061ℎ (1199021ℎ)

119909119889119909 + 1205720 int

Ω(120573minus2)21199011

ℎ1199011ℎ119889119909

minus 1205720 intΩ

1199021ℎ1199011ℎ119889119909 + 1205720 int

Ω1199011ℎ1199021ℎ119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

+ 1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+


(1199061ℎ)minus

119873+(12)

+ 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎ1199061

ℎ119889119909

(47)


ℎ (1199021ℎ)119909)119868119895 = 1199061ℎ1199021ℎ|119909minus119895+(12)

119909+119895minus(12)


1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909 + 1205720radic120598 int

Ω1199061ℎ (1199021ℎ)

119909119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= 1205720radic120598 119873sum119895=1

1199061ℎ1199021ℎ10038161003816100381610038161003816119909minus119895+(12)119909+

119895minus(12)

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= minus1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+

12+ 1205720radic120598 (1199021ℎ)minus119873+(12)

(1199061ℎ)minus

119873+(12)

(48)

Thus we obtain

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

= intΩ

1199060ℎ1199061

ℎ119889119909 (49)

From the fact that

intΩ

1199060ℎ1199061

ℎ119889119909 le 12 100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (50)

we have

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

le 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (51)


100381710038171003817100381710038171199061ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (52)


1003817100381710038171003817119906119898ℎ

1003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 119898 = 1 2 119875 (53)




(35) we obtain

10038171003817100381710038171003817119906119875+1ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)2119901119875+1ℎ 119901119875+1

ℎ )+ 1205720radic120598120583ℎ ((119906119875+1

ℎ )minus119873+(12)

)2


= 119875sum119894=1

(119887119894minus1 minus 119887119894) intΩ

119906119875+1minus119894ℎ 119906119875+1

ℎ 119889119909+ 119887119875 int

Ω1199060ℎ119906119875+1

ℎ 119889119909 le 119875sum119894=1


10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 119875sum119894=1

(119887119894minus1 minus 119887119894) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817= ( 119875sum

119894=1

(119887119894minus1 minus 119887119894) + 119887119875) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172+ 12 10038171003817100381710038171003817119906119875+1

ℎ

100381710038171003817100381710038172 (54)


ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (55)


ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 119862 (1003817100381710038171003817120601 (119909)1003817100381710038171003817 + 120591 1003817100381710038171003817120595 (119909)1003817100381710038171003817) 119899 = 0 1 119873 (56)


5 Error Estimates




ℎ1003817100381710038171003817






+119890119899119902 minus (P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) (59)


intΩ

119890119899119906V 119889119909 + 1205720radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1

(119887119894minus1 minus 119887119894) intΩ


1198900119906V 119889119909+ 1205720 int

Ω119903119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0

(60)


intΩP


+119890119899119902V119909119889119909+ radic120598 int

ΩP

minus119890119899119906119911119909119889119909 minus intΩ

(120573minus2)2Q119890119899119901119908 119889119909+ int

ΩP

+119890119899119902119908 119889119909 + intΩQ119890119899119901119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(P+119890119899119902)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1



119873+(12)



= 119899minus1sum119894=1



minus1198900119906V 119889119909minus 1205720 int

Ω119903119899 (119909) V 119889119909

+ intΩ


Ω(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) V119909119889119909

+ radic120598 intΩ


Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899)) 119908 119889119909

+ intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) 119908 119889119909+ int


119894=1





119895=1


119895=1


minus119873+(12)


(61)



119894=1


ΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ


+ 1205720radic120598 intΩ


Ω(Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899))

sdot (P+119890119899119902)119909

119889119909+ 1205720 int

Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899))Q119890119899119901119889119909

minus 1205720 intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899))Q119890119899119901119889119909+ 1205720 int

Ω(Q119901 (119909 119905119899) minus 119901 (119909 119905119899))P+119890119899119902119889119909


(119887119894minus1 minus 119887119894) intΩ


Ω(Pminus119906 (119909 1199050) minus 119906 (119909 1199050))


119895=1


119895=1






(62)


i=1(119887119894minus1 minus 119887119894)

sdot intΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ




119873+(12)







+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

= intΩP


Ω1199031 (119909)Pminus1198901119906119889119909

+ intΩ



minus intΩ


(64)



(65)


+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

le (10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

100381710038171003817100381710038171199031 (119909)10038171003817100381710038171003817


119873+(12)

(66)



+ 1205720120598119862 10038171003817100381710038171003817Q1198901119901100381710038171003817100381710038172 (67)






119873+(12)= 119897sum

119894=1


ΩP


ΩP


Ω119903119897+1 (119909)Pminus119890119897+1119906 119889119909

+ intΩ




119894=1

(119887119894minus1 minus 119887119894)


sdot intΩ



le ( 119897sum119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817+ 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817 + 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ ( 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(70)


then 10038171003817100381710038171003817Pminus119890119897+1119906


119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le (( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

(72)



100381710038171003817100381710038172+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 (73)


10038171003817100381710038171003817 le 119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) (74)





(76)







119873+(12)le ( 119897sum

119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817 + 119897sum

119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)


+ 1205912 + 1205911205722ℎ119896+1)) 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(78)



sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1))2 + 119862ℎ2119896+2120591120572

10038171003817100381710038171003817Pminus119890119897+1119906


sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le sum119897


(79)



(80)





