Power Series Method for Linear Partial Differential Equations of Fractional Order

Muhammet Kurulay,Mustafa Bayram

doi:10.26713/cma.v1i2.113

Abstract

In this article, a novel numerical method is proposed for linear partial differential equations with time-fractional derivatives. This method is based on power series and generalized Taylor's formula. The fractional derivatives are considered in the Caputo sense. Several illustrative examples are given to demonstrate the effectiveness of the present method. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. The obtained results are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, efficient and easy to implement.

Highlights

In the last several decades, many researchers have found that derivatives of noninteger order are very suitable for the description of various physical phenomena such as rheology, damping laws and diffusion process
We have considered the time-fractional linear partial differential equation, where the unknown function u = u(x, t) is a assumed to be a causal function of time and the fractional derivatives are taken in Caputo sense as follows: Definition 2.5
A new generalization of the power series method has been developed for linear partial differential equations with time-fractional derivatives

Summary

Introduction

In the last several decades, many researchers have found that derivatives of noninteger order are very suitable for the description of various physical phenomena such as rheology, damping laws and diffusion process These findings have invoked a growing interest of studies of the fractal calculus in some various fields such as physics, fluid mechanics, biology, chemistry, acoustics, control theory, chemistry and engineering. These are Laplace transform method, Fractional Green’s function method, Mellin transform method and method of orthogonal polynomials [1] Among these solution techniques, the variational iteration method and the Adomian decomposition method are the most clear methods of solution of fractional differential and integral equations, because they provide immediate and visible symbolic terms of analytic solutions, as well as numerical approximate solutions to both linear and nonlinear differential equations without linearization or discretization. Homotopy analysis method is applied to solve fractional partial differential equations [11]

Fractional Calculus

Power Series Method

Example

Conclusions