Finite difference/spectral approximations for the distributed order time fractional reaction-diffusion equation on an unbounded domain

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• 作者：Hu Chen ; ^{chenhuwen@buaa.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Shujuan Lü ; ; ^{lsj@buaa.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Wenping Chen
• 关键词：Distributed order differential equation ; Fractional diffusion ; Spectral method ; Error estimate
• 刊名：Journal of Computational Physics
• 出版年：2016
• 出版时间：15 June 2016
• 年：2016
• 卷：315
• 期：Complete
• 页码：84-97
• 全文大小：389 K

文摘

The numerical approximation of the distributed order time fractional reaction–diffusion equation on a semi-infinite spatial domain is discussed in this paper. A fully discrete scheme based on finite difference method in time and spectral approximation using Laguerre functions in space is proposed. The scheme is unconditionally stable and convergent with order O(τ²+Δα²+N^(1−m)/2)$\mathrm{O}({\tau }^2+\mathrm{\Delta }{\alpha }^2+N^{(1-m)/2})$, where τ, Δα, N, and m are the time-step size, step size in distributed-order variable, polynomial degree, and regularity in the space variable of the exact solution, respectively. A pseudospectral scheme is also proposed and analyzed. Some numerical examples are presented to demonstrate the efficiency of the proposed scheme.