Singularly Perturbed Problems in Case of Exchange of Stabilities

Solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection type with data on the position of a reaction front

In this paper, approaches to the numerical recovering of the initial condition in the inverse problem for a nonlinear singularly perturbed reaction–diffusion–advection equation are considered The feature of the formulation of the inverse problem is the use of additional information about the value of the solution of the equation at the known position of a reaction front, measured experimentally with a delay relative to the initial moment of time In this case, for the numerical solution of the inverse problem, the gradient method of minimizing the cost functional is applied In the case when only the position of the reaction front is known, the method of deep machine learning is applied Numerical experiments demonstrated the possibility of solving such kinds of considered inverse problems

https://www.mdpi.com/2227-7390/9/4/342/pdf

Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay

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.

https://www.mdpi.com/2073-8994/13/5/860/pdf

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

The reversible A↔B reaction–diffusion process, when species A and B are initially mixed and diffuse with different diffusion coefficients, is usually considered as a diffusion-limited process. In this work the reaction rate in such a process is investigated using the boundary layer function method. It was shown that the reaction–diffusion process can be considered as a quasi-equilibrium process. Despite this fact the contribution of the changes in the species concentration of the reaction is comparable to that of the diffusion. Moreover, the ratios of the reaction and diffusion contributions are independent of time and coordinate. The dependence of the reaction rate on the initial species distribution is analyzed. It was demonstrated, for the first time, that the number of the reaction zones is determined by the initial conditions and changes with time. Also the asymptotic long-time behavior of the reaction rate depends on the initial species distribution.

A. B. Vasilieva

Papers

Singularly Perturbed Problems with Boundary and Interior Layers: Theory and Application