Symmetry properties and explicit solutions of the nonlinear time fractional KdV equation

Gangwei Wang,Tianzhou Xu

doi:10.1186/1687-2770-2013-232

Gangwei Wang, Tianzhou Xu

Open Access

https://doi.org/10.1186/1687-2770-2013-232

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Journal: Boundary Value Problems	Publication Date: Nov 8, 2013
Citations: 32	License type: CC BY 2.0

Affiliation: Beijing Institute of Technology

Abstract

The time fractional KdV equation in the sense of the Riemann-Liouville derivatives is considered. The symmetry properties of the time fractional KdV equation is investigated by using the Lie group analysis method. On the basis of the point symmetry, the vector fields of the time fractional KdV equation are presented. And then, the symmetry reductions are constructed. By right of the obtained Lie point symmetries, it is shown that this equation could transform into a nonlinear ordinary differential equation of fractional order with the new independent variable . The derivative is an Erdelyi-Kober derivative depending on a parameter α. At last, by means of the sub-equation method, some exact and explicit solutions of the time fractional KdV equation are constructed. MSC:22E70, 26A33.

Highlights

1 Introduction Lie point transformation groups have been extensively applied to the nonlinear partial differential equations arising in mathematics, physics and in many other scientific fields
It can be said that the method of Lie symmetry groups is the most important approach for constructing analytical solutions of nonlinear partial differential equations
It is a fact that the classical Lie theory applied to fractional differential equations (FDEs) for symmetry analysis should be modified since it fails in the case of fractional differential equations

Summary

Introduction

Lie point transformation groups have been extensively applied to the nonlinear partial differential equations arising in mathematics, physics and in many other scientific fields. Lie point transformation groups can deal with symmetry reductions, similarity solutions of nonlinear differential partial equations. It can be said that the method of Lie symmetry groups is the most important approach for constructing analytical solutions of nonlinear partial differential equations.

Results

Conclusion