Homotopy Series Solutions to Time-Space Fractional Coupled Systems

Jin Zhang,Hui Wei,Bochao Chen,Ming Cai

doi:10.1155/2017/3540364

Abstract

We apply the homotopy perturbation Sumudu transform method (HPSTM) to the time-space fractional coupled systems in the sense of Riemann-Liouville fractional integral and Caputo derivative. The HPSTM is a combination of Sumudu transform and homotopy perturbation method, which can be easily handled with nonlinear coupled system. We apply the method to the coupled Burgers system, the coupled KdV system, the generalized Hirota-Satsuma coupled KdV system, the coupled WBK system, and the coupled shallow water system. The simplicity and validity of the method can be shown by the applications and the numerical results.

Highlights

Fractional calculus, compared to integer calculus, was mentioned in a letter from L’Hospital to Leibniz in 1695
The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit. Motivated by these fruitful results, Singh et al [25] proposed the homotopy perturbation Sumudu transform method based on the homotopy perturbation method and Sumudu transform method and applied it to nonlinear partial differential equations
It is worth mentioning that the homotopy perturbation Sumudu transform method (HPSTM) is applied without any using of Adomian polynomials, over restrictive assumption or linearization, and is capable of reducing the volume of computational work as compared to the classical numerical methods while still maintaining the high accuracy of the result

Summary

Introduction

Fractional calculus, compared to integer calculus, was mentioned in a letter from L’Hospital to Leibniz in 1695. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit Motivated by these fruitful results, Singh et al [25] proposed the homotopy perturbation Sumudu transform method based on the homotopy perturbation method and Sumudu transform method and applied it to nonlinear partial differential equations. It is worth mentioning that the HPSTM is applied without any using of Adomian polynomials, over restrictive assumption or linearization, and is capable of reducing the volume of computational work as compared to the classical numerical methods while still maintaining the high accuracy of the result It is appropriate for strongly nonlinear system and for weakly nonlinear system.

Preliminaries

Homotopy Perturbation Sumudu Transform Method

Application of HPSTM to Time-Space Fractional Coupled Systems

Concluding Remarks