The Problem of the Non-Uniqueness of the Solution to the Inverse Problem of Recovering the Symmetric States of a Bistable Medium with Data on the Position of an Autowave Front

Abstract

The paper considers the question of the possibility of recovering symmetric stable states of a bistable medium in the inverse problem for a nonlinear singularly perturbed autowave equation by data given on the position of an autowave front propagating through it. It is shown that under certain conditions, this statement of the problem is ill-posed in the sense of the non-uniqueness of the solution. A regularizing approach to its solution was proposed.

Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduced Statement

“…Often, in the formulation of inverse problems for partial differential equations, additional information about the solution on a part of the boundary of the domain is used (see, for example, [24,[28][29][30][31][32][33][34][35][36][37]). However, one of the possible statements of inverse problems for equations of the type under consideration is a statement with additional information about the dynamics of the reaction front motion (see, for example, [38][39][40]). Additional data of this type are in demand in practice, since they are most easily to observe in an experiments (the front is an easily distinguishable contrast structure).…”

“…However, the reduced formulations of such inverse problems may have special features. It was shown that the reduced formulations can contain (1) algebraic equations for an unknown coefficient (see, for example, [44]), (2) differential equations for an unknown coefficient (see, for example, [40]), (3) integral equations for an unknown coefficient. The first case is the simplest and allows to restore the unknown function only at those points through which the reaction front passed during its experimental observation.…”

On Some Features of the Numerical Solving of Coefficient Inverse Problems for an Equation of the Reaction-Diffusion-Advection-Type with Data on the Position of a Reaction Front

“…This work is a continuation of [36,[38][39][40][41]. In these works, questions were considered about the possibility of applying the methods of asymptotic analysis [3,[42][43][44] to recover some coefficients in inverse problems for nonlinear singularly perturbed reaction-diffusionadvection equations with data on the position of a reaction front.…”

“…The problem obtained using the methods of asymptotic analysis is called the reduced formulation of the inverse problem. The authors already showed that the reduced formulations can contain (1) algebraic equations for an unknown coefficient (see, for example, [38,39]), (2) differential equations for an unknown coefficient (see, for example, [40]), and (3) integral equations for an unknown coefficient (see, for example, [41]). It was shown that in the case of sufficiently small values of the singular parameter included in the original partial differential equation, the reduced formulation provides an opportunity to obtain an approximate solution close to the solution of the inverse problem in the full statement.…”

“…This is due to the fact that the reduced formulation is reduced to the need to solve a nonlinear boundary value problem for the eigenvalues. Moreover, according to the results of [39][40][41], the question of what to do in the case when the value of the singular parameter is not small enough remained open. When solving real applied problems, it is often not known in advance whether the singular parameter contained in the problem statement can be considered small enough to apply the approaches from [36,[39][40][41].…”