A New Approach for Solving the Undamped Helmholtz ...downloads.hindawi.com/journals/mpe/2020/7876413.pdfspin spacecrafts [10], spacecraft motion about slowly rotating asteroids [11],

Research ArticleA New Approach for Solving the Undamped HelmholtzOscillator for the Given Arbitrary Initial Conditions and ItsPhysical Applications

Alvaro H Salas S 1 Jairo E Castillo H2 and Darin J Mosquera P3

1Universidad Nacional de Colombia Department of Mathematics and Statistics FIZMAKO Research Group Bogota Colombia2Universidad Distrital Francisco Jose de Caldas FIZMAKO Research Group Bogota Colombia3Universidad Distrital Francisco Jose de Caldas ORION Research Group Bogota Colombia

Correspondence should be addressed to Alvaro H Salas S ahsalassunaleduco

Received 2 May 2020 Revised 17 July 2020 Accepted 7 August 2020 Published 7 October 2020

Academic Editor Francesco Clementi

Copyright copy 2020 Alvaro H Salas S et al ampis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this paper a new analytical solution to the undampedHelmholtz oscillator equation in terms of theWeierstrass elliptic functionis reported ampe solution is given for any arbitrary initial conditions A comparison between our new solution and the numericalapproximate solution using the Range Kutta approach is performedWe think that the methodology employed here may be usefulin the study of several nonlinear problems described by a differential equation of the form zPrime F(z) in the sense that z z(t) Inthis context our solutions are applied to some physical applications such as the signal that can propagate in the LC series circuitsAlso these solutions were used to describe and investigate some oscillations in plasma physics such as oscillations in elec-tronegative plasma with Maxwellian electrons and negative ions

1 Introduction

Early descriptions of many of them date back to at least1892 when the book by Greenhill [1] appeared presentinga variety of such problems a simple pendulum catenariesand a uniform chain that rotates steadily with a constantangular velocity about an axis to which the chain is fixed attwo points Later applications include nonlinear plasmaoscillations [2] Duffing oscillators [3] rigid plates sat-isfying the Johansen yield criterion [4] nonlinear trans-verse vibrations of a plate carrying a concentrated mass[5] a beam supported at an axially oscillating mount [6]doubly periodic cracks subjected to concentrated forces[7] surface waves in a plasma column [8] coupled modesof nonlinear flexural vibrations of a circular ring [9] dual-spin spacecrafts [10] spacecraft motion about slowlyrotating asteroids [11] nonlinear vibration of buckledbeams [12] a nonlinear wave equation [13] deep-waterwaves with two-dimensional surface patterns [14]

oscillations of a body with an orbital tethered system [15]and nonlinear mathematical models of DNA [16 17]Numerical studies of phase spaces stability analysis so-lution by means of finite differences application of theBernoulli wavelet method for estimating a solution oflinear stochastic integral equations existence of periodicsolutions and numerical simulations can be found in[18ndash22] ampe search for new analytical methods that leadto the exact solution of the Helmholtz equations is thevital importance since the developed methods can beapplied to Schrodingerrsquos nonlinear differential equationwhich is known to have different applications in nonlinearoptics plasma physics fluid mechanics and BosendashEin-stein condensates [23 24] As a contribution to the lit-erature in this article we present the exact solution to theHelmholtz Oscillator for the given arbitrary initial con-ditions by means of the Weierstrass elliptic function

In this paper we will derive the exact solution to theHelmholtz oscillator

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 7876413 7 pageshttpsdoiorg10115520207876413

euroq + α + βq + cq2

0

q(0) q0

qprime(0) _q0

(1)

where q equiv q(t) denotes the displacement of the systemqprime(0) equiv ztq|t0 equiv _q0 euroq equiv z2t q β is the natural frequency c is anonlinear system parameter and α is a system parameterindependent of the time

Equation (1) is applied for mathematical modeling inphysics and engineering like general relativity betatronoscillations vibrations of shells vibrations of the acousticallydriven human eardrum and solid-state physics [24ndash28] ampeHelmholtz oscillator can be interpreted as a particle movingin a quadratic potential field and it has also been studied in anonlinear circuit theory

One of the possible interesting interpretations ofequation (1) is given by a simple electrical circuit

2 ThePhysicalModels andHelmholtz Equation

21 Alternation the LC Series Circuit Let us consider al-ternating LC series circuit consisting of a linear inductor anda capacitor of two terminals as a dipole as shown in Figure 1Also it is known that a functional relationship between theelectric charge q the capacitor voltage u equiv u(q) and thetime charging t has the form as follows

f(q u t) 0 (2)

ampe relationship between the charge q of the nonlinearcapacitor and the voltage drop across it may be approxi-mated by the following quadratic equation [29]

uc sq + aq2 (3)

where uc gives the potential across the plates of the nonlinearcapacitor and s and a are constants related to the capacitor

Now Kirchhoffrsquos voltage law for the LC series circuitcould be written as follows

Liprime + sq + aq2

VE (4)

where iprime equiv zti L is the inductor inductance measured byHenry unit and VE gives the voltage of the battery which isconstant ampe relation between the current and charge readsi dqdt and by rearrange equation (4) the Helmholtzequation is obtained as


0 (5)

where q equiv q(t) euroq equiv z2t q β sL 1(LC) ω20 where ω0

1LC

radicrepresents the resonant angular frequency and

c aL 1(Cq0L) where C donates the capacitance of thecapacitor in farad and q0 represent the initial value of chargeon the capacitor plates which has minimum or zero value forthe charging capacitor and for discharge it has maximumvalues ampe LC circuit is considered as an oscillating circuitwhich stores energy to oscillate at the resonant frequencyf0 1(2π

LC

radic) of the circuit where f0 could be calculated

from the resonance condition inductor impedance(2πf0L) capacitor impedance (1(2πf0C))

