Abstract
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.
Highlights
This paper discusses the inverse problem of numerical recovering of the initial condition for a nonlinear singularly perturbed reaction–diffusion–advection equation with data on the position of a reaction front, measured in an experiment with a delay relative to the initial time
A feature of this type of problem is the presence of multiscale processes. Mathematical formulations of these problems are described by nonlinear parabolic equations with a small parameter at the highest derivative
For different values of t0 (the other parameters in the model example (7) remain unchanged), we find an approximate solution to the inverse problem qinv ( x )
Summary
This paper discusses the inverse problem of numerical recovering of the initial condition for a nonlinear singularly perturbed reaction–diffusion–advection equation with data on the position of a reaction front, measured in an experiment with a delay relative to the initial time. A feature of this type of problem is the presence of multiscale processes. Mathematical formulations of these problems are described by nonlinear parabolic equations with a small parameter at the highest derivative. Solutions of these problems are able to contain narrow boundary and/or interior layers (stationary and/or moving fronts). In the formulations of inverse problems for partial differential equations additional information about the solution on a part of the boundary is often used (see, for example, [21,22,23,24,25,26,27,28,29,30])
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