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Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Yang, Qianqian, Moroney, Timothy J., Liu, Fawang, & Turner, Ian (2012) Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation. In 5th IFAC Symposium on Fractional Differentiation and its Applications, 14-17 May 2012, Hohai University, Nanjing, China.

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Abstract

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach.

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ID Code: 50613
Item Type: Conference Paper
Keywords: fractional derivative of variable order, time-space fractional reaction-diffusion equation, preconditioned Lanczos method, matrix transfer technique, Krylov subspace
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200)
Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > NUMERICAL AND COMPUTATIONAL MATHEMATICS (010300) > Numerical Solution of Differential and Integral Equations (010302)
Divisions: Current > Schools > School of Mathematical Sciences
Copyright Owner: Copyright 2012 Please consult the authors.
Deposited On: 28 May 2012 12:33
Last Modified: 23 Mar 2013 05:10

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