When 1 lt 120572 lt 21003817100381710038171003817119906 (119909 119905119899) minus 119906119899ℎ1003817100381710038171003817




intΩ

119890119899119906V 119889119909 + 1205721radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1




sdot intΩ


1198900119906V 119889119909+ ]119899minus1 int

Ω119890minus1119906 V 119889119909 + 1205721 int

Ω119877119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0(83)







119905ℎ)10038171003817100381710038171003817le 119862 (ℎ119896+1 + 120591ℎ119896+1) (85)










minus 120572) 1119909120572











2ln119909(119909minus1))(119909minus1)119909 (93)





lim119909rarrinfin

119891 (119909) = lim119909rarrinfin



119909rarrinfin


119909rarrinfin







(94)

Hence 119891 (119909) le 12 minus 120572 119909 ge 1 (95)



(96)





le 119879120591 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1)

(98)







(99)








intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)



119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873






)= 119865119899V + 119899minus1sum

119894=1


where 119872 = (120601119895119898(119909) 120601119895


119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)


know that


ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where


119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)






(106)




















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= 119899minus2sum119894=1




sdot intΩ


119906 (119909 119905minus1) V 119889119909minus 1205721 int

Ω119877119899 (119909) V 119889119909

intΩ

119902 (119909 119905119899) 119908 119889119909 minus intΩ

(120573minus2)2119901 (119909 119905119899) 119908 119889119909 = 0intΩ

119901 (119909 119905119899) 119911 119889119909 + radic120598 (intΩ

119906 (119909 119905119899) 119911119909119889119909minus 119873sum

119895=1

(119906 (119909 119905119899) 119911minus119895+(12) minus 119906 (119909 119905119899) 119911+

119895minus(12))) = 0intΩ

119906 (119909 1199050) V 119889119909 minus intΩ

120601 (119909) V 119889119909 = 0intΩ

119906119905 (119909 1199050) V 119889119909 minus intΩ

120595 (119909) V 119889119909 = 0(41)

where 1205721 = 120591120572Γ(3 minus 120572)Let 119906119899

ℎ 119901119899ℎ 119902119899ℎ isin 119881119896



ℎ

intΩ

119906119899ℎV 119889119909 + 1205721radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1


119894=1


119906119899minus119894minus1ℎ V 119889119909 + (2]0

minus ]1) intΩ


Ω1199060ℎV 119889119909


119906minus1ℎ V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0

intΩ

1199060ℎV 119889119909 minus int

Ω120601 (119909) V 119889119909 = 0

intΩ

(1199060ℎ)

119905V 119889119909 minus int

Ω120595 (119909) V 119889119909 = 0

(42)





119873+(12) = 0 (43)

4 Stability



ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 (44)


intΩ

119906119899ℎV 119889119909 + 1205720radic120598 int

Ω119902119899ℎV119909119889119909 + radic120598 int

Ω119906119899ℎ119911119909119889119909

minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 + int

Ω119902119899ℎ119908 119889119909 + int

Ω119901119899ℎ119911 119889119909

+ 1205720radic120598119873minus1sum119895=1


119895=1

(119906119899ℎ)minus119895+(12) [119911]119895+(12)


+ radic120598120583ℎ ((119906119899ℎ)minus119873+(12))2

= 119899minus1sum119894=1

(119887119894minus1 minus 119887119894) intΩ


Ω1199060ℎV 119889119909

(45)




intΩ

1199061ℎV 119889119909 + 1205720radic120598 int

Ω1199021ℎV119909119889119909 + radic120598 int

Ω1199061ℎz119909119889119909

minus intΩ

(120573minus2)21199011ℎ119908 119889119909 + int

Ω1199021ℎ119908 119889119909 + int

Ω1199011ℎ119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1

(1199061ℎ)minus

119895+(12)[119911]119895+(12)

+ 1205720radic120598 (1199021ℎ)+12


Vminus119873+(12)

+ radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎV 119889119909

(46)


ℎ 119911 = 12057201199021ℎ we getintΩ

1199061ℎ1199061

ℎ119889119909 + 1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909

+ 1205720radic120598 intΩ

1199061ℎ (1199021ℎ)

119909119889119909 + 1205720 int

Ω(120573minus2)21199011

ℎ1199011ℎ119889119909

minus 1205720 intΩ

1199021ℎ1199011ℎ119889119909 + 1205720 int

Ω1199011ℎ1199021ℎ119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

+ 1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+


(1199061ℎ)minus

119873+(12)

+ 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎ1199061

ℎ119889119909

(47)


ℎ (1199021ℎ)119909)119868119895 = 1199061ℎ1199021ℎ|119909minus119895+(12)

119909+119895minus(12)


1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909 + 1205720radic120598 int

Ω1199061ℎ (1199021ℎ)

119909119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= 1205720radic120598 119873sum119895=1

1199061ℎ1199021ℎ10038161003816100381610038161003816119909minus119895+(12)119909+

119895minus(12)

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= minus1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+

12+ 1205720radic120598 (1199021ℎ)minus119873+(12)

(1199061ℎ)minus

119873+(12)

(48)

Thus we obtain

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

= intΩ

1199060ℎ1199061

ℎ119889119909 (49)

From the fact that

intΩ

1199060ℎ1199061

ℎ119889119909 le 12 100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (50)

we have

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

le 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (51)


100381710038171003817100381710038171199061ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (52)


1003817100381710038171003817119906119898ℎ

1003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 119898 = 1 2 119875 (53)




