Work

Time-fractional diffusion equations with randomness, and efficient numerical estimations of expected values

Year: 2025

Type: article

Abstract: In this work, we explore a time-fractional diffusion equation of order $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) with a stochastic diffusivity coefficient $${\kappa }$$ κ . We focus on efficient estimation of the expected values of linear functionals acting on the solution of our model problem. To estimate the expected value computationally, the infinite expansions of $${\kappa }$$ κ need to be truncated. Then we approximate the high-dimensional integral over the random field using a high-order quasi-Monte Carlo method. This follows by approximating the deterministic solution over the space-time domain via a second-order accurate time-stepping scheme in combination with a spatial discretization by Galerkin finite elements. We show some regularity properties of the parametric solution and investigate the errors from estimating the expected value. We report on numerical experiments that complement the theoretical results.

Source: Numerical Algorithms

DOI: 10.1007/s11075-025-02066-0

Authors: Josef Dick, Hecong Gao, William McLean, Kassem Mustapha

Institutions: UNSW Sydney, Harbin Institute of Technology

Language: English

Topic: Fractional Differential Equations Solutions

Subfield: Modeling and Simulation

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,290

Funders: University of New South Wales, Australian Research Council