22 Electronegative Plasmas and the KdVndashHelmholtzEquation Let us consider the propagation of electrostaticnonlinear ion-acoustic structures in a collisionless electro-negative plasma consisting of thermal particles (includingMaxwellian electrons and light negative ions) and fluid coldpositive ions ampe dynamics of nonlinear electrostaticstructures are governed by the following dimensionless fluidequations [30ndash33]

ztn + zx(nu) 0 (6)

ztu + uzxu + zxϕ 0 (7)

z2xϕ μe exp[ϕ] + μn exp σnϕ1113858 1113859 minus n (8)

where n and u are the normalized positive ion numberdensity and fluid velocity respectively and ϕ gives thenormalized electrostatic potential [30] Here σn TeTn

gives the temperature ratio of electron-to-negative ionμe n(0)

e n(0) gives the electron concentration andμn n(0)

n n(0) refers to the negative concentration wheren(0) n(0)

e and n(0)n are the unperturbed equilibrium number

densities of the positive ion electron and negative ionrespectively Accordingly the neutrality condition at equi-librium reads μe + μn 1 with μn αn(1 + αn) andμe 1(1 + αn) where αn n(0)

n n(0)e

By applying the RPT the independent variables are givenby ζ ε12(x minus Vpht) and τ ε32t and the dependentquantities are expanded as n 1 + εn(1) + ε2n(2) + middot middot middotu εu(1) + ε2u(2) + middot middot middot and ϕ εϕ(1) + ε2ϕ(2) + middot middot middot HereVph is the normalized wave phase velocity and ε is a real andsmall parameter (ε≪ 1) Inserting stretching and expansionvalues of the independent and dependent quantities intoequations (6)ndash(8) we get a system of reduced equations withdifferent orders of ε ampe first-order dependent quantities(n(1) u(1) ϕ(1)) could be obtained from solving the system ofthe lowest orders in ε as n(1) ϕ(1)V2

ph u(1) ϕ(1)Vph andVph

1S1

1113968 By solving the system of reduced equations to

the next order in ε the following KdV equation is obtained

zτφ + K1φzζφ + K2z3ζφ 0 (9)

where the coefficient of the nonlinear termK1 V3

ph(3V4ph minus 2S2)2 and the coefficient of the disper-

sion term K2 V3ph2 where S1 (μe + μnσ) and

L

Vinsine

+

ndashC

Figure 1 ampe LC series circuit

2 Mathematical Problems in Engineering

S2 (μe + μnσ2)2 Note that for K1 gt 0(K1 lt 0) the posi-tive (negative) pulses can exist and propagate in the presentplasma model ampe sign and the values of K1 are related tothe relevant plasma parameters

Using the travelling wave transform ξ (ζ + λτ) andφ(ζ τ) q(ξ) into equation (9) and integrating the obtainedresult over ξ we get

qPrime + α + βq + cq2

0 (10)

where q equiv q(ξ) _q equiv zξq euroq equiv z2ξq and λ represents thenormalized velocity of the moving frame Here α ckK2where ck is the integrating constant β λK2 andc K1(2K2)

qPrime + αq + βq2

0 (11)

3 Exact Solution

In this section we are going to solve in general theequation


0

q(0) q0

qprime(0) _q0

(12)

To do that the following ansatz is suggested

q(t) A +B

1 + Cweierp t minus t0 g2 g3( 1113857 (13)

where BCne 0 weierp(t) stands for the Weierstrass ellipticfunction ampis function satisfies the following ordinarydifferential equation (ODE)

weierpprime F g2 g3( 11138572

4weierp3 F g2 g3( 1113857 minus g1weierp F g2 g3( 1113857 minus g3

(14)

weierpPrime F g2 g3( 1113857 minusg2

2+ 6weierp2 F g2 g3( 1113857 (15)

Inserting ansatz (13) into (12) and making use of (15)gives R(t) 0 where

R(t) C2

A2Cc + ACβ + 2B + Cα1113872 1113873weierp3 t minus t0 g2 g3( 1113857

+ C 3A2Cc + 2ABCc + 3ACβ + BCβ minus 6B + 3Cα1113872 1113873

weierp2 t minus t0 g2 g3( 1113857

+12

C 6A2c + 8ABc + 6Aβ + 2B

2c minus 3BCg21113872

+ 4Bβ + 6α1113857weierp t minus t0 g2 g3( 1113857

+ A2c + 2ABc + Aβ + B

2c minus 2BC

2g3 +

12

BCg2 + Bβ + α

(16)

Equating the coefficients of weierpj(t minus t0 g2 g3) to zerogives an algebraic system and by solving it we obtain

B minus6

2Ac + βA2c + Aβ + α1113872 1113873

C 12

2Ac + β

g2 112

β2 minus 4αc1113872 1113873

g3 1216

(2Ac + β) minus 2A2c2

minus 2Aβc minus 6αc + β21113872 1113873

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(17)

and thus we get

q(t) A minus6(A(Ac + β) + α)

12weierp t minus t0 (112) β2 minus 4αc1113872 1113873 (1216)(β + 2Ac) β2 minus 2Acβ minus 2c cA2

+ 3α1113872 11138731113872 11138731113872 1113873 + 2Ac + β (18)

Now for determining the values of t0 and A the initialconditions (q(0) q0 and qprime(0) _q0) must be applied inaddition to the following formula

weierp z + y g2 g3( 1113857 14weierpprime z g2 g3( 1113857 minus weierpprime y g2 g3( 1113857

weierp z g2 g3( 1113857 minus weierp y g2 g3( 11138571113888 1113889

2

minus weierp z g2 g3( 1113857 minus weierp y g2 g3( 1113857

(19)

where z t and y minus t0 for our case According to thisrelation we get

weierp t minus t0 g2 g3( 1113857 14weierpprime t g2 g3( 1113857 minus c1

weierp t g2 g3( 1113857 minus c01113888 1113889

2

minus weierp t g2 g3( 1113857 minus c0

(20)

where c0 weierp(minus t0 g2 g3) and c1 weierpprime(minus t0 g2 g3)ampe initial conditions (q(0) q0 and qprime(0) _q0) give

c0 minusminus q0(2A + B) + A(A + B) + B _q0 + q

20

C A minus q0( 11138572

c1 minusB _q0

C A minus q0( 11138572

(21)

where the values of B and C are defined in equation (17)

