Convergence analysis for forward and inverse problems in singularly perturbed time-dependent reaction-advection-diffusion equations

TL;DR

This paper develops an asymptotic expansion method to analyze singularly perturbed reaction-advection-diffusion equations, proving solution existence, constructing approximate solutions, and introducing an efficient algorithm for inverse source problems with proven convergence.

Contribution

It introduces the asymptotic expansion regularization (AER) method for nonlinear inverse problems and provides convergence and error estimates, advancing solution techniques for singularly perturbed PDEs.

Findings

Proved existence and uniqueness of smoothing solutions.

Constructed high-order accurate approximate solutions.

Demonstrated efficiency of AER through numerical examples.

Abstract

In this paper, by employing the asymptotic expansion method, we prove the existence and uniqueness of a smoothing solution for a time-dependent nonlinear singularly perturbed partial differential equation (PDE) with a small-scale parameter. As a by-product, we obtain an approximate smooth solution, constructed from a sequence of reduced stationary PDEs with vanished high-order derivative terms. We prove that the accuracy of the constructed approximate solution can be in any order of this small-scale parameter in the whole domain, except a negligible transition layer. Furthermore, based on a simpler link equation between this approximate solution and the source function, we propose an efficient algorithm, called the asymptotic expansion regularization (AER), for solving nonlinear inverse source problems governed by the original PDE. The convergence-rate results of AER are proven, and the a posteriori error estimation of AER is also studied under some a priori assumptions of source functions. Various numerical examples are provided to demonstrate the efficiency of our new approach.