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Galerkin finite element method for a two-dimensional tempered time–space fractional diffusion equation with application to a Bloch–Torrey equation retaining Larmor precession

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  • Feng, Libo
  • Liu, Fawang
  • Anh, Vo V.

Abstract

This paper considers a two-dimensional tempered time–space fractional diffusion equation with a reaction term on convex domains. Firstly, the analytical solution to a one-dimensional tempered time–space diffusion equation is derived in terms of the Fox H function. However, for a two-dimensional counterpart, an explicit expression for its analytical solution does not seem tractable. This motivates us to resort to numerical methods. Next, the L1 formula on a graded mesh is modified to approximate the Caputo tempered time-fractional derivative. A fast evaluation for the tempered time-fractional operator is developed based on a sum-of-exponentials approximation, reducing the computational work and storage significantly. Furthermore, the Galerkin finite element method based on an unstructured mesh is utilised to solve the problem. Its stability and convergence are established. Finally, two numerical examples in different convex domains are investigated to demonstrate the effectiveness of the numerical method. As an application, the tempered fractional Bloch–Torrey equation retaining Larmor precession in a human brain-like domain is illustrated to observe the evolution of the transverse magnetisation. An interesting finding is that the tempered parameter has a significant impact on the decay of magnetisation.

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  • Feng, Libo & Liu, Fawang & Anh, Vo V., 2023. "Galerkin finite element method for a two-dimensional tempered time–space fractional diffusion equation with application to a Bloch–Torrey equation retaining Larmor precession," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 517-537.
    • Handle: RePEc:eee:matcom:v:206:y:2023:i:c:p:517-537
    DOI: 10.1016/j.matcom.2022.11.024
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    1. She, Mianfu & Li, Dongfang & Sun, Hai-wei, 2022. "A transformed L1 method for solving the multi-term time-fractional diffusion problem," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 584-606.
    2. Bu, Weiping & Zhao, Yanmin & Shen, Chen, 2021. "Fast and efficient finite difference/finite element method for the two-dimensional multi-term time-space fractional Bloch-Torrey equation," Applied Mathematics and Computation, Elsevier, vol. 398(C).
    3. Alvaro Cartea & Diego del-Castillo-Negrete, 2007. "On the Fluid Limit of the Continuous-Time Random Walk with General Lévy Jump Distribution Functions," Birkbeck Working Papers in Economics and Finance 0708, Birkbeck, Department of Economics, Mathematics & Statistics.
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    1. Hu, Zesen & Li, Xiaolin, 2024. "Analysis of a fast element-free Galerkin method for the multi-term time-fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 223(C), pages 677-692.

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