Fast finite difference methods for space-time fractional partial differential equations in three space dimensions with nonlocal boundary conditions

Fractional partial differential equations (FPDEs) provide very competitive tools to model challenging phenomena involving anomalous diffusion or long-range memory and spatial interactions. However, numerical methods for space-time FPDEs generate dense stiffness matrices and involve numerical solutions at all the previous time steps, and so have O(N-2 + MN) memory requirement and O(MN3 + (MN)-N-2) computational complexity where N and M are the numbers of spatial unknowns and time steps, respectively. We develop and analyze a finite difference method (FDM) for space-time FPDEs in three space dimensions with a combination of Dirichlet and fractional Neumann boundary conditions, in which a shifted Grfinwald discretization is used in space and an L-1 discretization is used in time so the derived FDM has virtually first-order accuracy. We then derive different fast FDMs by carefully analyzing the structure of the stiffness matrix of numerical discretization as well as the coupling in the time direction. The resulting fast FDMs have an almost linear computational complexity and linear storage with respect to the number of spatial unknowns as well as time steps. Numerical experiments are presented to demonstrate the utility of the methods. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.