The Allen-Cahn equation with a time Caputo-Hadamard derivative: Mathematical and Numerical Analysis

Zhen Wang; Luhan Sun; Zhen Wang; Luhan Sun

doi:10.3934/cam.2023031

Communications in Analysis and Mechanics

2023, Volume 15, Issue 4: 611-637. doi: 10.3934/cam.2023031

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The Allen-Cahn equation with a time Caputo-Hadamard derivative: Mathematical and Numerical Analysis

Zhen Wang ^,,
Luhan Sun

School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013, P.R. China

Received: 10 July 2023 Revised: 26 September 2023 Accepted: 27 September 2023 Published: 12 October 2023
65M60, 65N15

In this paper, we investigate the local discontinuous Galerkin (LDG) finite element method for the fractional Allen-Cahn equation with Caputo-Hadamard derivative in the time domain. First, the regularity of the solution is analyzed, and the results indicate that the solution of this equation generally possesses initial weak regularity in the time dimension. Due to this property, a logarithmic nonuniform L1 scheme is adopted to approximate the Caputo-Hadamard derivative, while the LDG method is used for spatial discretization. The stability and convergence of this fully discrete scheme are proven using a discrete fractional Gronwall inequality. Numerical examples demonstrate the effectiveness of this method.
- fractional Allen-Cahn equation,
- Caputo-Hadamard derivative,
- LDG method,
- error estimate
Citation: Zhen Wang, Luhan Sun. The Allen-Cahn equation with a time Caputo-Hadamard derivative: Mathematical and Numerical Analysis[J]. Communications in Analysis and Mechanics, 2023, 15(4): 611-637. doi: 10.3934/cam.2023031

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Abstract

In this paper, we investigate the local discontinuous Galerkin (LDG) finite element method for the fractional Allen-Cahn equation with Caputo-Hadamard derivative in the time domain. First, the regularity of the solution is analyzed, and the results indicate that the solution of this equation generally possesses initial weak regularity in the time dimension. Due to this property, a logarithmic nonuniform L1 scheme is adopted to approximate the Caputo-Hadamard derivative, while the LDG method is used for spatial discretization. The stability and convergence of this fully discrete scheme are proven using a discrete fractional Gronwall inequality. Numerical examples demonstrate the effectiveness of this method.

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