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

Author

Listed:
  • Zhengang Zhao

    (Department of Fundamental Courses, Shanghai Customs University, Shanghai 201204, China
    These authors contributed equally to this work.)

  • Yunying Zheng

    (School of Mathematics and Statistics, Huaibei Normal University, Huaibei 235026, China
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Zhengang Zhao & Yunying Zheng, 2024. "Galerkin Finite Element Method for Caputo–Hadamard Time-Space Fractional Diffusion Equation," Mathematics, MDPI, vol. 12(23), pages 1-14, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:23:p:3786-:d:1533473
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    References listed on IDEAS

    as
    1. Sahoo, Sanjay Ku & Gupta, Vikas & Dubey, Shruti, 2024. "A robust higher-order finite difference technique for a time-fractional singularly perturbed problem," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 215(C), pages 43-68.
    2. Garra, Roberto & Mainardi, Francesco & Spada, Giorgio, 2017. "A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 333-338.
    3. Zhao, Zhengang & Zheng, Yunying, 2023. "A Galerkin finite element method for the space Hadamard fractional partial differential equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 214(C), pages 272-289.
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