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Research Papers

An Efficient Legendre Spectral Tau Matrix Formulation for Solving Fractional Subdiffusion and Reaction Subdiffusion Equations

[+] Author and Article Information
E. H. Doha

Department of Mathematics,
Faculty of Science,
Cairo University,
Giza 12613, Egypt
e-mail: eiddoha@frcu.eun.eg

A. H. Bhrawy

Department of Mathematics,
Faculty of Science,
King Abdulaziz University,
Jeddah 21589, Saudi Arabia;
Department of Mathematics,
Faculty of Science,
Beni-Suef University,
Beni-Suef 62511, Egypt
e-mail: alibhrawy@yahoo.co.uk

S. S. Ezz-Eldien

Department of Basic Science,
Institute of Information Technology,
Modern Academy,
Cairo 11731, Egypt
e-mail: s_sezeldien@yahoo.com

Manuscript received January 26, 2014; final manuscript received June 19, 2014; published online January 14, 2015. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 10(2), 021019 (Mar 01, 2015) (8 pages) Paper No: CND-14-1030; doi: 10.1115/1.4027944 History: Received January 26, 2014; Revised June 19, 2014; Online January 14, 2015

In this work, we discuss an operational matrix approach for introducing an approximate solution of the fractional subdiffusion equation (FSDE) with both Dirichlet boundary conditions (DBCs) and Neumann boundary conditions (NBCs). We propose a spectral method in both temporal and spatial discretizations for this equation. Our approach is based on the space-time shifted Legendre tau-spectral method combined with the operational matrix of fractional integrals, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. In addition, this approach is also investigated for solving the FSDE with the variable coefficients and the fractional reaction subdiffusion equation (FRSDE). For conforming the validity and accuracy of the numerical scheme proposed, four numerical examples with their approximate solutions are presented. Also, comparisons between our numerical results and those obtained by compact finite difference method (CFDM), Box-type scheme (B-TS), and FDM with Fourier analysis (FA) are introduced.

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Figures

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Fig. 1

Error function at n = m = 12 and γ = 0.25 for example 1

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Fig. 2

Error function at n = m = 12 and γ = 0.75 for example 1

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Fig. 3

Error function at n = m = 16 and γ = 0.30 for example 2

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Fig. 4

Error function at n = m = 16 and γ = 0.50 for example 2

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Fig. 5

Error function at n = m = 16 and γ = 0.70 for example 2

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Fig. 6

Error function at n = m = 16 and γ = 0.90 for example 2

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Fig. 7

Error function at n = m = 16 and γ = 0.50 for example 3

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Fig. 8

Error function at n = m = 16 and γ = 0.75 for example 3

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