A numerical scheme for the time-fractional diffusion equation by layer potentials

Abstract

We develop a numerical scheme for solving an initial-boundary value problem for the time-fractional diffusion equation, based on the boundary integral equation method. By expressing the solution as a single-layer potential, the initial-boundary value problem is transformed into a boundary integral equation for the unknown density function. To numerically solve the resulting boundary integral equation, we develop a stable discretization scheme for layer potentials. First, we rewrite the layer potential operators as generalized Abel integral operators in time, where the kernels are time-dependent boundary integrals. Then, the asymptotic expansions of those kernels at the initial time are derived by carefully analyzing the fundamental solution of the time-fractional diffusion equation. Consequently, we establish a stable time discretization scheme, using the composite trapezoidal rule with a correction at the endpoint to deal with the singularity of the integrand. The spatial discretization is performed by a standard quadrature rule for boundary integrals of smooth functions. Finally, we present several numerical examples to show the efficiency and accuracy of the proposed numerical scheme.