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Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Shen, Shujun, Liu, Fawang, Anh, Vo, & Turner, Ian (2006) Detailed analysis of a conservative difference approximation for the time fractional diffusion equation. Journal of Applied Mathematics and Computing, 22(3), pp. 1-19.

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Abstract

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

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6 citations in Scopus
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ID Code: 23024
Item Type: Journal Article
Additional URLs:
Keywords: Time Fractional Diffusion Equation, Conservative Difference Approximation, Non-Markovian Random Walk, Stability Analysis, Convergence Analysis
ISSN: 1598 - 5865
Subjects: 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)
Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
Deposited On: 17 Jun 2009 23:34
Last Modified: 17 Jun 2013 07:08

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