Mathematical Problems in Engineering 3

Finally the value of A can be estimated from thecondition

euroq (0) + α + βq(0) + cq2(0) 0 (22)

which gives2 2cA

3+ 3βA

2+ 6αA minus 6αq0 minus 3βq

20 minus 2cq

30 minus 3q

201113872 1113873

3 A minus q0( 1113857 0

(23)

It is clear that A must obey the cubic equation

cA3

+ 3βA2

+ 6αA minus 6αq0 + 3βq20 + 2cq

30 + 3q

201113872 1113873 0

(24)

We will choose the first root of equation (24) ie

A 12c

Δ27

+ 6c2 2αq0 + βq

20 + q

201113872 1113873 + 6αβc minus β3 + 4c

3q30

3

1113970

+β2 minus 4αc

(Δ27) + 6c

2 2αq0 + βq20 + q

201113872 1113873 + 6αβc minus β3 + 4c

3q30

31113969 minus β

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(25)

with

Δ

729 β + 2cq0( 1113857 minus 6αc + β2 minus 2cq0 β + cq0( 11138571113872 1113873 minus 6c2q201113872 1113873

2+ 36αc minus 9β21113872 1113873

31113970

(26)

For _q0 0 ie q(0) q0 and qprime(0) 0 the followingsolution is obtained

q(t) q0 minus6 α + βq0 + cq

201113872 1113873

β + 2cq0 + 12weierp q (112) β2 minus 4αc1113872 1113873 (1216) β + 2cq0( 1113857 β2 minus 2cq0β minus 2c2q20 minus 6αc1113872 11138731113872 1113873

(27)

In the case when _q0 ne 0 the solution to initial valueproblem (1) is given by

q(t) A +B

1 + C (14) c1 minus weierpprime t g2 g3( 1113857( 1113857 c0 minus weierp t g2 g3( 1113857( 1113857( 11138572

minus c0 minus c1 minus weierp t g2 g3( 11138571113872 1113873 (28)

or in a more compact form

q(t) A +B


where t0 minus weierpminus 1(c0 g2 g3)Expression weierpminus 1(c0 g2 g3) stands for the inverse of the

Weierstrass elliptic function which is defined as

weierpminus 1x g2 g3( 1113857 1113946

x

infin

1

4ζ3 minus g2ζ minus g3

1113969 dζ (30)

for real x and Re(4ζ3 minus g2ζ minus g3)gt 0

ampe obtained solution (29) is periodic with period

T plusmn21113946infin

e1

1

4z3

minus g2z minus g3

1113969 dz (31)

where e1 is the greatest real root of the cubic4z

3minus g2z minus g3 0 (32)

Taking (30) into account the period T in (29) may alsobe expressed as

T plusmn2weierpminus 1e1 g2 g3( 1113857 (33)

Remark 1 ampe Weierstrass elliptic function weierp(z g2 g3) isrelated to the Jacobian elliptic functions sn cd and dn as follows


dn(ω

radiczm)

2 1 minus

3mω3weierp z (43) m

2minus m + 11113872 1113873ω2

(427)(m minus 2)(m + 1)(2m minus 1)ω31113872 1113873

(34)

weierp z g2 g3( 1113857 minusmω

dn(ω

radicz | m)

2minus 1

minus13

(m + 1)ω

ω

sn(ω

radicz | m)

2 minus13

(m + 1)ω

ω

1 minus cn(ω

radicz | m)

2 minus13

(m + 1)ω

(35)

where

16ω6minus 24ω4

g2 + 9ω2g22 minus g

32 + 27g

23 0 (36)

4 g32 minus 27g

231113872 1113873ω6

minus 12 g32 minus 27g

231113872 1113873ω5

minus 3 g32 + 216g

231113872 1113873ω4

+ 2 13g32 + 378g

231113872 1113873ω3

minus 3 g32 + 216g

231113872 1113873ω2


231113872 1113873ω + 4g

32 minus 108g

23 0

(37)

In view of (35) the fundamental period of the Weier-strass function weierp(z g2 g3) may also be expressed in thefollowing form

T 2K(m)

ω

radic (38)

Further properties of the Weierstrass function withapplications in quantum theory of Lame potential may befound in [26]

4 Results and Discussions

Let us consider a nonlinear LC series circuit with L 1HC 01F qo 1C and let the circuit without externalbattery ie VE 0(α 0) ampus the exact solution to theinitial value problem

euroq + 10q + q2

0

q(0) q0 1

qprime(0) _q0 0

(39)

according to formula (27) is given by

q(t) 1 minus11

2(1 + weierp(t (253) (133))) (40)

ampis solution is periodic and to find its period accordingto equation (32) the following cubic equation is solved

4z3

minus253

z minus133

0 (41)

which is given the following three real roots

z1 minus 1

z2 3 + 4

3

radic

6

z3 3 minus 4

3

radic

6

(42)

ampe greatest root is e1 z2Using formula (31) we obtain

T 1113946infin

z2

1

4z3

minus (253)z minus (133)

1113969 dz

2211

(9 + 43

radic)

1113970

Re K111

(43 + 243

radic)1113874 11138751113874 1113875

(43)

Also we can use equations (34)ndash(38) to obtain

q(t) minus 4(2 +3

radic) +(9 + 4

3

radic)dn

middot

32

+23

radic

1113971

t111

(43 minus 243

radic)

1113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

2

(44)

and then the period of the solution is given by

T 2K((111)(43 minus 24

3

radic))

