Abstract
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.
Highlights
Nonlinear fractional differential equations (FDEs) as a special category of nonlinear partial differential equations (PDEs) have its variety of applications in physics, biology, chemistry, fluid flow, electrical networks, signal and image processing, acoustics, and so on [1,2,3,4,5,6,7,8]
Owing to widely applications and further properties in various fields of natural sciences, seeking the solutions of fractional PDEs has drawing the attention of scholars
Long has generated the steady-state version of the Korteweg-de Vries (KdV) equation [12] and an integral expression for the coefficients of the KdV equation in fluid is given by Benney [13]
Summary
Nonlinear fractional differential equations (FDEs) as a special category of nonlinear partial differential equations (PDEs) have its variety of applications in physics, biology, chemistry, fluid flow, electrical networks, signal and image processing, acoustics, and so on [1,2,3,4,5,6,7,8]. Owing to widely applications and further properties in various fields of natural sciences, seeking the solutions of fractional PDEs has drawing the attention of scholars Analyzing their solutions can help us understand and explain the nonlinear phenomena. In [10], the author applies the modified fractional subequation method to obtain the exact solution of the fractional coupled KdV equation [10]. Application of modified sine-cosine method to solve the fractional fifth-order KdV equation’s traveling wave solutions is shown in [19]. This article is committed to seeking the new exact solutions for nonlinear time fractional fifth-order KdV via the generalized exp(−Φ(ξ))-expansion method [30].
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