Finite difference/finite element method for two-dimensional time and
space fractional Bloch-Torrey equations with variable coefficients on
irregular convex domains
Abstract
In magnetic resonance imaging of the human brain, the diffusion process
of tissue water is considered in the complex tissue environment of
cells, membranes and connective tissue. Models based on fractional order
Bloch-Torrey equations can provide a new insights into further
investigations of tissue structures and the microenvironment. In this
paper, we consider new two-dimensional multi-term time and space
fractional Bloch-Torrey equations with variable coefficients on
irregular convex domains, which involves the Caputo time fractional
derivative and the Riemann-Liouville space fractional derivative. An
unstructured-mesh Galerkin finite element method is used to discretise
the spatial fractional derivative, while for each time fractional
derivative we use a novel $L1$ scheme on a temporal graded mesh. The
stability and convergence of the fully discrete scheme are proved.
Numerical examples are given to verify the efficiency of our method.