(32) +(2

3

radic)

1113968 (45)

From equation (43) and (45) the period T is obtained as

T 2211

(9 + 43

radic)

1113970

Re K111

(43 + 243

radic)1113874 11138751113874 1113875

2K((111)(43 minus 24

3

radic))

(32) +(2

3

radic)

1113968

asymp 199590288250676993

(46)

In Figure 2 the analytical solution (40) is compared withthe RungendashKutta (RK) numerical solution and the agree-ment between the two solutions is found to be very good Ifthe initial current value of the LC circuit is not equal to zeroie i qprime(0) _q0 ne 0 but has a specified value _q0 03A inthis case we can consider solution (28) or (29) instead of thefirst one (27) It is observed that the results obtained by the


exact solution are in an excellent agreement with the RKnumerical solution as shown in Figure 3 For the plasmaapplication our analyses are based on the experimental

observations data where electron and negative ion tem-peratures are respectively given by Te asymp 069 eV andTn asymp (006 plusmn 002) eV and ne 38 times 109 cmminus 3 gives theelectron number density According to these data we getσn 87 115 and 1725 and the negative ions concentration0le μn le 1 [26 27 30] Using these data we get(β c) (4433 044) and by plotting these data accordingto solution (27) we note from Figure 4 the excellentmatching between our solution and the RK numericalsolution

5 Conclusions

ampe quadratic nonlinear Helmholtz differential equation issolved analytically for any arbitrary initial conditionsthrough theWeierstrass elliptic function Accordingly somenew analytical solutions are obtained for the first time usingansatz (12) ampe proposed methodology is of great impor-tance in solving several physics problems in plasma physicsand electronic circuits For instance our obtained solutionsare devoted for describing the dynamics of nonlinear os-cillations that propagate in electronegative complex plasmasMoreover our solutions are applied for studying thecharacteristic behavior of signal oscillations in the LC seriescircuits

Data Availability

No data were used to support this research

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

ampe authors thank the University Francisco Jose de Caldasfor the support to carry out this work Also the authorsthank Associate Prof SA El-Tantawy the chief of ResearchCenter for Physics (RCP) Faculty of Science and Arts Al-Baha University Saudi Arabia and Faculty of Science PortSaid University Egypt for fruitful discussions carefulreading and improving their manuscript very well

References

[1] A G Greenhill e Applications of Elliptic FunctionsMacmillan London UK 1892

[2] T OrsquoNeil ldquoCollisionless damping of nonlinear plasma os-cillationsrdquo Physics of Fluids vol 8 pp 2255ndash2262 1965

[3] P G D Barkham and A C Soudack ldquoAn extension to themethod of Kryloff and Bogoliuboffdaggerrdquo International Journal ofControl vol 10 no 4 pp 377ndash392 1969

[4] I F Collins ldquoOn the theory of rigidperfectly plastic platesunder uniformly distributed loadsrdquo Acta Mechanica vol 18no 3-4 pp 233ndash254 1973

[5] B M Karmakar ldquoNonlinear vibrations of orthotropic platescarrying concentrated massrdquo Journal of Engineering for In-dustry vol 100 no 2 pp 293-294 1978

[6] J Zajaczkowski ldquoDestabilizing effect of Coulomb friction onvibration of a beam supported at an axially oscillating mountrdquo

10

05

00

ndash05

ndash10

Amplitu

de

0 2 4 6 8Time

Figure 2 (Color online) A comparison between the analyticalsolution (40) (dashed curve) and the RungendashKutta (RK) numericalsolution (dotted curve) for the LC series circuit with L 1HC 01F q0 1C and _q0 0

10

05

00

ndash05

ndash10

Amplitu

de

0 2 4 6 8 10Time

Figure 3 (Color online) A comparison between the analyticalsolution (28) (dashed curve) and the RungendashKutta (RK) numericalsolution (dotted curve) for the LC series circuit with L 1HC 01F q0 1 and _q0 03A

Amplitu

de

Time

05

00

ndash05

ndash100 5 10 15 20

Figure 4 (Color online) A comparison between the analyticalsolution (28) (dashed curve) and the RungendashKutta (RK) numericalsolution (dotted curve) for the electronegative plasma dataαn 01 σn 87 q0 051 and _q0 01


Journal of Sound and Vibration vol 79 no 4 pp 575ndash5801981

[7] S S Chang ldquoampe general solutions of the doubly periodiccracksrdquo Engineering Fracture Mechanics vol 18 no 4pp 887ndash893 1983

[8] D Grozev A Shivarova and A D Boardman ldquoEnvelopesolitons of surface waves in a plasma columnrdquo Journal ofPlasma Physics vol 38 no 3 pp 427ndash437 1987

[9] A I Manevich ldquoInteraction of coupled modes accompanyingnon-linear flexural vibrations of a circular ringrdquo Journal ofApplied Mathematics and Mechanics vol 58 no 6pp 1061ndash1068 1994

[10] R H Rand R J Kinsey and D L Mingori ldquoDynamics ofspinup through resonancerdquo International Journal of Non-Linear Mechanics vol 27 no 3 pp 489ndash502 1992

[11] W Hu and D J Scheeres ldquoSpacecraft motion about slowlyrotating asteroidsrdquo Advances in the Astronautical Sciencesvol 105 pp 839ndash848 2000

[12] W Lestari and S Hanagud ldquoNonlinear vibration of buckledbeams some exact solutionsrdquo International Journal of Solidsand Structures vol 38 no 26-27 pp 4741ndash4757 2001

[13] S Liu Z Fu S Liu and Q Zhao ldquoJacobi elliptic functionexpansion method and periodic wave solutions of nonlinearwave equationsrdquo Physics Letters A vol 289 no 1-2 pp 69ndash742001

