A second–order approximation scheme for Caputo–Hadamard derivative and its application in fractional Allen–Cahn equation

Luhan Sun; Zhen Wang; Yabing Wei; Luhan Sun; Zhen Wang; Yabing Wei

doi:10.3934/cam.2025025

Communications in Analysis and Mechanics

2025, Volume 17, Issue 2: 630-661. doi: 10.3934/cam.2025025

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Research article

A second–order approximation scheme for Caputo–Hadamard derivative and its application in fractional Allen–Cahn equation

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

Received: 17 February 2025 Revised: 22 May 2025 Accepted: 28 May 2025 Published: 18 June 2025
35R11, 65M12, 65M60

In light of the advantages of the Caputo–Hadamard fractional derivative in characterizing ultra-slow diffusion phenomena, this paper proposes a second-order approximation scheme to approximate it. Then, for the Allen–Cahn equation with the Caputo–Hadamard fractional derivative in time, a numerical algorithm is designed. This algorithm employs the proposed second-order formula for time discretization. Considering the potential anisotropic behavior of the solution in space, the anisotropic nonconforming quasi-Wilson finite element method is utilized for spatial approximation. The error in the $ L^2 $-norm and the superclose error in the $ H^1 $-norm of this algorithm are analyzed. The global superconvergence in the $ H^1 $-norm is demonstrated through interpolation postprocessing techniques. Numerical examples are given to verify the theoretical results and further investigate the influence of different time derivatives on the dynamic behavior of the solution.
- Caputo–Hadamard derivative,
- L1⁺ scheme,
- fractional Allen–Cahn equation,
- nonconforming FEM,
- superconvergence
Citation: Luhan Sun, Zhen Wang, Yabing Wei. A second–order approximation scheme for Caputo–Hadamard derivative and its application in fractional Allen–Cahn equation[J]. Communications in Analysis and Mechanics, 2025, 17(2): 630-661. doi: 10.3934/cam.2025025

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Abstract

In light of the advantages of the Caputo–Hadamard fractional derivative in characterizing ultra-slow diffusion phenomena, this paper proposes a second-order approximation scheme to approximate it. Then, for the Allen–Cahn equation with the Caputo–Hadamard fractional derivative in time, a numerical algorithm is designed. This algorithm employs the proposed second-order formula for time discretization. Considering the potential anisotropic behavior of the solution in space, the anisotropic nonconforming quasi-Wilson finite element method is utilized for spatial approximation. The error in the $ L^2 $-norm and the superclose error in the $ H^1 $-norm of this algorithm are analyzed. The global superconvergence in the $ H^1 $-norm is demonstrated through interpolation postprocessing techniques. Numerical examples are given to verify the theoretical results and further investigate the influence of different time derivatives on the dynamic behavior of the solution.

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