A new fourth-order grouping iterative method for the time fractional sub-diffusion equation having a weak singularity at initial time

Muhammad Asim Khan; Norma Alias; Umair Ali; Muhammad Asim Khan; Norma Alias; Umair Ali

doi:10.3934/math.2023697

AIMS Mathematics

2023, Volume 8, Issue 6: 13725-13746. doi: 10.3934/math.2023697

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Research article

A new fourth-order grouping iterative method for the time fractional sub-diffusion equation having a weak singularity at initial time

1.
Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi, Malaysia
2.
Department of Applied Mathematics and Statistics, Institute of Space Technology, Pakistan

Received: 09 October 2022 Revised: 25 January 2023 Accepted: 07 February 2023 Published: 10 April 2023
MSC : 35R11, 65N06

A new fourth-order explicit grouping iterative method is constructed for the numerical solution of the fractional sub-diffusion equation. The discretization of the equation is based on fourth-order finite difference method. Captive fractional discretization having functions with a weak singularity at $ t = 0 $ is used for time and similarly, the space derivative is approximated with the help of fourth-order approximation. Furthermore, the convergence and stability of the scheme are analyzed. Finally, the accuracy and validity are investigated by some numerical examples.
- sub-diffusion equation,
- Crank-Nicolson,
- grouping method,
- convergence and stability,
- fourth-order FDM,
- captive fractional discretisation
Citation: Muhammad Asim Khan, Norma Alias, Umair Ali. A new fourth-order grouping iterative method for the time fractional sub-diffusion equation having a weak singularity at initial time[J]. AIMS Mathematics, 2023, 8(6): 13725-13746. doi: 10.3934/math.2023697

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Abstract

A new fourth-order explicit grouping iterative method is constructed for the numerical solution of the fractional sub-diffusion equation. The discretization of the equation is based on fourth-order finite difference method. Captive fractional discretization having functions with a weak singularity at $ t = 0 $ is used for time and similarly, the space derivative is approximated with the help of fourth-order approximation. Furthermore, the convergence and stability of the scheme are analyzed. Finally, the accuracy and validity are investigated by some numerical examples.

References

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