Reconstruction of space-dependence and nonlinearity of a reaction term in a subdiffusion equation

TL;DR

This paper develops methods for reconstructing coupled nonlinear and space-dependent coefficients in a reaction-subdiffusion equation using various observational data, proving convergence and uniqueness, with numerical validation.

Contribution

It introduces fixed point and Newton methods for joint reconstruction of reaction coefficients in subdiffusion equations, addressing the challenge of multiplicative coupling.

Findings

Convergence of fixed point scheme proven for final time data.

Uniqueness of reconstructed coefficients established.

Numerical experiments demonstrate method effectiveness and dependence on subdiffusion order.

Abstract

In this paper we study the simultaneous reconstruction of two coefficients in a reaction-subdiffusion equation, namely a nonlinearity and a space dependent factor. The fact that these are coupled in a multiplicative matter makes the reconstruction particularly challenging. Several situations of overposed data are considered: boundary observations over a time interval, interior observations at final time, as well as a combination thereof. We devise fixed point schemes and also describe application of a frozen Newton method. In the final time data case we prove convergence of the fixed point scheme as well as uniqueness of both coefficients. Numerical experiments illustrate performance of the reconstruction methods, in particular dependence on the differentiation order in the subdiffusion equation.