Sinc-Chebyshev collocation method for a class of fractional diffusion-wave equations.

Zhi Mao,Long Shi,Aiguo Xiao,Zuguo Yu

doi:10.1155/2014/143983

Zhi Mao, Long Shi + Show 2 more

Open Access

https://doi.org/10.1155/2014/143983

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Abstract

This paper is devoted to investigating the numerical solution for a class of fractional diffusion-wave equations with a variable coefficient where the fractional derivatives are described in the Caputo sense. The approach is based on the collocation technique where the shifted Chebyshev polynomials in time and the sinc functions in space are utilized, respectively. The problem is reduced to the solution of a system of linear algebraic equations. Through the numerical example, the procedure is tested and the efficiency of the proposed method is confirmed.

Highlights

Fractional models have been increasingly shown by many scientists to describe adequately the problems with memory and nonlocal properties in fluid mechanics, viscoelasticity, physics, biology, chemistry, finance, and other areas of applications [1,2,3,4,5,6]
The fractional diffusion-wave equation has been used to model many important physical phenomena ranging from amorphous, colloid, glassy, and porous materials through fractals, percolation clusters, and random and disordered media to comb structures, dielectrics and semiconductors, polymers, and biological systems [7,8,9,10]
It is a generalization of the classical diffusion-wave equation by replacing the integer-order time derivative with a fractional derivative of order α (1 < α < 2)

Summary

Introduction

Fractional models have been increasingly shown by many scientists to describe adequately the problems with memory and nonlocal properties in fluid mechanics, viscoelasticity, physics, biology, chemistry, finance, and other areas of applications [1,2,3,4,5,6]. The fractional diffusion-wave equation has been used to model many important physical phenomena ranging from amorphous, colloid, glassy, and porous materials through fractals, percolation clusters, and random and disordered media to comb structures, dielectrics and semiconductors, polymers, and biological systems [7,8,9,10]. It is a generalization of the classical diffusion-wave equation by replacing the integer-order time derivative with a fractional derivative of order α (1 < α < 2).

Notations and Some Preliminary Results

The Derivation of the Sinc-Chebyshev Collocation Method

Numerical Examples

Conclusion