Identification of space-dependent coefficients in two competing terms of a nonlinear subdiffusion equation

TL;DR

This paper develops a fixed point method to reconstruct space-dependent coefficients in nonlinear subdiffusion equations, with proven convergence and local uniqueness, supported by numerical experiments.

Contribution

It introduces a novel fixed point scheme for coefficient reconstruction in nonlinear subdiffusion equations with proven convergence and local uniqueness.

Findings

The scheme successfully reconstructs coefficients from interior observations.

Convergence and local uniqueness are theoretically established.

Numerical experiments demonstrate the scheme's effectiveness.

Abstract

We consider a (sub)diffusion equation with a nonlinearity of the form $pf(u)-qu$, where $p$ and $q$ are space dependent functions. Prominent examples are the Fisher-KPP, the Frank-Kamenetskii-Zeldovich and the Allen-Cahn equations. We devise a fixed point scheme for reconstructing the spatially varying coefficients from interior observations a) at final time under two different excitations b) at two different time instances under a single excitation. Convergence of the scheme as well as local uniqueness of these coefficients is proven. Numerical experiments illustrate the performance of the reconstruction scheme.