(35) we obtain

10038171003817100381710038171003817119906119875+1ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)2119901119875+1ℎ 119901119875+1

ℎ )+ 1205720radic120598120583ℎ ((119906119875+1

ℎ )minus119873+(12)

)2


= 119875sum119894=1

(119887119894minus1 minus 119887119894) intΩ

119906119875+1minus119894ℎ 119906119875+1

ℎ 119889119909+ 119887119875 int

Ω1199060ℎ119906119875+1

ℎ 119889119909 le 119875sum119894=1


10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 119875sum119894=1

(119887119894minus1 minus 119887119894) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817= ( 119875sum

119894=1

(119887119894minus1 minus 119887119894) + 119887119875) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172+ 12 10038171003817100381710038171003817119906119875+1

ℎ

100381710038171003817100381710038172 (54)


ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (55)


ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 119862 (1003817100381710038171003817120601 (119909)1003817100381710038171003817 + 120591 1003817100381710038171003817120595 (119909)1003817100381710038171003817) 119899 = 0 1 119873 (56)


5 Error Estimates




ℎ1003817100381710038171003817






+119890119899119902 minus (P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) (59)


intΩ

119890119899119906V 119889119909 + 1205720radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1

(119887119894minus1 minus 119887119894) intΩ


1198900119906V 119889119909+ 1205720 int

Ω119903119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0

(60)


intΩP


+119890119899119902V119909119889119909+ radic120598 int

ΩP

minus119890119899119906119911119909119889119909 minus intΩ

(120573minus2)2Q119890119899119901119908 119889119909+ int

ΩP

+119890119899119902119908 119889119909 + intΩQ119890119899119901119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(P+119890119899119902)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1



119873+(12)



= 119899minus1sum119894=1



minus1198900119906V 119889119909minus 1205720 int

Ω119903119899 (119909) V 119889119909

+ intΩ


Ω(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) V119909119889119909

+ radic120598 intΩ


Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899)) 119908 119889119909

+ intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) 119908 119889119909+ int


119894=1





119895=1


119895=1


minus119873+(12)


(61)



119894=1


ΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ


+ 1205720radic120598 intΩ


Ω(Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899))

sdot (P+119890119899119902)119909

119889119909+ 1205720 int

Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899))Q119890119899119901119889119909

minus 1205720 intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899))Q119890119899119901119889119909+ 1205720 int

Ω(Q119901 (119909 119905119899) minus 119901 (119909 119905119899))P+119890119899119902119889119909


(119887119894minus1 minus 119887119894) intΩ


Ω(Pminus119906 (119909 1199050) minus 119906 (119909 1199050))


119895=1


119895=1






(62)


i=1(119887119894minus1 minus 119887119894)

sdot intΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ




119873+(12)







+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

= intΩP


Ω1199031 (119909)Pminus1198901119906119889119909

+ intΩ



minus intΩ


(64)



(65)


+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

le (10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

100381710038171003817100381710038171199031 (119909)10038171003817100381710038171003817


119873+(12)

(66)



+ 1205720120598119862 10038171003817100381710038171003817Q1198901119901100381710038171003817100381710038172 (67)






119873+(12)= 119897sum

119894=1


ΩP


ΩP


Ω119903119897+1 (119909)Pminus119890119897+1119906 119889119909

+ intΩ




119894=1

(119887119894minus1 minus 119887119894)


sdot intΩ



le ( 119897sum119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817+ 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817 + 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ ( 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(70)


then 10038171003817100381710038171003817Pminus119890119897+1119906


119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le (( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

(72)



100381710038171003817100381710038172+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 (73)


10038171003817100381710038171003817 le 119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) (74)





(76)







119873+(12)le ( 119897sum

119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817 + 119897sum

119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)


+ 1205912 + 1205911205722ℎ119896+1)) 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(78)



sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1))2 + 119862ℎ2119896+2120591120572

10038171003817100381710038171003817Pminus119890119897+1119906


sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le sum119897


(79)



(80)





When 1 lt 120572 lt 21003817100381710038171003817119906 (119909 119905119899) minus 119906119899ℎ1003817100381710038171003817




intΩ

119890119899119906V 119889119909 + 1205721radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1




sdot intΩ


1198900119906V 119889119909+ ]119899minus1 int

Ω119890minus1119906 V 119889119909 + 1205721 int

Ω119877119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0(83)







119905ℎ)10038171003817100381710038171003817le 119862 (ℎ119896+1 + 120591ℎ119896+1) (85)










minus 120572) 1119909120572











2ln119909(119909minus1))(119909minus1)119909 (93)





lim119909rarrinfin

119891 (119909) = lim119909rarrinfin



119909rarrinfin


119909rarrinfin







(94)

Hence 119891 (119909) le 12 minus 120572 119909 ge 1 (95)



(96)





le 119879120591 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1)

(98)







(99)








intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)



119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873






)= 119865119899V + 119899minus1sum

119894=1


where 119872 = (120601119895119898(119909) 120601119895


119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)


know that


ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where


119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)






(106)




















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











intΩ

1199061ℎV 119889119909 + 1205720radic120598 int

Ω1199021ℎV119909119889119909 + radic120598 int

Ω1199061ℎz119909119889119909

minus intΩ

(120573minus2)21199011ℎ119908 119889119909 + int

Ω1199021ℎ119908 119889119909 + int

Ω1199011ℎ119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1

(1199061ℎ)minus

119895+(12)[119911]119895+(12)