[14] J L Hammack and D M Henderson ldquoExperiments on deep-water waves with two-dimensional surface patternsrdquo Journalof Offshore Mechanics and Arctic Engineering vol 125 no 1pp 48ndash53 2003

[15] V S Aslanov ldquoampe oscillations of a body with an orbitaltethered systemrdquo Journal of Applied Mathematics and Me-chanics vol 71 no 6 pp 926ndash932 2007

[16] A H Salas and J E Castillo ldquoExact solutions for a nonlinearmodelrdquo Applied Mathematics and Computations vol 217no 4 pp 1646ndash1651 2010

[17] A H Salas and J E Castillo ldquoLa ecuacion Seno-Gordonperturbada en la Dinamica no lineal del ADNrdquo RevistaMexicana de Fisica vol 58 pp 481ndash487 2012

[18] P J Jolmes ldquoA nonlinear oscillator with a strange attractorrdquoProceedings of the Royal Society vol 292 pp 419ndash448 1979

[19] S A Khuri and S Xie ldquoOn the numerical verification of theasymptotic expansion of duffingrsquos equationrdquo InternationalJournal of Computer Mathematics vol 72 no 3 pp 325ndash3301999

[20] F Mirzaee and N Samadyar ldquoCombination of nite differencemethod and meshless method based on radial basis functionsto solve fractional stochastic advection diffusion equationsrdquoEngineering with Computers 2019

[21] N Samadyar and F Mirzaee ldquoNumerical solution of two-dimensional weakly singular stochastic integral equations onnon-rectangular domains via radial basis functionsrdquo Engi-neering Analysis with Boundary Elements vol 101 pp 27ndash362019

[22] F Mirzaee and N Samadyar ldquoNumerical solution based ontwo-dimensional orthonormal Bernstein polynomials forsolving some classes of two-dimensional nonlinear integralequations of fractional orderrdquo Applied Mathematics andComputation vol 344-345 pp 191ndash203 2019

[23] Y Liu and G-R Li ldquoMatter wave soliton solutions of thecubic-quintic nonlinear Schrodinger equation with ananharmonic potentialrdquo Applied Mathematics and Computa-tion vol 219 no 9 pp 4847ndash4852 2013

[24] Y Geng J Li and L Zhang ldquoExact explicit traveling wavesolutions for two nonlinear Schrodinger type equationsrdquo

Applied Mathematics and Computation vol 217 no 4pp 1509ndash1521 2010

[25] N Nayfeth and D T Mook Non-linear Oscillations JohnWiley New York NY USA 1973

[26] J A Almendral and M A F Sanjun ldquoIntegrability andsymmetries for the Helmholtz oscillator with frictionrdquoJournal of Physics A Mathematical and General vol 36 no 3p 695 2003

[27] S Morfa and J C Comte ldquoA nonlinear oscilators netwokdevoted to image processingrdquo International Journal of Bi-furcation and Chaos vol 14 no 4 pp 1385ndash1394 2009

[28] A H Salas and J E Castillo ldquoExact solutions to cubic Duffingequation for a nonlinear electrical circuitrdquo Vision Electronica-Algo mas que un estado solido vol 8 no 1 2014

[29] E Gluskin ldquoA nonlinear resistor and nonlinear inductorusing a nonlinear capacitorrdquo Journal of the Franklin Institutevol 336 no 7 pp 1035ndash1047 1999

[30] S A El-Tantawy ldquoNonlinear dynamics of soliton collisions inelectronegative plasmas the phase shifts of the planar KdV-and mkdV-soliton collisionsrdquo Chaos Solitons amp Fractalsvol 93 pp 162ndash168 2016

[31] S A El-Tantawy and T Aboelenen ldquoSimulation study ofplanar and nonplanar super rogue waves in an electronegativeplasma local discontinuous Galerkin methodrdquo Physics ofPlasmas vol 24 Article ID 052118 2017

[32] S A El-Tantawy A M Wazwaz and S Ali Shan ldquoOn thenonlinear dynamics of breathers waves in electronegativeplasmas with Maxwellian negative ionsrdquo Physics of Plasmasvol 24 Article ID 022105 2017

[33] S A El-Tantawy ldquoTarek Aboelenen and Sherif M E Ismaeellocal discontinuous Galerkin method for modeling thenonplanar structures (solitons and shocks) in an electro-negative plasmardquo Physics of Plasmas vol 26 Article ID022115 2019



0

q(0) q0

qprime(0) _q0

(1)

where q equiv q(t) denotes the displacement of the systemqprime(0) equiv ztq|t0 equiv _q0 euroq equiv z2t q β is the natural frequency c is anonlinear system parameter and α is a system parameterindependent of the time

Equation (1) is applied for mathematical modeling inphysics and engineering like general relativity betatronoscillations vibrations of shells vibrations of the acousticallydriven human eardrum and solid-state physics [24ndash28] ampeHelmholtz oscillator can be interpreted as a particle movingin a quadratic potential field and it has also been studied in anonlinear circuit theory

One of the possible interesting interpretations ofequation (1) is given by a simple electrical circuit

2 ThePhysicalModels andHelmholtz Equation

21 Alternation the LC Series Circuit Let us consider al-ternating LC series circuit consisting of a linear inductor anda capacitor of two terminals as a dipole as shown in Figure 1Also it is known that a functional relationship between theelectric charge q the capacitor voltage u equiv u(q) and thetime charging t has the form as follows

f(q u t) 0 (2)

ampe relationship between the charge q of the nonlinearcapacitor and the voltage drop across it may be approxi-mated by the following quadratic equation [29]

uc sq + aq2 (3)

where uc gives the potential across the plates of the nonlinearcapacitor and s and a are constants related to the capacitor

Now Kirchhoffrsquos voltage law for the LC series circuitcould be written as follows

Liprime + sq + aq2

VE (4)

where iprime equiv zti L is the inductor inductance measured byHenry unit and VE gives the voltage of the battery which isconstant ampe relation between the current and charge readsi dqdt and by rearrange equation (4) the Helmholtzequation is obtained as


