Galerkin Finite Element Method for Caputo–Hadamard Time-Space Fractional Diffusion Equation

Zhengang Zhao and Yunying Zheng ()
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Zhengang Zhao: Department of Fundamental Courses, Shanghai Customs University, Shanghai 201204, China
Yunying Zheng: School of Mathematics and Statistics, Huaibei Normal University, Huaibei 235026, China

Mathematics, 2024, vol. 12, issue 23, 1-14

Abstract: In this paper, we study the Caputo–Hadamard time-space fractional diffusion equation, where the Caputo derivative is defined in the temporal direction and the Hadamard derivative is defined in the spatial direction separately. We first use the Laplace transform and the modified Fourier transform to study the analytical solution of the Cauchy problem. Then, using the Galerkin finite element method in space, we generate a semi-discrete scheme and study the convergence analysis. Furthermore, using the L1 scheme of the Caputo derivative in time, we construct a fully discrete scheme and then discuss the stability and error estimation in detail. Finally, the numerical experiments are displaced to verify the theoretical results.

Keywords: Hadamard fractional diffusion equation; Hadamard fractional derivative; modified Fourier transform; Galerkin finite element method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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