A finite volume element method for the multi-term time fractional reaction-diffusion models with variable coefficients

Jie Zhao; Zhichao Fang; Jiahuan Ren; Jie Zhao; Zhichao Fang; Jiahuan Ren

doi:10.3934/math.2025924

AIMS Mathematics

2025, Volume 10, Issue 9: 20689-20714. doi: 10.3934/math.2025924

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Research article

A finite volume element method for the multi-term time fractional reaction-diffusion models with variable coefficients

1.
School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China
2.
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010030, China

Received: 12 June 2025 Revised: 24 August 2025 Accepted: 01 September 2025 Published: 09 September 2025
MSC : 65M08, 65M15, 65M60

In this paper, a fully discrete finite volume element (FVE) scheme is proposed to treat the linear multi-term time fractional reaction-diffusion models with variable coefficients on triangular grids, where the Caputo fractional derivative of time variable is approximated by the $ L1 $ formula. The existence and uniqueness of solution for the proposed scheme is proved based on the matrix theories, the stability results are derived in detail by using a multi-term fractional Gronwall inequality, and the optimal error estimates under the $ L^2(\Omega) $ and $ H^1(\Omega) $ norms are also obtained. Owing to the variable coefficients in the models, a special technique should be used to analyze the stability and error estimates. Finally, three numerical examples with different fractional derivative terms are given to validate the feasibility and effectiveness.
- finite volume element method,
- $ L1 $ formula,
- multi-term time fractional reaction-diffusion model,
- stability result,
- optimal error estimate
Citation: Jie Zhao, Zhichao Fang, Jiahuan Ren. A finite volume element method for the multi-term time fractional reaction-diffusion models with variable coefficients[J]. AIMS Mathematics, 2025, 10(9): 20689-20714. doi: 10.3934/math.2025924

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Abstract

In this paper, a fully discrete finite volume element (FVE) scheme is proposed to treat the linear multi-term time fractional reaction-diffusion models with variable coefficients on triangular grids, where the Caputo fractional derivative of time variable is approximated by the $ L1 $ formula. The existence and uniqueness of solution for the proposed scheme is proved based on the matrix theories, the stability results are derived in detail by using a multi-term fractional Gronwall inequality, and the optimal error estimates under the $ L^2(\Omega) $ and $ H^1(\Omega) $ norms are also obtained. Owing to the variable coefficients in the models, a special technique should be used to analyze the stability and error estimates. Finally, three numerical examples with different fractional derivative terms are given to validate the feasibility and effectiveness.

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