NOVEL NUMERICAL METHODS FOR SOLVING THE TIME-SPACE FRACTIONAL DIFFUSION EQUATION IN TWO DIMENSIONS

In this paper, a time-space fractional diffusion equation in two dimensions (TSFDE-2D) with homogeneous Dirichlet boundary conditions is considered. The TSFDE-2D is obtained from the standard diffusion equation by replacing the first-order time derivative with the Caputo fractional derivative ... (0...

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Bibliographic Details
Published in:SIAM journal on scientific computing 2011-01, Vol.33 (3-4), p.1159-1180
Main Authors: YANG, Qianqian, TURNER, Ian, LIU, Fawang, ILICT, Milos
Format: Article
Language:English
Subjects:
Derivatives
Diffusion
Dimensional analysis
Dirichlet problem
Exact sciences and technology
Laplace transforms
Mathematical analysis
Mathematical models
Mathematics
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis
Numerical analysis. Scientific computation
Numerical methods in probability and statistics
Ordinary differential equations
Real functions
Sciences and techniques of general use
Studies
Symbols
Two dimensional
Vectors (mathematics)
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Summary:In this paper, a time-space fractional diffusion equation in two dimensions (TSFDE-2D) with homogeneous Dirichlet boundary conditions is considered. The TSFDE-2D is obtained from the standard diffusion equation by replacing the first-order time derivative with the Caputo fractional derivative ... (0,1), and the second-order space derivatives with the fractional Laplacian ... Traditional approximation of ... requires diagonalization of A, which is very time-consuming for large sparse matrices. The novelty of the authors' proposed numerical schemes is that, using either the finite difference method or the Laplace transform to handle the Caputo time fractional derivative, the solution of the TSFDE-2D is written in terms of a matrix function vector product f(A)b at each time step, where b is a suitably defined vector. They give error bounds for the new methods and illustrate their roles in solving the TSFDE-2D. They also derive the analytical solution of the TSFDE-2D in terms of the Mittag--Leffler function.(ProQuest: ... denotes formulae/symbols omitted.)
ISSN:1064-8275
1095-7197
DOI:10.1137/100800634