An implicit Keller Box numerical scheme for the solution of fractional subdiffusion equations

Article


Osman, S. A and Langlands, T. A. M.. 2019. "An implicit Keller Box numerical scheme for the solution of fractional subdiffusion equations." Applied Mathematics and Computation. 348, pp. 609-626. https://doi.org/10.1016/j.amc.2018.12.015
Article Title

An implicit Keller Box numerical scheme for the solution of fractional subdiffusion equations

ERA Journal ID49
Article CategoryArticle
AuthorsOsman, S. A (Author) and Langlands, T. A. M. (Author)
Journal TitleApplied Mathematics and Computation
Journal Citation348, pp. 609-626
Number of Pages18
Year2019
Place of PublicationUnited States
ISSN0096-3003
1873-5649
Digital Object Identifier (DOI)https://doi.org/10.1016/j.amc.2018.12.015
Web Address (URL)https://www.sciencedirect.com/science/article/pii/S0096300318310622
Abstract

In this work, we present a new implicit numerical scheme for fractional subdiffusion equations. In this approach we use the Keller Box method [1] to spatially discretise the fractional subdiffusion equation and we use a modified L1 scheme (ML1), similar to the L1 scheme originally developed by Oldham and Spanier [2], to approximate the fractional derivative. The stability of the proposed method was investigated by using Von-Neumann stability analysis. We have proved the method is unconditionally stable when 0<λq<min([Formula presented],2γ) and 0 < γ ≤ 1, and demonstrated the method is also stable numerically in the case [Formula presented]<λq≤2γ and log32 ≤ γ ≤ 1. The accuracy and convergence of the scheme was also investigated and found to be of order O(Δt1+γ) in time and O(Δx2) in space. To confirm the accuracy and stability of the proposed method we provide three examples with one including a linear reaction term.

Keywordsfractional subdiffusion equation, Keller Box method, fractional calculus, L1 scheme, linear reaction
ANZSRC Field of Research 2020490303. Numerical solution of differential and integral equations
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Byline AffiliationsSchool of Agricultural, Computational and Environmental Sciences
Computational Engineering and Science Research Centre
Institution of OriginUniversity of Southern Queensland
Funding source
Australian Research Council (ARC)
Grant ID
DP130100595
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