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

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. ISSN 0096-3003

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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 $0<{\lambda}_q <\min(\frac{1}{\mu_0},2^\gamma)$ and $0<\gamma \le 1$, and demonstrated the method is also stable numerically in the case $\frac{1}{\mu_0}<{\lambda}_q \le 2^\gamma$ and $\log_3{2} \le \gamma \le 1$. The accuracy and convergence of the scheme was also investigated and found to be of order $O(\Delta t^{1+\gamma})$ in time and $O(\Delta x^2)$ in space. To confirm the accuracy and stability of the proposed method we provide three examples with one including a linear reaction term.


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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Additional Information: Published version cannot be displayed due to copyright restrictions.
Faculty/School / Institute/Centre: Current - Faculty of Health, Engineering and Sciences - School of Agricultural, Computational and Environmental Sciences
Date Deposited: 21 May 2019 04:33
Last Modified: 13 Jun 2019 02:02
Uncontrolled Keywords: fractional subdiffusion equation, Keller Box method, fractional calculus, L1 scheme, linear reaction
Fields of Research : 01 Mathematical Sciences > 0103 Numerical and Computational Mathematics > 010302 Numerical Solution of Differential and Integral Equations
Socio-Economic Objective: E Expanding Knowledge > 97 Expanding Knowledge > 970101 Expanding Knowledge in the Mathematical Sciences
Funding Details:
Identification Number or DOI: 10.1016/j.amc.2018.12.015
URI: http://eprints.usq.edu.au/id/eprint/36062

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