+ 1205720radic120598 (1199021ℎ)+12


Vminus119873+(12)

+ radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎV 119889119909

(46)


ℎ 119911 = 12057201199021ℎ we getintΩ

1199061ℎ1199061

ℎ119889119909 + 1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909

+ 1205720radic120598 intΩ

1199061ℎ (1199021ℎ)

119909119889119909 + 1205720 int

Ω(120573minus2)21199011

ℎ1199011ℎ119889119909

minus 1205720 intΩ

1199021ℎ1199011ℎ119889119909 + 1205720 int

Ω1199011ℎ1199021ℎ119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

+ 1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+


(1199061ℎ)minus

119873+(12)

+ 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2 = int

Ω1199060ℎ1199061

ℎ119889119909

(47)


ℎ (1199021ℎ)119909)119868119895 = 1199061ℎ1199021ℎ|119909minus119895+(12)

119909+119895minus(12)


1205720radic120598 intΩ

1199021ℎ (1199061ℎ)

119909119889119909 + 1205720radic120598 int

Ω1199061ℎ (1199021ℎ)

119909119889119909

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= 1205720radic120598 119873sum119895=1

1199061ℎ1199021ℎ10038161003816100381610038161003816119909minus119895+(12)119909+

119895minus(12)

+ 1205720radic120598119873minus1sum119895=1

(1199021ℎ)+119895+(12)

[1199061ℎ]

119895+(12)

+ 1205720radic120598119873minus1sum119895=1

(1199061ℎ)minus

119895+(12)[1199021ℎ]

119895+(12)

= minus1205720radic120598 (1199021ℎ)+12

(1199061ℎ)+

12+ 1205720radic120598 (1199021ℎ)minus119873+(12)

(1199061ℎ)minus

119873+(12)

(48)

Thus we obtain

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

= intΩ

1199060ℎ1199061

ℎ119889119909 (49)

From the fact that

intΩ

1199060ℎ1199061

ℎ119889119909 le 12 100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (50)

we have

100381710038171003817100381710038171199061ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)21199011ℎ 1199011

ℎ) + 1205720radic120598120583ℎ ((1199061ℎ)minus

119873+(12))2

le 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172 (51)


100381710038171003817100381710038171199061ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (52)


1003817100381710038171003817119906119898ℎ

1003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 119898 = 1 2 119875 (53)




(35) we obtain

10038171003817100381710038171003817119906119875+1ℎ

100381710038171003817100381710038172 + 1205720 ((120573minus2)2119901119875+1ℎ 119901119875+1

ℎ )+ 1205720radic120598120583ℎ ((119906119875+1

ℎ )minus119873+(12)

)2


= 119875sum119894=1

(119887119894minus1 minus 119887119894) intΩ

119906119875+1minus119894ℎ 119906119875+1

ℎ 119889119909+ 119887119875 int

Ω1199060ℎ119906119875+1

ℎ 119889119909 le 119875sum119894=1


10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 119875sum119894=1

(119887119894minus1 minus 119887119894) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817= ( 119875sum

119894=1

(119887119894minus1 minus 119887119894) + 119887119875) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172+ 12 10038171003817100381710038171003817119906119875+1

ℎ

100381710038171003817100381710038172 (54)


ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (55)


ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 119862 (1003817100381710038171003817120601 (119909)1003817100381710038171003817 + 120591 1003817100381710038171003817120595 (119909)1003817100381710038171003817) 119899 = 0 1 119873 (56)


5 Error Estimates




ℎ1003817100381710038171003817






+119890119899119902 minus (P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) (59)


intΩ

119890119899119906V 119889119909 + 1205720radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1

(119887119894minus1 minus 119887119894) intΩ


1198900119906V 119889119909+ 1205720 int

Ω119903119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0

(60)


intΩP


+119890119899119902V119909119889119909+ radic120598 int

ΩP

minus119890119899119906119911119909119889119909 minus intΩ

(120573minus2)2Q119890119899119901119908 119889119909+ int

ΩP

+119890119899119902119908 119889119909 + intΩQ119890119899119901119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(P+119890119899119902)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1



119873+(12)



= 119899minus1sum119894=1



minus1198900119906V 119889119909minus 1205720 int

Ω119903119899 (119909) V 119889119909

+ intΩ


Ω(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) V119909119889119909

+ radic120598 intΩ


Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899)) 119908 119889119909

+ intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) 119908 119889119909+ int


119894=1





119895=1


119895=1


minus119873+(12)


(61)



119894=1


ΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ


+ 1205720radic120598 intΩ


Ω(Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899))

sdot (P+119890119899119902)119909

119889119909+ 1205720 int

Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899))Q119890119899119901119889119909

minus 1205720 intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899))Q119890119899119901119889119909+ 1205720 int

Ω(Q119901 (119909 119905119899) minus 119901 (119909 119905119899))P+119890119899119902119889119909


(119887119894minus1 minus 119887119894) intΩ


Ω(Pminus119906 (119909 1199050) minus 119906 (119909 1199050))


119895=1


119895=1






(62)


i=1(119887119894minus1 minus 119887119894)

sdot intΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ




119873+(12)







+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

= intΩP


Ω1199031 (119909)Pminus1198901119906119889119909

+ intΩ



minus intΩ


(64)



(65)


+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

le (10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

100381710038171003817100381710038171199031 (119909)10038171003817100381710038171003817


