$ \alpha $-robust error analysis of two nonuniform schemes for Caputo-Hadamard fractional reaction sub-diffusion problems

Xingyang Ye; Xiaoyue Liu; Tong Lyu; Chunxiu Liu; Xingyang Ye; Xiaoyue Liu; Tong Lyu; Chunxiu Liu

doi:10.3934/era.2025018

Electronic Research Archive

2025, Volume 33, Issue 1: 353-380. doi: 10.3934/era.2025018

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Research article

$ \alpha $-robust error analysis of two nonuniform schemes for Caputo-Hadamard fractional reaction sub-diffusion problems

School of Science, Jimei University, Xiamen 361021, China

Received: 26 October 2024 Revised: 29 December 2024 Accepted: 14 January 2025 Published: 23 January 2025

In this paper, we focused on the Caputo-Hadamard fractional reaction sub-diffusion equations. By using the nonuniform L1 scheme and nonuniform Alikhanov scheme in the temporal domain, we formulated two efficient numerical schemes, where the second order difference method was used in the spatial dimension. Furthermore, we derived the stability and convergence of these proposed schemes. Remarkably, both derived numerical methods exhibited $ \alpha $-robustness, that is, it remained valid when $ \alpha\rightarrow 1^- $. Numerical experiments were given to demonstrate the theoretical statements.
- Caputo-Hadamard derivative,
- reaction subdiffusion equations,
- the nonuniform L1 scheme,
- the nonuniform Alikhanov scheme,
- $ \alpha $-robust error estimates
Citation: Xingyang Ye, Xiaoyue Liu, Tong Lyu, Chunxiu Liu. $ \alpha $-robust error analysis of two nonuniform schemes for Caputo-Hadamard fractional reaction sub-diffusion problems[J]. Electronic Research Archive, 2025, 33(1): 353-380. doi: 10.3934/era.2025018

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Abstract

In this paper, we focused on the Caputo-Hadamard fractional reaction sub-diffusion equations. By using the nonuniform L1 scheme and nonuniform Alikhanov scheme in the temporal domain, we formulated two efficient numerical schemes, where the second order difference method was used in the spatial dimension. Furthermore, we derived the stability and convergence of these proposed schemes. Remarkably, both derived numerical methods exhibited $ \alpha $-robustness, that is, it remained valid when $ \alpha\rightarrow 1^- $. Numerical experiments were given to demonstrate the theoretical statements.

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