Applications of the Fractional Calculus: On a Discretization of Fractional Diffusion Equation in One Dimension

Tomas Kisela

doi:10.26552/com.c.2010.1.5-11

Abstract

The paper discusses the problem of classical and fractional diffusion models. It is known that the classical model fails in heterogeneous structures with locations where particles move at a large speed over a long distance. If we replace the second derivative in the space variable in the classical diffusion equation by a fractional derivative of order less than two, we obtain the fractional diffusion equation (FDE) which is more useful in this case. In this paper we introduce a discretization of FDE based on the theory of the difference fractional calculus and we sketch a basic numerical scheme of its solution. Finally, we present some examples comparing classical and fractional diffusion models.

Highlights

Fractional calculus is a mathematical discipline dealing with derivatives and integrals of non-integer orders
We focus on one of such phenomena – anomalous diffusion which is faster than the classical model predicts
The main purpose of this paper is to obtain a numerical solution of the fractional diffusion equation (FDE)

Summary

Introduction

Fractional calculus is a mathematical discipline dealing with derivatives and integrals of non-integer orders. During centuries many mathematicians have contributed to the discussion on the “right” definition of a fractional derivative and various approaches have been proposed (see [13]). This concept of fractional derivatives and integrals has been still considered only as a pure theoretical matter because no real applications were known at that time. The main purpose of this paper is to obtain a numerical solution of the fractional diffusion equation (FDE). We present only the example of the FDE with boundary conditions, we believe that the numerical scheme proposed in this paper allows to involve initial or boundary conditions of any type and any values.

Fundamentals of the Fractional Calculus

Discrete Fractional Calculus

The Origin of the Fractional Diffusion Equation

Examples

Summary