119873+(12)

(66)



+ 1205720120598119862 10038171003817100381710038171003817Q1198901119901100381710038171003817100381710038172 (67)






119873+(12)= 119897sum

119894=1


ΩP


ΩP


Ω119903119897+1 (119909)Pminus119890119897+1119906 119889119909

+ intΩ




119894=1

(119887119894minus1 minus 119887119894)


sdot intΩ



le ( 119897sum119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817+ 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817 + 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ ( 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(70)


then 10038171003817100381710038171003817Pminus119890119897+1119906


119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le (( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

(72)



100381710038171003817100381710038172+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 (73)


10038171003817100381710038171003817 le 119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) (74)





(76)







119873+(12)le ( 119897sum

119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817 + 119897sum

119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)


+ 1205912 + 1205911205722ℎ119896+1)) 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(78)



sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1))2 + 119862ℎ2119896+2120591120572

10038171003817100381710038171003817Pminus119890119897+1119906


sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le sum119897


(79)



(80)





When 1 lt 120572 lt 21003817100381710038171003817119906 (119909 119905119899) minus 119906119899ℎ1003817100381710038171003817




intΩ

119890119899119906V 119889119909 + 1205721radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1




sdot intΩ


1198900119906V 119889119909+ ]119899minus1 int

Ω119890minus1119906 V 119889119909 + 1205721 int

Ω119877119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0(83)







119905ℎ)10038171003817100381710038171003817le 119862 (ℎ119896+1 + 120591ℎ119896+1) (85)










minus 120572) 1119909120572











2ln119909(119909minus1))(119909minus1)119909 (93)





lim119909rarrinfin

119891 (119909) = lim119909rarrinfin



119909rarrinfin


119909rarrinfin







(94)

Hence 119891 (119909) le 12 minus 120572 119909 ge 1 (95)



(96)





le 119879120591 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1)

(98)







(99)








intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)



119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873






)= 119865119899V + 119899minus1sum

119894=1


where 119872 = (120601119895119898(119909) 120601119895


119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)


know that


ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where


119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)






(106)




















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= 119875sum119894=1

(119887119894minus1 minus 119887119894) intΩ

119906119875+1minus119894ℎ 119906119875+1

ℎ 119889119909+ 119887119875 int

Ω1199060ℎ119906119875+1

ℎ 119889119909 le 119875sum119894=1


10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 119875sum119894=1

(119887119894minus1 minus 119887119894) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817+ 119887119875 100381710038171003817100381710038171199060

ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817= ( 119875sum

119894=1

(119887119894minus1 minus 119887119894) + 119887119875) 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 10038171003817100381710038171003817119906119875+1ℎ

10038171003817100381710038171003817 le 12 100381710038171003817100381710038171199060ℎ

100381710038171003817100381710038172+ 12 10038171003817100381710038171003817119906119875+1

ℎ

100381710038171003817100381710038172 (54)


ℎ

10038171003817100381710038171003817 le 100381710038171003817100381710038171199060ℎ

10038171003817100381710038171003817 (55)


ℎ satisfies1003817100381710038171003817119906119899ℎ1003817100381710038171003817 le 119862 (1003817100381710038171003817120601 (119909)1003817100381710038171003817 + 120591 1003817100381710038171003817120595 (119909)1003817100381710038171003817) 119899 = 0 1 119873 (56)


5 Error Estimates




ℎ1003817100381710038171003817






+119890119899119902 minus (P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) (59)


intΩ

119890119899119906V 119889119909 + 1205720radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1

(119887119894minus1 minus 119887119894) intΩ


1198900119906V 119889119909+ 1205720 int

Ω119903119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0

(60)


intΩP


+119890119899119902V119909119889119909+ radic120598 int

ΩP

minus119890119899119906119911119909119889119909 minus intΩ

(120573minus2)2Q119890119899119901119908 119889119909+ int

ΩP

+119890119899119902119908 119889119909 + intΩQ119890119899119901119911 119889119909

+ 1205720radic120598119873minus1sum119895=1

(P+119890119899119902)+119895+(12)

[V]119895+(12)+ radic120598119873minus1sum

119895=1



119873+(12)



= 119899minus1sum119894=1



minus1198900119906V 119889119909minus 1205720 int

Ω119903119899 (119909) V 119889119909

+ intΩ


Ω(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) V119909119889119909

+ radic120598 intΩ


Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899)) 119908 119889119909

+ intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) 119908 119889119909+ int


119894=1





119895=1


119895=1


minus119873+(12)


(61)



119894=1


ΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ


+ 1205720radic120598 intΩ


Ω(Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899))

sdot (P+119890119899119902)119909

119889119909+ 1205720 int

Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899))Q119890119899119901119889119909

minus 1205720 intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899))Q119890119899119901119889119909+ 1205720 int

Ω(Q119901 (119909 119905119899) minus 119901 (119909 119905119899))P+119890119899119902119889119909


(119887119894minus1 minus 119887119894) intΩ


Ω(Pminus119906 (119909 1199050) minus 119906 (119909 1199050))


119895=1


119895=1






(62)


i=1(119887119894minus1 minus 119887119894)

sdot intΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ




119873+(12)







+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

= intΩP


Ω1199031 (119909)Pminus1198901119906119889119909

+ intΩ



minus intΩ


(64)



(65)


+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

le (10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

100381710038171003817100381710038171199031 (119909)10038171003817100381710038171003817


