Internal layer solutions and coefficient recovery in time-periodic reaction-diffusion-advection equations

TL;DR

This paper develops asymptotic and numerical methods for solving inverse problems in reaction-diffusion-advection equations with internal layers, enabling accurate coefficient reconstruction in periodic, nonlinear PDEs.

Contribution

It introduces a novel asymptotic solution with internal layers and a simple numerical algorithm for reconstructing coefficients in nonlinear, periodic PDEs, with proven existence and uniqueness.

Findings

Numerical experiments show high accuracy in coefficient reconstruction.

The method effectively handles ill-posed inverse problems with noise.

Existence and uniqueness of solutions are rigorously established.

Abstract

This article investigates the non-stationary reaction-diffusion-advection equation, emphasizing solutions with internal layers and the associated inverse problems. We examine a nonlinear singularly perturbed partial differential equation (PDE) within a bounded spatial domain and an infinite temporal domain, subject to periodic temporal boundary conditions. A periodic asymptotic solution featuring an inner transition layer is proposed, advancing the mathematical modeling of reaction-diffusion-advection dynamics. Building on this asymptotic analysis, we develop a simple yet effective numerical algorithm to address ill-posed nonlinear inverse problems aimed at reconstructing coefficient functions that depend solely on spatial or temporal variables. Conditions ensuring the existence and uniqueness of solutions for both forward and inverse problems are established. The proposed method's effectiveness is validated through numerical experiments, demonstrating high accuracy in reconstructing coefficient functions under varying noise conditions.