# Implicit difference approximation for the time fractional diffusion equation

Zhuang, Pinghui & Liu, Fawang (2006) Implicit difference approximation for the time fractional diffusion equation. Journal of Applied Mathematics and Computing, 22(3), pp. 87-99.

## Abstract

In this paper, we consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order $0<\alpha<1$ ). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent with $O(\tau+h^2)$, where $\tau$ and $h$ are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.

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62 citations in Scopus

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ID Code: 23700 Journal Article Fractional Differential Equation, Implicit Difference Approximation, Stability, Convergence 1598-5865 Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > PURE MATHEMATICS (010100) > Ordinary Differential Equations Difference Equations and Dynamical Systems (010109)Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > NUMERICAL AND COMPUTATIONAL MATHEMATICS (010300) > Numerical Analysis (010301)Australian and New Zealand Standard Research Classification > INFORMATION AND COMPUTING SCIENCES (080000) > COMPUTATION THEORY AND MATHEMATICS (080200) Past > QUT Faculties & Divisions > Faculty of Science and Technology 17 Jun 2009 14:05 29 Feb 2012 13:21

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