119873+(12)

(66)



+ 1205720120598119862 10038171003817100381710038171003817Q1198901119901100381710038171003817100381710038172 (67)






119873+(12)= 119897sum

119894=1


ΩP


ΩP


Ω119903119897+1 (119909)Pminus119890119897+1119906 119889119909

+ intΩ




119894=1

(119887119894minus1 minus 119887119894)


sdot intΩ



le ( 119897sum119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817+ 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817 + 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ ( 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(70)


then 10038171003817100381710038171003817Pminus119890119897+1119906


119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le (( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

(72)



100381710038171003817100381710038172+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 (73)


10038171003817100381710038171003817 le 119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) (74)





(76)







119873+(12)le ( 119897sum

119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817 + 119897sum

119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)


+ 1205912 + 1205911205722ℎ119896+1)) 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(78)



sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1))2 + 119862ℎ2119896+2120591120572

10038171003817100381710038171003817Pminus119890119897+1119906


sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le sum119897


(79)



(80)





When 1 lt 120572 lt 21003817100381710038171003817119906 (119909 119905119899) minus 119906119899ℎ1003817100381710038171003817




intΩ

119890119899119906V 119889119909 + 1205721radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1




sdot intΩ


1198900119906V 119889119909+ ]119899minus1 int

Ω119890minus1119906 V 119889119909 + 1205721 int

Ω119877119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0(83)







119905ℎ)10038171003817100381710038171003817le 119862 (ℎ119896+1 + 120591ℎ119896+1) (85)










minus 120572) 1119909120572











2ln119909(119909minus1))(119909minus1)119909 (93)





lim119909rarrinfin

119891 (119909) = lim119909rarrinfin



119909rarrinfin


119909rarrinfin







(94)

Hence 119891 (119909) le 12 minus 120572 119909 ge 1 (95)



(96)





le 119879120591 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1)

(98)







(99)








intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)



119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873






)= 119865119899V + 119899minus1sum

119894=1


where 119872 = (120601119895119898(119909) 120601119895


119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)


know that


ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where


119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)






(106)




















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= 119899minus1sum119894=1



minus1198900119906V 119889119909minus 1205720 int

Ω119903119899 (119909) V 119889119909

+ intΩ


Ω(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) V119909119889119909

+ radic120598 intΩ


Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899)) 119908 119889119909

+ intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899)) 119908 119889119909+ int


119894=1





119895=1


119895=1


minus119873+(12)


(61)



119894=1


ΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ


+ 1205720radic120598 intΩ


Ω(Pminus119906 (119909 119905119899) minus 119906 (119909 119905119899))

sdot (P+119890119899119902)119909

119889119909+ 1205720 int

Ω(120573minus2)2 (Q119901 (119909 119905119899) minus 119901 (119909 119905119899))Q119890119899119901119889119909

minus 1205720 intΩ

(P+119902 (119909 119905119899) minus 119902 (119909 119905119899))Q119890119899119901119889119909+ 1205720 int

Ω(Q119901 (119909 119905119899) minus 119901 (119909 119905119899))P+119890119899119902119889119909


(119887119894minus1 minus 119887119894) intΩ


Ω(Pminus119906 (119909 1199050) minus 119906 (119909 1199050))


119895=1


119895=1






(62)


i=1(119887119894minus1 minus 119887119894)

sdot intΩP


ΩP


Ω119903119899 (119909)Pminus119890119899119906119889119909

+ intΩ




119873+(12)







+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

= intΩP


Ω1199031 (119909)Pminus1198901119906119889119909

+ intΩ



minus intΩ


(64)



(65)


+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

le (10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

100381710038171003817100381710038171199031 (119909)10038171003817100381710038171003817


119873+(12)

(66)



+ 1205720120598119862 10038171003817100381710038171003817Q1198901119901100381710038171003817100381710038172 (67)






119873+(12)= 119897sum

119894=1


ΩP


ΩP


Ω119903119897+1 (119909)Pminus119890119897+1119906 119889119909

+ intΩ




119894=1

(119887119894minus1 minus 119887119894)


sdot intΩ



le ( 119897sum119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817+ 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817 + 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ ( 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(70)


then 10038171003817100381710038171003817Pminus119890119897+1119906


119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le (( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

(72)



100381710038171003817100381710038172+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 (73)


10038171003817100381710038171003817 le 119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) (74)





(76)







119873+(12)le ( 119897sum

119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817 + 119897sum

119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)


+ 1205912 + 1205911205722ℎ119896+1)) 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(78)



sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1))2 + 119862ℎ2119896+2120591120572

10038171003817100381710038171003817Pminus119890119897+1119906


sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le sum119897


(79)



(80)





When 1 lt 120572 lt 21003817100381710038171003817119906 (119909 119905119899) minus 119906119899ℎ1003817100381710038171003817




intΩ

119890119899119906V 119889119909 + 1205721radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1




sdot intΩ


1198900119906V 119889119909+ ]119899minus1 int

Ω119890minus1119906 V 119889119909 + 1205721 int

Ω119877119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0(83)







119905ℎ)10038171003817100381710038171003817le 119862 (ℎ119896+1 + 120591ℎ119896+1) (85)










minus 120572) 1119909120572











2ln119909(119909minus1))(119909minus1)119909 (93)





lim119909rarrinfin

119891 (119909) = lim119909rarrinfin



119909rarrinfin


119909rarrinfin







(94)

