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

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.