Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time-space fractional Bloch-Torrey equations on irregular convex domains

Models based on partial differential equations containing time–space fractional derivatives have attracted considerable interest in the past decade because of their ability to model anomalous transport phenomena. These phenomena are strongly connected to the interactions within complex and non-homogeneous media exhibiting spatial heterogeneity. The class of equations with multi-term time–space derivatives of fractional orders has been found to be very useful in the description of such interactions. This motivates the extension of the classical Bloch–Torrey equation through the application of the operators of fractional calculus to new multi-term time–space fractional Bloch–Torrey equations with Riesz fractional operators. In this paper, we firstly propose an unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time–space fractional diffusion equation with Riesz fractional operators on irregular convex domains. Secondly, we rigorously establish the stability and convergence of the numerical scheme. Thirdly, we extend the computational model to solve a system of coupled two-dimensional multi-term time–space fractional Bloch–Torrey equations. Finally, some numerical results are given to demonstrate the versatility and application of the models.