Hence 119891 (119909) le 12 minus 120572 119909 ge 1 (95)



(96)





le 119879120591 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1)

(98)







(99)








intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)



119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873






)= 119865119899V + 119899minus1sum

119894=1


where 119872 = (120601119895119898(119909) 120601119895


119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)


know that


ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where


119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)






(106)




















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119873+(12)







+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

= intΩP


Ω1199031 (119909)Pminus1198901119906119889119909

+ intΩ



minus intΩ


(64)



(65)


+ 1205720radic120598120583ℎ (Pminus1198901119906)2119873+(12)

le (10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

100381710038171003817100381710038171199031 (119909)10038171003817100381710038171003817


119873+(12)

(66)



+ 1205720120598119862 10038171003817100381710038171003817Q1198901119901100381710038171003817100381710038172 (67)






119873+(12)= 119897sum

119894=1


ΩP


ΩP


Ω119903119897+1 (119909)Pminus119890119897+1119906 119889119909

+ intΩ




119894=1

(119887119894minus1 minus 119887119894)


sdot intΩ



le ( 119897sum119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817+ 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817 + 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ ( 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(70)


then 10038171003817100381710038171003817Pminus119890119897+1119906


119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le (( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

(72)



100381710038171003817100381710038172+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 (73)


10038171003817100381710038171003817 le 119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) (74)





(76)







119873+(12)le ( 119897sum

119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817 + 119897sum

119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)


+ 1205912 + 1205911205722ℎ119896+1)) 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(78)



sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1))2 + 119862ℎ2119896+2120591120572

10038171003817100381710038171003817Pminus119890119897+1119906


sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le sum119897


(79)



(80)





When 1 lt 120572 lt 21003817100381710038171003817119906 (119909 119905119899) minus 119906119899ℎ1003817100381710038171003817




intΩ

119890119899119906V 119889119909 + 1205721radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1




sdot intΩ


1198900119906V 119889119909+ ]119899minus1 int

Ω119890minus1119906 V 119889119909 + 1205721 int

Ω119877119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0(83)







119905ℎ)10038171003817100381710038171003817le 119862 (ℎ119896+1 + 120591ℎ119896+1) (85)










minus 120572) 1119909120572











2ln119909(119909minus1))(119909minus1)119909 (93)





lim119909rarrinfin

119891 (119909) = lim119909rarrinfin



119909rarrinfin


119909rarrinfin







(94)

Hence 119891 (119909) le 12 minus 120572 119909 ge 1 (95)



(96)





le 119879120591 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1)

(98)







(99)








intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)



119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873






)= 119865119899V + 119899minus1sum

119894=1


where 119872 = (120601119895119898(119909) 120601119895


119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)


know that


ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where


119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)






(106)




















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










sdot intΩ



le ( 119897sum119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817+ 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817 + 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817+ ( 119897sum

119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(70)


then 10038171003817100381710038171003817Pminus119890119897+1119906


119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

le (( 119897sum119894=1


10038171003817100381710038171003817+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)

(72)



100381710038171003817100381710038172+ 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 (73)


10038171003817100381710038171003817 le 119887minus1119897 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1) (74)





(76)







119873+(12)le ( 119897sum

119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817 + 119897sum

119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)


+ 1205912 + 1205911205722ℎ119896+1)) 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(78)



sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1))2 + 119862ℎ2119896+2120591120572

10038171003817100381710038171003817Pminus119890119897+1119906


sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le sum119897


(79)



(80)





When 1 lt 120572 lt 21003817100381710038171003817119906 (119909 119905119899) minus 119906119899ℎ1003817100381710038171003817




intΩ

119890119899119906V 119889119909 + 1205721radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1




sdot intΩ


1198900119906V 119889119909+ ]119899minus1 int

Ω119890minus1119906 V 119889119909 + 1205721 int

Ω119877119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0(83)







119905ℎ)10038171003817100381710038171003817le 119862 (ℎ119896+1 + 120591ℎ119896+1) (85)










minus 120572) 1119909120572











2ln119909(119909minus1))(119909minus1)119909 (93)





lim119909rarrinfin

119891 (119909) = lim119909rarrinfin



119909rarrinfin


119909rarrinfin







(94)

Hence 119891 (119909) le 12 minus 120572 119909 ge 1 (95)



(96)





le 119879120591 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1)

(98)







(99)








intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)



119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873






)= 119865119899V + 119899minus1sum

119894=1


where 119872 = (120601119895119898(119909) 120601119895


119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)


know that


ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where


119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)






(106)




















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




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Journal of


Function Spaces






Algebra














119873+(12)le ( 119897sum

119894=1


10038171003817100381710038171003817 + 119887119897 10038171003817100381710038171003817Pminus119890011990610038171003817100381710038171003817 + 1205720

10038171003817100381710038171003817119903119897+1 (119909)10038171003817100381710038171003817+ 1003817100381710038171003817Pminus119906 (119909 119905119897+1) minus 119906 (119909 119905119897+1)1003817100381710038171003817 + 119897sum

119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

le ( 119897sum119894=1


10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572 + 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172+ 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2

119873+(12)


+ 1205912 + 1205911205722ℎ119896+1)) 10038171003817100381710038171003817Pminus119890119897+1119906

10038171003817100381710038171003817 + 119862ℎ2119896+2120591120572+ 1205720120598119862 10038171003817100381710038171003817Q119890119897+1119901

