Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart

TL;DR

This paper introduces a linear time mapping technique that transforms solutions of integer order diffusion equations into solutions of fractional order equations, significantly accelerating computations for fractional diffusion problems.

Contribution

The authors develop a novel linear mapping method that reduces the computational complexity of solving fractional diffusion equations from quadratic to near-linear time.

Findings

Achieves up to three orders of magnitude speedup in simulations.

Reduces computational complexity from O(N^2) to O(N^α).

Successfully applies to fractional advection diffusion modeling.

Abstract

The solution of a Caputo time fractional diffusion equation of order $0<\alpha<1$ is expressed in terms of the solution of a corresponding integer order diffusion equation. We demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem. In the context of an $N$-point finite difference time discretisation, the mapping allows for an improvement in time computational complexity from $O\left(N^2\right)$ to $O\left(N^\alpha\right)$, given a precomputation of $O\left(N^{1+\alpha}\ln N\right)$. The mapping is applied successfully to the least-squares fitting of a fractional advection diffusion model for the current in a time-of-flight experiment, resulting in a computational speed up in the range of one to three orders of magnitude for realistic problem sizes.