Solving Some Fractional Partial Differential Equations by Invariant Subspace and  Double Sumudu Transform Methods

Sameer Qasim Hasan,Hasan Shather Kadhem

doi:10.30526/33.3.2485

Abstract

In this paper, several types of space-time fractional partial differential equations has been solved by using most of special double linear integral transform ”double Sumudu ”. Also, we are going to argue the truth of these solutions by another analytically method “invariant subspace method”. All results are illustrative numerically and graphically.

Highlights

As known, linear integral transformation is used to solve differential equations, by convert the Linear partial differential equation into algebraic equation which can be solved
Numerical Examples we consider that the inverse double Sumudu transform is exist. We apply this transform and technique of invariant subspace method to solve some of the fractional diffusion heat and wave equations in one dimension with initial and boundary conditions and fractional parabolic-hyperbolic differential equations
Solution by invariant subspace method The inhomogeneous fractional partial differential equation (10), cannot be solved by invariant subspace method directly, in this case we find the solution of homogeneous equation directly by invariant subspace method and we can be use any suitable numerical method ”Variation iteration method”, i. e, coupled invariant subspace method with variation iteration method(ISVIM) to find an approximate solution of the original equation

Summary

Introduction

Linear integral transformation is used to solve differential equations, by convert the Linear partial differential equation into algebraic equation which can be solved . The Riemann–Liouville fractional integral of order α for a function f is defined as:. The Caputo fractional derivative of positive order α for a function f is defined as:. Which can be used to solve the ordinary and partial differential equations with ordinary and fractional order. The following definitions and properties of single and double Sumudu transform are necessary for our work. We state here some of the important properties of double Sumudu transform which are needed 1. If the double Sumudu transform of the function f (x; t) given by S f x, t T u, v , : 1. T u, v m k a , for b where a 1, a 1, The double Sumudu transform of the partial Caputo fractional derivatives are given in the following theorem: Theorem 2: [4]. Let the exponent order function f x, t has a continuous partial derivative on R R and n 1 α n, m 1 β m, : 3. S D f x, t u T u, v ∑ u T 0, v

On Invariant Subspace Method

Conclusion