Multiresolution Galerkin method for solving the functional distribution of anomalous diffusion described by time-space fractional diffusion equation

TL;DR

This paper develops multiresolution Galerkin methods using wavelet B-splines for efficiently computing functional distributions of anomalous diffusion modeled by time-space fractional diffusion equations, with proven stability and convergence.

Contribution

It introduces novel multiresolution Galerkin schemes with wavelet B-splines for fractional diffusion, enhancing computational efficiency and stability over traditional methods.

Findings

Schemes are unconditionally stable and convergent.

Wavelet B-spline bases preserve Toeplitz structure, facilitating computation.

Numerical experiments confirm theoretical results.

Abstract

The functional distributions of particle trajectories have wide applications, including the occupation time in half-space, the first passage time, and the maximal displacement, etc. The models discussed in this paper are for characterizing the distribution of the functionals of the paths of anomalous diffusion described by time-space fractional diffusion equation. This paper focuses on providing effective computation methods for the models. Two kinds of time stepping schemes are proposed for the fractional substantial derivative. The multiresolution Galerkin method with wavelet B-spline is used for space approximation. Compared with the finite element or spectral polynomial bases, the wavelet B-spline bases have the advantage of keeping the Toeplitz structure of the stiffness matrix, and being easy to generate the matrix elements and to perform preconditioning. The unconditional stability and convergence of the provided schemes are theoretically proved and numerically verified. Finally, we also discuss the efficient implementations and some extensions of the schemes, such as the wavelet preconditioning and the non-uniform time discretization.