The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations

A new algorithm is presented to find exact traveling wave solutions of differentialdifference equations in terms of tanh functions. For systems with parameters, the algorithm determines the conditions on the parameters so that the equations might admit polynomial solutions in tanh.Examples illustrate the key steps of the algorithm. Parallels are drawn through discussion and example to the tanh-method for partial differential equations.The new algorithm is implemented in Mathematica. The package DDESpecialSolutions.m can be used to automatically compute traveling wave solutions of nonlinear polynomial differential-difference equations. Use of the package, implementation issues, scope, and limitations of the software are addressed. Program summary

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“…Substituting (27) into (37), clearing the denominator and setting coefficients of power terms in T n to zero, gives…”

Section: Example Of a Differential-difference Equationmentioning

confidence: 99%

Symbolic computation of hyperbolic tangent solutions for nonlinear differential–difference equations

Baldwin

Göktaş

Hereman

2004

Computer Physics Communications

149

113

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“…The higher-order KdV equation can be derived for magnetized plasmas by using the reductive perturbation technique. Traveling wave solutions of Kawahara equation and modified Kawahara equation have been studied [9,48,49]. Moreover, the solitary wave solutions of nonlinear equations with arbitrary odd-order derivatives were studied by many authors [47,51].…”

Section: The Modified Fifth Order Kdv Equationmentioning

confidence: 99%

“…Also, we can study the exact solution of the generalized KdV equation (δ = 0) which studied by many authors [22,23,31]. The generalized ZK equation was first derived for describing weakly nonlinear ion-acoustic waves in strongly magnetized lossless plasma in two dimensions and governs the behavior of weakly nonlinear ion-acoustic waves in plasma comprising cold ions and This equation appears in the theory of shallow water waves with surface tension and the theory of magneto-acoustic waves in plasmas [9]. Wazwaz [55] studied soliton solutions of fifth-order KdV equation.…”

Section: The Modified Fifth Order Kdv Equationmentioning

confidence: 99%

“…The tanh-function method is the well known method as a direct selection of the function F(ξ)=tanh( ξ). Recently, many exact solutions expressed by various Jacobi elliptic functions (JEFs) of many NLEEs have been obtained by Jacobi elliptic function expansion method [7][8][9][10], mapping method [11,12], F-expansion method [13], extended F-expansion method [14], the generalized Jacobi elliptic function method [15] and other methods [16][17][18][19][20]. Various exact solutions were obtained by using these methods, including the solitary wave solutions, shock wave solutions and periodic wave solutions.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Exact Solutions Expressible in Hyperbolic and Jacobi Elliptic Functions of Some Important Equations of Ion-Acoustic Waves

Khater¹,

Hassan²

2011

Acoustic Waves - From Microdevices to Helioseismology

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“…Yousefi [17] used Legendre multi wavelet Galerkin method for solving the hyperbolic telegraph equation. Parkes et al [18] used the Jacobi elliptic function expansion method, and proposed double periodic solutions.…”

Section: Introductionmentioning

confidence: 99%

Comparison of Adomian Decomposition Method and Taylor Matrix Method in Solving Different Kinds of Partial Differential Equations

Deni̇z¹,

Bıldık²

2014

IJMO

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Abstract-In this paper, we will present a comparison between the Adomian Decomposition Method (ADM) and Taylor Matrix Method by solving some well-known partial differential equations (PDEs). In order to illustrate the analysis we examined the Telegraph equation, which is considered one of the most significant partial differential equations, describe wave propagation of electric signals in a cable transmission line and Klein-Gordon equation which is encountered in several applied physics fields such as, quantum field theory , fluid dynamics , optoelectronic devices design and numerical analysis. Our study shows that the decomposition method is faster and easy to use from a computational viewpoint.Index Terms-Adomian decomposition method, Klein gordon equation, taylor matrix method, telegraph equation.

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The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations

Cited by 260 publications

References 14 publications

Symbolic computation of hyperbolic tangent solutions for nonlinear differential–difference equations

Symbolic computation of hyperbolic tangent solutions for nonlinear differential–difference equations

Exact Solutions Expressible in Hyperbolic and Jacobi Elliptic Functions of Some Important Equations of Ion-Acoustic Waves

Comparison of Adomian Decomposition Method and Taylor Matrix Method in Solving Different Kinds of Partial Differential Equations

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