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The Numerical Simulation of Space-Time Variable Fractional Order Diffusion Equation

Published online by Cambridge University Press:  28 May 2015

Hongmei Zhang  and
Shujun Shen
Hongmei Zhang*
Affiliation:
School of Mathematical and Computer Sciences, Fuzhou University, Fuzhou, China
Shujun Shen*
Affiliation:
School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian, China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
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Abstract

Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. Numerical methods and analysis of stability and convergence of numerical scheme for the variable fractional order partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the space-time variable fractional order diffusion equation on a finite domain. It is worth mentioning that here we use the Coimbra-definition variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation is proposed and then the stability and convergence of the numerical scheme are investigated. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Keywords

26A3334K2865M1260J70Variable fractional derivativediffusion equationimplicit Euler schemestabilityconvergence
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 4 , November 2013 , pp. 571 - 585
Copyright
Copyright © Global Science Press Limited 2013

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