A space-time spectral collocation algorithm for the variable order fractional wave equation.

A. H. Bhrawy,M. A. Abdelkawy,J. F. Alzaidy,E. H. Doha

doi:10.1186/s40064-016-2899-5

Abstract

The variable order wave equation plays a major role in acoustics, electromagnetics, and fluid dynamics. In this paper, we consider the space–time variable order fractional wave equation with variable coefficients. We propose an effective numerical method for solving the aforementioned problem in a bounded domain. The shifted Jacobi polynomials are used as basis functions, and the variable-order fractional derivative is described in the Caputo sense. The proposed method is a combination of shifted Jacobi–Gauss–Lobatto collocation scheme for the spatial discretization and the shifted Jacobi–Gauss–Radau collocation scheme for temporal discretization. The aforementioned problem is then reduced to a problem consists of a system of easily solvable algebraic equations. Finally, numerical examples are presented to show the effectiveness of the proposed numerical method.

Highlights

The subject of fractional calculus is one of the branches of applied mathematics which deals with derivatives and integrals of any arbitrary order (Hilfer 2000; Kilbas and Trujillo 2002; Kilbas et al 2006)
The aim of this paper is to find the numerical solution of the space–time variable order fractional wave equation subject to initial-boundary conditions
This paper extends the SJ–GL-C and shifted Jacobi Gauss– Radau (SJ–GR)-C schemes in order to solve the space-time variable order fractional wave equation

Summary

Introduction

The subject of fractional calculus is one of the branches of applied mathematics which deals with derivatives and integrals of any arbitrary order (Hilfer 2000; Kilbas and Trujillo 2002; Kilbas et al 2006). Few numerical methods have been introduced and discussed to solve the variable-order fractional problems (Sun et al 2012; Ma et al 2012; Zeng et al 2015; Fu et al 2015; Abdelkawy et al 2015). Bhrawy and Zaky (2015a) proposed a new algorithm for solving one-and two-dimensional variable-order cable equations based on Jacobi spectral collocation approximation together with the Jacobi operational matrix for variable-order fractional derivative. Chen et al (2014) proposed an implicit alternating direct method for the two-dimensional variable-order fractional percolation equation discussed the stability and convergence of the implicit alternating direct method

Objectives

Conclusion