0 (5)

where q equiv q(t) euroq equiv z2t q β sL 1(LC) ω20 where ω0

1LC

radicrepresents the resonant angular frequency and

c aL 1(Cq0L) where C donates the capacitance of thecapacitor in farad and q0 represent the initial value of chargeon the capacitor plates which has minimum or zero value forthe charging capacitor and for discharge it has maximumvalues ampe LC circuit is considered as an oscillating circuitwhich stores energy to oscillate at the resonant frequencyf0 1(2π

LC

radic) of the circuit where f0 could be calculated

from the resonance condition inductor impedance(2πf0L) capacitor impedance (1(2πf0C))

22 Electronegative Plasmas and the KdVndashHelmholtzEquation Let us consider the propagation of electrostaticnonlinear ion-acoustic structures in a collisionless electro-negative plasma consisting of thermal particles (includingMaxwellian electrons and light negative ions) and fluid coldpositive ions ampe dynamics of nonlinear electrostaticstructures are governed by the following dimensionless fluidequations [30ndash33]

ztn + zx(nu) 0 (6)

ztu + uzxu + zxϕ 0 (7)

z2xϕ μe exp[ϕ] + μn exp σnϕ1113858 1113859 minus n (8)

where n and u are the normalized positive ion numberdensity and fluid velocity respectively and ϕ gives thenormalized electrostatic potential [30] Here σn TeTn

gives the temperature ratio of electron-to-negative ionμe n(0)

e n(0) gives the electron concentration andμn n(0)

n n(0) refers to the negative concentration wheren(0) n(0)

e and n(0)n are the unperturbed equilibrium number

densities of the positive ion electron and negative ionrespectively Accordingly the neutrality condition at equi-librium reads μe + μn 1 with μn αn(1 + αn) andμe 1(1 + αn) where αn n(0)

n n(0)e

By applying the RPT the independent variables are givenby ζ ε12(x minus Vpht) and τ ε32t and the dependentquantities are expanded as n 1 + εn(1) + ε2n(2) + middot middot middotu εu(1) + ε2u(2) + middot middot middot and ϕ εϕ(1) + ε2ϕ(2) + middot middot middot HereVph is the normalized wave phase velocity and ε is a real andsmall parameter (ε≪ 1) Inserting stretching and expansionvalues of the independent and dependent quantities intoequations (6)ndash(8) we get a system of reduced equations withdifferent orders of ε ampe first-order dependent quantities(n(1) u(1) ϕ(1)) could be obtained from solving the system ofthe lowest orders in ε as n(1) ϕ(1)V2

ph u(1) ϕ(1)Vph andVph

1S1

1113968 By solving the system of reduced equations to

the next order in ε the following KdV equation is obtained

zτφ + K1φzζφ + K2z3ζφ 0 (9)

where the coefficient of the nonlinear termK1 V3

ph(3V4ph minus 2S2)2 and the coefficient of the disper-

sion term K2 V3ph2 where S1 (μe + μnσ) and

L

Vinsine

+

ndashC

Figure 1 ampe LC series circuit





0 (10)


qPrime + αq + βq2

0 (11)

3 Exact Solution



0

q(0) q0

qprime(0) _q0

(12)


q(t) A +B





(14)


2+ 6weierp2 F g2 g3( 1113857 (15)


R(t) C2




+12

C 6A2c + 8ABc + 6Aβ + 2B



+ A2c + 2ABc + Aβ + B

2c minus 2BC

2g3 +

12

BCg2 + Bβ + α

(16)


B minus6

2Ac + βA2c + Aβ + α1113872 1113873

C 12

2Ac + β

g2 112

β2 minus 4αc1113872 1113873

g3 1216



⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(17)

and thus we get



+ 3α1113872 11138731113872 11138731113872 1113873 + 2Ac + β (18)




2


(19)




2


(20)



20

C A minus q0( 11138572

c1 minusB _q0

C A minus q0( 11138572

(21)




euroq (0) + α + βq(0) + cq2(0) 0 (22)

which gives2 2cA

3+ 3βA


20 minus 2cq

30 minus 3q

201113872 1113873

3 A minus q0( 1113857 0

(23)


cA3

+ 3βA2


30 + 3q

201113872 1113873 0

(24)


A 12c

Δ27

+ 6c2 2αq0 + βq

20 + q

201113872 1113873 + 6αβc minus β3 + 4c

3q30

3

1113970

+β2 minus 4αc

(Δ27) + 6c

2 2αq0 + βq20 + q

201113872 1113873 + 6αβc minus β3 + 4c

3q30

31113969 minus β

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(25)

with

Δ


2+ 36αc minus 9β21113872 1113873

31113970

(26)



201113872 1113873


(27)


q(t) A +B




q(t) A +B




weierpminus 1x g2 g3( 1113857 1113946

x

infin

1


1113969 dζ (30)




e1

1

4z3

minus g2z minus g3

1113969 dz (31)







dn(ω

radiczm)

2 1 minus


2minus m + 11113872 1113873ω2

(427)(m minus 2)(m + 1)(2m minus 1)ω31113872 1113873

(34)


dn(ω

radicz | m)

2minus 1

minus13

(m + 1)ω

ω

sn(ω

radicz | m)

2 minus13

(m + 1)ω

ω

1 minus cn(ω

radicz | m)

2 minus13

(m + 1)ω

(35)

where

16ω6minus 24ω4


32 + 27g

23 0 (36)

4 g32 minus 27g

231113872 1113873ω6


231113872 1113873ω5

minus 3 g32 + 216g

231113872 1113873ω4

+ 2 13g32 + 378g

231113872 1113873ω3

minus 3 g32 + 216g

231113872 1113873ω2


231113872 1113873ω + 4g

32 minus 108g

23 0

(37)


