Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations

Yongjin Li,Kamal Shah

doi:10.1155/2017/1535826

Abstract

We develop a numerical method by using operational matrices of fractional order integrations and differentiations to obtain approximate solutions to a class of coupled systems of fractional order partial differential equations (FPDEs). We use shifted Legendre polynomials in two variables. With the help of the aforesaid matrices, we convert the system under consideration to a system of easily solvable algebraic equation of Sylvester type. During this process, we need no discretization of the data. We also provide error analysis and some test problems to demonstrate the established technique.

Highlights

Fractional calculus has attracted the attention of many researchers as it has many applications in various disciplines of applied sciences and engineering
We develop a numerical method by using operational matrices of fractional order integrations and differentiations to obtain approximate solutions to a class of coupled systems of fractional order partial differential equations (FPDEs)
The area of fractional calculus devoted to the existence and uniqueness of positive solution to (FDEs) and (FPDEs) is well studied and a lot of research work is available on it

Summary

Introduction

Fractional calculus has attracted the attention of many researchers as it has many applications in various disciplines of applied sciences and engineering (we refer to [1,2,3,4] and the references therein). All the operational matrices methods are used to solve FDEs and FPDEs. As much as we know they are not well studied and neither is properly applied to solve coupled systems of PFDEs. To the best of our knowledge very few articles are devoted to the numerical solutions of coupled systems of integral equations and ordinary and partial differential equations of fractional order; for details see [39,40,41]. With the use of shifted Legendre polynomials in two variables, some operational matrices corresponding to fractional order differentiations and integrations are developed Thanks to these operational matrices, the coupled system under consideration is transformed to a system of Sylvester type algebraic equations.

Preliminaries

Operational Matrices of Integrations and Differentiations

Solutions of the Coupled Systems of Equations

Numerical Test Problems

Conclusion