Generalized ( G ′/ G ) -Expansion Method For Generalized Fifth Order KdV Equation with Time-Dependent Coefficients

The Korteweg-de Vries (KdV) equation, especially the fractional higher order one, provides a relatively accurate description of motions of long waves in shallow water under gravity and wave propagation in one-dimensional nonlinear lattice. In this article, the generalizedexp⁡(-Φ(ξ))-expansion method is proposed to construct exact solutions of space-time fractional generalized fifth-order KdV equation with Jumarie’s modified Riemann-Liouville derivatives. At the end, three types of exact traveling wave solutions are obtained which indicate that the method is very practical and suitable for solving nonlinear fractional partial differential equations.

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Soliton solutions to the time-dependent coupled KdV–Burgers’ equation

Advances in Difference Equations ◽

10.1186/s13662-019-2429-1 ◽

2019 ◽

Vol 2019 (1) ◽

Author(s):

Aisha Alqahtani ◽

Vikas Kumar

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Burgers Equation ◽

Kdv Equation ◽

Soliton Solutions ◽

Rational Functions ◽

Expansion Method ◽

Time Dependent ◽

Nonlinear Ordinary Differential Equations ◽

Similarity Transformations

AbstractIn this article, the authors apply the Lie symmetry approach and the modified $( G'/G )$(G′/G)-expansion method for seeking the solutions of time-dependent coupled KdV–Burgers equation. Using suitable similarity transformations, the time-dependent coupled KdV–Burgers equation is reduced to a system of nonlinear ordinary differential equations. Further, the reduced system of nonlinear ordinary differential equations for the coupled KdV equation is solved with the help of the modified $( G'/G )$(G′/G)-expansion method to obtain soliton solutions which are expressed by hyperbolic functions, trigonometric functions, and rational functions.

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Approximate Solutions of Generalized Fifth-Order Korteweg-De Vries (KdV) Equation by the Standard Truncated Expansion Method

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.201-202.246 ◽

2012 ◽

Vol 201-202 ◽

pp. 246-250

Author(s):

Jiang Long Wu ◽

Wei Rong Yang

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Kdv Equation ◽

Nonlinear Partial Differential Equations ◽

Expansion Method ◽

Approximate Solutions ◽

Partial Differential ◽

Special Cases ◽

De Vries ◽

Fifth Order

It is difficult to obtain exact solutions of the nonlinear partial differential equations (PDEs) due to the complexity and nonlinearity, especially for non-integrable systems. In this case, some reasonable approximations of real physics are considered, by means of the standard truncated expansion approach to solve real nonlinear system is proposed. In this paper, a simple standard truncated expansion approach with a quite universal pseudopotential is used for generalized fifth-order Korteweg-de Vries (KdV) equation, we can get two kinds of approximate solutions of the above equation, in some special cases, the approximate solutions may become exact. The same idea can also used to find approximate solutions of other well known nonlinear equations. We find a quite universal expansion approach which is valid for various nonlinear partial differential equations (PDEs).

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