T 2K(m)

ω

radic (38)




euroq + 10q + q2

0

q(0) q0 1

qprime(0) _q0 0

(39)


q(t) 1 minus11

2(1 + weierp(t (253) (133))) (40)


4z3

minus253

z minus133

0 (41)


z1 minus 1

z2 3 + 4

3

radic

6

z3 3 minus 4

3

radic

6

(42)


T 1113946infin

z2

1

4z3


1113969 dz

2211

(9 + 43

radic)

1113970

Re K111

(43 + 243

radic)1113874 11138751113874 1113875

(43)


q(t) minus 4(2 +3

radic) +(9 + 4

3

radic)dn

middot

32

+23

radic

1113971

t111

(43 minus 243

radic)

1113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

2

(44)


T 2K((111)(43 minus 24

3

radic))

(32) +(2

3

radic)

1113968 (45)


T 2211

(9 + 43

radic)

1113970

Re K111

(43 + 243

radic)1113874 11138751113874 1113875

2K((111)(43 minus 24

3

radic))

(32) +(2

3

radic)

1113968

asymp 199590288250676993

(46)





5 Conclusions


Data Availability




Acknowledgments


References







10

05

00

ndash05

ndash10

Amplitu

de

0 2 4 6 8Time


10

05

00

ndash05

ndash10

Amplitu

de

0 2 4 6 8 10Time


Amplitu

de

Time

05

00

ndash05

ndash100 5 10 15 20




































0 (10)


qPrime + αq + βq2

0 (11)

3 Exact Solution



0

q(0) q0

qprime(0) _q0

(12)


q(t) A +B





(14)


2+ 6weierp2 F g2 g3( 1113857 (15)


R(t) C2




+12

C 6A2c + 8ABc + 6Aβ + 2B



+ A2c + 2ABc + Aβ + B

2c minus 2BC

2g3 +

12

BCg2 + Bβ + α

(16)


B minus6

2Ac + βA2c + Aβ + α1113872 1113873

C 12

2Ac + β

g2 112

β2 minus 4αc1113872 1113873

g3 1216



⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(17)

and thus we get



+ 3α1113872 11138731113872 11138731113872 1113873 + 2Ac + β (18)




2


(19)




2


(20)



20

C A minus q0( 11138572

c1 minusB _q0

C A minus q0( 11138572

(21)




euroq (0) + α + βq(0) + cq2(0) 0 (22)

which gives2 2cA

3+ 3βA


20 minus 2cq

30 minus 3q

201113872 1113873

3 A minus q0( 1113857 0

(23)


cA3

+ 3βA2


30 + 3q

201113872 1113873 0

(24)


A 12c

Δ27

+ 6c2 2αq0 + βq

20 + q

201113872 1113873 + 6αβc minus β3 + 4c

3q30

3

1113970

+β2 minus 4αc

(Δ27) + 6c

2 2αq0 + βq20 + q

201113872 1113873 + 6αβc minus β3 + 4c

3q30

31113969 minus β

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(25)

with

Δ


2+ 36αc minus 9β21113872 1113873

31113970

(26)



201113872 1113873


(27)


q(t) A +B




q(t) A +B




weierpminus 1x g2 g3( 1113857 1113946

x

infin

1


1113969 dζ (30)




e1

1

4z3

minus g2z minus g3

1113969 dz (31)







dn(ω

radiczm)

2 1 minus


2minus m + 11113872 1113873ω2

(427)(m minus 2)(m + 1)(2m minus 1)ω31113872 1113873

(34)


dn(ω

radicz | m)

2minus 1

minus13

(m + 1)ω

ω

sn(ω

radicz | m)

2 minus13

(m + 1)ω

ω

1 minus cn(ω

radicz | m)

2 minus13

(m + 1)ω

(35)

where

16ω6minus 24ω4


32 + 27g

23 0 (36)

4 g32 minus 27g

231113872 1113873ω6


231113872 1113873ω5

minus 3 g32 + 216g

231113872 1113873ω4

+ 2 13g32 + 378g

231113872 1113873ω3

minus 3 g32 + 216g

231113872 1113873ω2


231113872 1113873ω + 4g

32 minus 108g

23 0

(37)


T 2K(m)

ω

radic (38)




euroq + 10q + q2

0

q(0) q0 1

qprime(0) _q0 0

(39)


q(t) 1 minus11

2(1 + weierp(t (253) (133))) (40)


4z3

minus253

z minus133

0 (41)


z1 minus 1

z2 3 + 4

3

radic

6

z3 3 minus 4

3

radic

6

(42)


T 1113946infin

z2

1

4z3


1113969 dz

2211

(9 + 43

radic)

1113970

Re K111

(43 + 243

radic)1113874 11138751113874 1113875

(43)


q(t) minus 4(2 +3

radic) +(9 + 4

3

radic)dn

middot

32

+23

radic

1113971

t111

(43 minus 243

radic)

1113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

2

(44)


T 2K((111)(43 minus 24

3

radic))

(32) +(2

3

radic)

1113968 (45)


T 2211

(9 + 43

radic)

1113970

Re K111

(43 + 243

radic)1113874 11138751113874 1113875

2K((111)(43 minus 24

3

radic))

(32) +(2

3

radic)

1113968

asymp 199590288250676993

(46)





5 Conclusions


Data Availability




Acknowledgments


References







10

05

00

ndash05

ndash10

Amplitu

de

0 2 4 6 8Time


10

05

00

ndash05

ndash10

Amplitu

de

0 2 4 6 8 10Time


Amplitu

de

Time

05

00

ndash05

ndash100 5 10 15 20


































euroq (0) + α + βq(0) + cq2(0) 0 (22)

which gives2 2cA

3+ 3βA


20 minus 2cq

30 minus 3q

201113872 1113873

3 A minus q0( 1113857 0

(23)


cA3

+ 3βA2


30 + 3q

201113872 1113873 0

(24)


