Legendre spectral element method for solving time fractional modified anomalous sub-diffusion equation

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• 作者：Mehdi Dehghan ; ^a ; ^{mdehghan@aut.ac.ir" class="auth_mail" title="E-mail the corresponding author} ; ^{mdehghan.aut@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; ^{mdehghan_aut@yahoo.com" class="auth_mail" title="E-mail the corresponding author} ; Mostafa Abbaszadeh^a ; ^{m.abbaszadeh@aut.ac.ir" class="auth_mail" title="E-mail the corresponding author} ; Akbar Mohebbi^b ; ^{a_mohebbi@kashanu.ac.ir" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fractional modified anomalous sub-diffusion ; Finite difference scheme ; Spectral element method ; Legendre polynomial ; Unconditional stability ; Error estimate and convergence
• 刊名：Applied Mathematical Modelling
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：40
• 期：5-6
• 页码：3635-3654
• 全文大小：2247 K

文摘

In this paper, an efficient numerical method is proposed for the solution of time fractional modified anomalous sub-diffusion equation. The proposed method is based on a finite difference scheme in time variable and Legendre spectral element method for space component. The fractional derivative of equation is described in the Riemann–Liouville sense. Firstly, for obtaining a semi-discrete scheme, the time fractional derivative of the mentioned equation has been discretized by integrating both sides of it. Secondly, we use the Legendre spectral element method for full discretization in one- and two-dimensional cases. In this approach the time fractional derivative of mentioned equation is approximated by a scheme of order g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0307904X15006988&_mathId=si101.gif&_user=111111111&_pii=S0307904X15006988&_rdoc=1&_issn=0307904X&md5=11ae2dd6a7f52d93e12aa6e2f75b1ba5" title="Click to view the MathML source">O(τ1+γ)gif" overflow="scroll">O(τ1+γ) for 0 < γ < 1. We prove the stability and convergence of time discrete scheme using energy method, and show the time discrete scheme is convergent. Also, we propose an error estimate for the full discretization scheme. Numerical examples confirm the high accuracy and efficiency of the proposed numerical scheme.