Error analysis of a finite difference scheme on a modified graded mesh for a nonlocal-in-time parabolic equation

Shujun Liu; Li-Bin Liu; Xiaobing Bao; Shujun Liu; Li-Bin Liu; Xiaobing Bao

doi:10.3934/nhm.2025042

Networks and Heterogeneous Media

2025, Volume 20, Issue 3: 970-986. doi: 10.3934/nhm.2025042

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Error analysis of a finite difference scheme on a modified graded mesh for a nonlocal-in-time parabolic equation

1.
Center for Applied Mathematics of Guangxi, Nanning Normal University, Nanning 530100, China
2.
School of Big Data and Artificial Intelligence Chizhou University Chizhou, Anhui 247000, China

Received: 11 July 2025 Revised: 30 August 2025 Accepted: 09 September 2025 Published: 15 September 2025

In this paper, a novel numerical method is developed to solve a parabolic diffusion equation with a nonlocal-in-time operator. First, drawing upon the mesh equidistribution principle, a modified graded mesh along with its properties is introduced. Additionally, the spatial derivative is discretized using the standard central difference scheme on a uniform mesh, whereas the time nonlocal operator is approximated by employing a finite difference scheme on the aforementioned modified graded mesh. Furthermore, the stability and convergence of our proposed scheme are demonstrated. Finally, the theoretical findings are corroborated through a series of numerical experiments.
- nonlocal-in-time operator,
- parabolic diffusion equation,
- finite difference scheme,
- modified graded mesh
Citation: Shujun Liu, Li-Bin Liu, Xiaobing Bao. Error analysis of a finite difference scheme on a modified graded mesh for a nonlocal-in-time parabolic equation[J]. Networks and Heterogeneous Media, 2025, 20(3): 970-986. doi: 10.3934/nhm.2025042

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Abstract

In this paper, a novel numerical method is developed to solve a parabolic diffusion equation with a nonlocal-in-time operator. First, drawing upon the mesh equidistribution principle, a modified graded mesh along with its properties is introduced. Additionally, the spatial derivative is discretized using the standard central difference scheme on a uniform mesh, whereas the time nonlocal operator is approximated by employing a finite difference scheme on the aforementioned modified graded mesh. Furthermore, the stability and convergence of our proposed scheme are demonstrated. Finally, the theoretical findings are corroborated through a series of numerical experiments.

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