A 12c

Δ27

+ 6c2 2αq0 + βq

20 + q

201113872 1113873 + 6αβc minus β3 + 4c

3q30

3

1113970

+β2 minus 4αc

(Δ27) + 6c

2 2αq0 + βq20 + q

201113872 1113873 + 6αβc minus β3 + 4c

3q30

31113969 minus β

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(25)

with

Δ


2+ 36αc minus 9β21113872 1113873

31113970

(26)



201113872 1113873


(27)


q(t) A +B




q(t) A +B




weierpminus 1x g2 g3( 1113857 1113946

x

infin

1


1113969 dζ (30)




e1

1

4z3

minus g2z minus g3

1113969 dz (31)







dn(ω

radiczm)

2 1 minus


2minus m + 11113872 1113873ω2

(427)(m minus 2)(m + 1)(2m minus 1)ω31113872 1113873

(34)


dn(ω

radicz | m)

2minus 1

minus13

(m + 1)ω

ω

sn(ω

radicz | m)

2 minus13

(m + 1)ω

ω

1 minus cn(ω

radicz | m)

2 minus13

(m + 1)ω

(35)

where

16ω6minus 24ω4


32 + 27g

23 0 (36)

4 g32 minus 27g

231113872 1113873ω6


231113872 1113873ω5

minus 3 g32 + 216g

231113872 1113873ω4

+ 2 13g32 + 378g

231113872 1113873ω3

minus 3 g32 + 216g

231113872 1113873ω2


231113872 1113873ω + 4g

32 minus 108g

23 0

(37)


T 2K(m)

ω

radic (38)




euroq + 10q + q2

0

q(0) q0 1

qprime(0) _q0 0

(39)


q(t) 1 minus11

2(1 + weierp(t (253) (133))) (40)


4z3

minus253

z minus133

0 (41)


z1 minus 1

z2 3 + 4

3

radic

6

z3 3 minus 4

3

radic

6

(42)


T 1113946infin

z2

1

4z3


1113969 dz

2211

(9 + 43

radic)

1113970

Re K111

(43 + 243

radic)1113874 11138751113874 1113875

(43)


q(t) minus 4(2 +3

radic) +(9 + 4

3

radic)dn

middot

32

+23

radic

1113971

t111

(43 minus 243

radic)

1113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

2

(44)


T 2K((111)(43 minus 24

3

radic))

(32) +(2

3

radic)

1113968 (45)


T 2211

(9 + 43

radic)

1113970

Re K111

(43 + 243

radic)1113874 11138751113874 1113875

2K((111)(43 minus 24

3

radic))

(32) +(2

3

radic)

1113968

asymp 199590288250676993

(46)





5 Conclusions


Data Availability




Acknowledgments


References







10

05

00

ndash05

ndash10

Amplitu

de

0 2 4 6 8Time


10

05

00

ndash05

ndash10

Amplitu

de

0 2 4 6 8 10Time


Amplitu

de

Time

05

00

ndash05

ndash100 5 10 15 20

































dn(ω

radiczm)

2 1 minus


2minus m + 11113872 1113873ω2

(427)(m minus 2)(m + 1)(2m minus 1)ω31113872 1113873

(34)


dn(ω

radicz | m)

2minus 1

minus13

(m + 1)ω

ω

sn(ω

radicz | m)

2 minus13

(m + 1)ω

ω

1 minus cn(ω

radicz | m)

2 minus13

(m + 1)ω

(35)

where

16ω6minus 24ω4


32 + 27g

23 0 (36)

4 g32 minus 27g

231113872 1113873ω6


231113872 1113873ω5

minus 3 g32 + 216g

231113872 1113873ω4

+ 2 13g32 + 378g

231113872 1113873ω3

minus 3 g32 + 216g

231113872 1113873ω2


231113872 1113873ω + 4g

32 minus 108g

23 0

(37)


T 2K(m)

ω

radic (38)




euroq + 10q + q2

0

q(0) q0 1

qprime(0) _q0 0

(39)


q(t) 1 minus11

2(1 + weierp(t (253) (133))) (40)


4z3

minus253

z minus133

0 (41)


z1 minus 1

z2 3 + 4

3

radic

6

z3 3 minus 4

3

radic

6

(42)


T 1113946infin

z2

1

4z3


1113969 dz

2211

(9 + 43

radic)

1113970

Re K111

(43 + 243

radic)1113874 11138751113874 1113875

(43)


q(t) minus 4(2 +3

radic) +(9 + 4

3

radic)dn

middot

32

+23

radic

1113971

t111

(43 minus 243

radic)

1113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

2

(44)


T 2K((111)(43 minus 24

3

radic))

(32) +(2

3

radic)

1113968 (45)


T 2211

(9 + 43

radic)

1113970

Re K111

(43 + 243

radic)1113874 11138751113874 1113875

2K((111)(43 minus 24

3

radic))

(32) +(2

3

radic)

1113968

asymp 199590288250676993

(46)





5 Conclusions


Data Availability




Acknowledgments


References







10

05

00

ndash05

ndash10

Amplitu

de

0 2 4 6 8Time


10

05

00

ndash05

ndash10

Amplitu

de

0 2 4 6 8 10Time


Amplitu

de

Time

05

00

ndash05

ndash100 5 10 15 20



































5 Conclusions


Data Availability




Acknowledgments


References







10

05

00

ndash05

ndash10

Amplitu

de

0 2 4 6 8Time


10

05

00

ndash05

ndash10

Amplitu

de

0 2 4 6 8 10Time


Amplitu

de

Time

05

00

ndash05

ndash100 5 10 15 20































































Documents

A New Approach for Solving the Undamped Helmholtz ...downloads.hindawi.com/journals/mpe/2020/7876413.pdfspin spacecrafts [10], spacecraft motion about slowly rotating asteroids [11],