100381710038171003817100381710038172 + 1205720radic120598120583ℎ (Pminus119890119897+1119906 )2119873+(12)

(78)



sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1))2 + 119862ℎ2119896+2120591120572

10038171003817100381710038171003817Pminus119890119897+1119906


sdot 119862 (ℎ119896+1 + 1205912 + 1205911205722ℎ119896+1)le sum119897


(79)



(80)





When 1 lt 120572 lt 21003817100381710038171003817119906 (119909 119905119899) minus 119906119899ℎ1003817100381710038171003817




intΩ

119890119899119906V 119889119909 + 1205721radic120598 (intΩ

119890119899119902V119909119889119909minus 119873sum

119895=1


119894=1




sdot intΩ


1198900119906V 119889119909+ ]119899minus1 int

Ω119890minus1119906 V 119889119909 + 1205721 int

Ω119877119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0(83)







119905ℎ)10038171003817100381710038171003817le 119862 (ℎ119896+1 + 120591ℎ119896+1) (85)










minus 120572) 1119909120572











2ln119909(119909minus1))(119909minus1)119909 (93)





lim119909rarrinfin

119891 (119909) = lim119909rarrinfin



119909rarrinfin


119909rarrinfin







(94)

Hence 119891 (119909) le 12 minus 120572 119909 ge 1 (95)



(96)





le 119879120591 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1)

(98)







(99)








intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)



119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873






)= 119865119899V + 119899minus1sum

119894=1


where 119872 = (120601119895119898(119909) 120601119895


119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)


know that


ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where


119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)






(106)




















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










sdot intΩ


1198900119906V 119889119909+ ]119899minus1 int

Ω119890minus1119906 V 119889119909 + 1205721 int

Ω119877119899 (119909) V 119889119909 + int

Ω119890119899119902119908 119889119909

minus intΩ

(120573minus2)2119890119899119901119908 119889119909 + intΩ

119890119899119901119911 119889119909+ radic120598 (int

Ω119890119899119906119911119909119889119909

minus 119873sum119895=1


119895minus(12))) = 0(83)







119905ℎ)10038171003817100381710038171003817le 119862 (ℎ119896+1 + 120591ℎ119896+1) (85)










minus 120572) 1119909120572











2ln119909(119909minus1))(119909minus1)119909 (93)





lim119909rarrinfin

119891 (119909) = lim119909rarrinfin



119909rarrinfin


119909rarrinfin







(94)

Hence 119891 (119909) le 12 minus 120572 119909 ge 1 (95)



(96)





le 119879120591 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1)

(98)







(99)








intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)



119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873






)= 119865119899V + 119899minus1sum

119894=1


where 119872 = (120601119895119898(119909) 120601119895


119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)


know that


ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where


119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)






(106)




















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2ln119909(119909minus1))(119909minus1)119909 (93)





lim119909rarrinfin

119891 (119909) = lim119909rarrinfin



119909rarrinfin


119909rarrinfin







(94)

Hence 119891 (119909) le 12 minus 120572 119909 ge 1 (95)



(96)





le 119879120591 119862 (ℎ119896+1 + 1205911+120572 + 1205911205722ℎ119896+1)le 119879119862119906 (120591minus1ℎ119896+1 + 1205912 + ℎ119896+1)

(98)







(99)








intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)



119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873






)= 119865119899V + 119899minus1sum

119894=1


where 119872 = (120601119895119898(119909) 120601119895


119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)


know that


ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where


119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)






(106)




















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











intΩ

119906119899ℎV 119889119909 + 1205720radic120598 (int

Ω119902119899ℎV119909119889119909

minus 119873sum119895=1




Ω1199060ℎV 119889119909 + int

Ω119891 (119909 119905) V 119889119909

intΩ

119902119899ℎ119908 119889119909 minus intΩ

(120573minus2)2119901119899ℎ119908 119889119909 = 0

intΩ

119901119899ℎ119911 119889119909 + radic120598 (int

Ω119906119899ℎ119911119909119889119909

minus 119873sum119895=1

(119906119899ℎ119911minus

119895+(12) minus 119906119899ℎ119911+

119895minus(12))) = 0intΩ

1199060ℎV 119889119909 minus int

Ω1199060 (119909) V 119889119909 = 0

(100)



119894=1 119906119895(119905)120601119895(119909) 119901119899ℎ =sum119873






)= 119865119899V + 119899minus1sum

119894=1


where 119872 = (120601119895119898(119909) 120601119895


119860 = (120601119895119898 (119909) 120601119895

119896

1015840 (119909))119895=1119873

11986711 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895+12))119895=1119873

11986712 = (120601119895

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

11986713 = (120601119895

119898 (119909119895+12) 120601119895

119896 (119909119895+12))119895=1119873

11986714 = (120601119895minus1

119898 (119909119895minus12) 120601119895

119896 (119909119895minus12))119895=1119873

(102)


know that


ℎ+1199091198682minus120573119887 119901119899ℎ2 cos (1205731205872) (103)

where


119886(119909 minus 120585)1minus120573 119901119899

ℎ (120585 119905) 1198891205851199091198682minus120573119887 119901119899

ℎ (119909 119905) = 1Γ (2 minus 120573) int119887

119909(120585 minus 119909)1minus120573 119901119899

ℎ (120585 119905) 119889120585 (104)






(106)




















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















(108)




7 Conclusions








Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Competing Interests


Acknowledgments


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









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