Inverse source in two-parameter anomalous diffusion, numerical algorithms and simulations over graded time-meshes

TL;DR

This paper develops a numerical method using graded time-meshes to solve an inverse source problem in two-parameter anomalous diffusion models, with applications to physical processes like microwave heating.

Contribution

It introduces a novel numerical algorithm employing bi-orthogonal bases and graded meshes for better convergence in inverse two-parameter sub-diffusion problems.

Findings

The algorithm effectively approximates the unknown source term.

Numerical experiments confirm the expected convergence order.

The method handles solution singularities near t=0 efficiently.

Abstract

We consider an inverse source two-parameter sub-diffusion model subject to a nonlocal initial condition. The problem models several physical processes, among them are the microwave heating and light propagation in photoelectric cells. A bi-orthogonal pair of bases is employed to construct a series representation of the solution and a Volterra integral equation for the source term. We develop a numerical algorithm for approximating the unknown time-dependent source term. Due to the singularity of the solution near $t=0$, a graded mesh is used to improve the convergence rate. Numerical experiments are provided to illustrate the expected analytical order of convergence.