Numerical solutions of fractional wave equations using an efficient class of FDM based on the Hermite formula

Mohamed M Khader,Mohamed H Adel

doi:10.1186/s13662-015-0731-0

Abstract

In this article, a numerical study is introduced for solving the fractional wave equations by using an efficient class of finite difference methods. The proposed scheme is based on the Hermite formula. The stability and the convergence analysis of the proposed methods are given by a recently proposed procedure similar to the standard von Neumann stability analysis. A simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, are given and checked numerically. Finally, a numerical example is presented to confirm the theoretical results.

Highlights

In recent years, it has turned out that many phenomena in engineering, physics, chemistry, and other sciences can be described very successfully by models using mathematical tools from fractional calculus, i.e. the theory of derivatives and integrals of fractional order
We study the time fractional case and use an efficient class of finite difference methods based on the Hermite formula to solve this model
5 Conclusion and remarks This paper presents a class of numerical methods for solving the fractional wave equations

Summary

Introduction

It has turned out that many phenomena in engineering, physics, chemistry, and other sciences can be described very successfully by models using mathematical tools from fractional calculus, i.e. the theory of derivatives and integrals of fractional (non-integer) order. Several numerical methods to solve FDEs have been given such as the variational iteration method [ ], the Adomian’s decomposition method [ ], the collocation method [ – ], and the finite difference method [ , , ]. We introduce the Riemann-Liouville definitions of the fractional derivative operator Dα [ , ]. Definition The Riemann-Liouville derivative Dα of order α of the function y(x) is defined by. We study the time fractional case and use an efficient class of finite difference methods based on the Hermite formula to solve this model. In Section , we give some approximate formulas of the fractional derivatives and numerical finite difference scheme. In Section , we introduce numerical solutions of fractional wave equation.

Finite difference scheme of the fractional wave equation

Conclusion and remarks