Lie Symmetry Analysis and Explicit Solutions of the Time Fractional Fifth-Order KdV Equation

Gang Wei Wang,Tao Feng,Tian Zhou Xu,Duccio Rocchini

doi:10.1371/journal.pone.0088336

Gang Wei Wang, Tao Feng + Show 2 more

Open Access

https://doi.org/10.1371/journal.pone.0088336

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Journal: PloS one	Publication Date: Feb 11, 2014
Citations: 71	License type: CC BY 4.0

Affiliation: Beijing Institute of Technology, Yanshan University

Abstract

In this paper, using the Lie group analysis method, we study the invariance properties of the time fractional fifth-order KdV equation. A systematic research to derive Lie point symmetries to time fractional fifth-order KdV equation is performed. In the sense of point symmetry, all of the vector fields and the symmetry reductions of the fractional fifth-order KdV equation are obtained. At last, by virtue of the sub-equation method, some exact solutions to the fractional fifth-order KdV equation are provided.

Highlights

It is well known that the Lie symmetries, originally advocated by the Norwegian mathematician Sophus Lie in the beginning of the 19th century, are widely applied to investigate nonlinear differential equations (including multi-component systems of partial differential equations (PDEs) and ordinary differential equations (ODEs)), notably, for constructing their exact and explicit solutions
It has been showed that how the Lie symmetry analysis have been effectively used to look for exact and explicit solutions to both ODEs and PDEs
There is no general method for dealing with exact explicit solutions to fractional differential equations (FDEs)

Summary

Introduction

It is well known that the Lie symmetries, originally advocated by the Norwegian mathematician Sophus Lie in the beginning of the 19th century, are widely applied to investigate nonlinear differential equations (including multi-component systems of partial differential equations (PDEs) and ordinary differential equations (ODEs)), notably, for constructing their exact and explicit solutions. It is important to note, that a very small number of them involve Lie symmetries to solve problems for fractional differential equations (FDEs). Many powerful methods have been established and developed to construct exact, explicit and numerical solutions of nonlinear FDEs, such as Adomian decomposition method [24,25], the invariant subspace method [26], transform method [27,28], homotopy perturbation method [29], variational iteration method [30], sub-equation method [31,32,33], Lie symmetry group method [10,11,38,47,48] and so on

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The corresponding invariants are

Conclusions