The space-time generalized finite difference method for the time fractional mobile/immobile diffusion equation

Haili Qiao; Haili Qiao

doi:10.3934/nhm.2025035

Networks and Heterogeneous Media

2025, Volume 20, Issue 3: 818-843. doi: 10.3934/nhm.2025035

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The space-time generalized finite difference method for the time fractional mobile/immobile diffusion equation

Haili Qiao ^,

School of Mathematical Sciences, Liaocheng University, Liaocheng, Shandong, China

Received: 01 April 2025 Revised: 22 May 2025 Accepted: 23 June 2025 Published: 16 July 2025

The space-time generalized finite difference method was used to solve the time fractional mobile/immobile diffusion equation. Traditionally, partial differential equations are solved by separating time and space dimensions, and applying distinct numerical methods to each. Here, the time dimension was treated as an additional spatial dimension, thus transforming the $ d $-dimensional spatial problem into a new $ (d+1) $-dimensional problem in the space-time domain. In the space-time generalized finite difference method, the complexity of the numerical solution of the problem was effectively reduced by simultaneously discretizing the space and time dimensions in the space-time domain, while retaining all the advantages of the generalized finite difference method. We took the second-order Taylor series expansion as an example to show the numerical algorithm of the 2D time fractional mobile/immobile diffusion equation, and numerically simulated the one- and two-dimensional problems where the solution was a sufficiently smooth or weak singularity at the initial moment, thereby verifying the effectiveness of the algorithm.
Citation: Haili Qiao. The space-time generalized finite difference method for the time fractional mobile/immobile diffusion equation[J]. Networks and Heterogeneous Media, 2025, 20(3): 818-843. doi: 10.3934/nhm.2025035

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Abstract

The space-time generalized finite difference method was used to solve the time fractional mobile/immobile diffusion equation. Traditionally, partial differential equations are solved by separating time and space dimensions, and applying distinct numerical methods to each. Here, the time dimension was treated as an additional spatial dimension, thus transforming the $ d $-dimensional spatial problem into a new $ (d+1) $-dimensional problem in the space-time domain. In the space-time generalized finite difference method, the complexity of the numerical solution of the problem was effectively reduced by simultaneously discretizing the space and time dimensions in the space-time domain, while retaining all the advantages of the generalized finite difference method. We took the second-order Taylor series expansion as an example to show the numerical algorithm of the 2D time fractional mobile/immobile diffusion equation, and numerically simulated the one- and two-dimensional problems where the solution was a sufficiently smooth or weak singularity at the initial moment, thereby verifying the effectiveness of the algorithm.

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