Solving forward and inverse problems involving a nonlinear three-dimensional partial differential equation via asymptotic expansions

Abstract

This paper concerns the use of asymptotic expansions for the efficient solving of forward and inverse problems involving a nonlinear singularly perturbed time-dependent reaction–diffusion–advection equation. By using an asymptotic expansion with the local coordinates in the transition-layer region, we prove the existence and uniqueness of a smooth solution with a sharp transition layer for a 3D partial differential equation. Moreover, with the help of asymptotic expansion, a simplified model is derived for the corresponding inverse source problem, which is close to the original inverse problem over the entire region except for a narrow transition layer. We show that such simplification does not reduce the accuracy of the inversion results when the measurement data contain noise. Based on this simpler inversion model, an asymptotic-expansion regularization algorithm is proposed for efficiently solving the inverse source problem in the 3D case. A model problem shows the feasibility of the proposed numerical approach.

An improved squirrel search algorithm for a coefficient inversion problem of a singularly perturbed reaction-diffusion equation

“…When the highest spatial derivative of these partial differential equations is multiplied by a small parameter, such problems are called singularly perturbed inverse problems. Recently, a lot researchers have paid attention to develop some numerical methods for solving singularly perturbed inverse problems (see [31][32][33][34][35][36][37] and their references). It should be pointed out that the authors in [31][32][33][34][35][36][37] used the gradient method to find the solution of inverse problems for singularly perturbed partial differential equations.…”

“…Recently, a lot researchers have paid attention to develop some numerical methods for solving singularly perturbed inverse problems (see [31][32][33][34][35][36][37] and their references). It should be pointed out that the authors in [31][32][33][34][35][36][37] used the gradient method to find the solution of inverse problems for singularly perturbed partial differential equations. To this day, there is no any report by using swarm intelligent optimization algorithms to solve these inverse problems.…”