Abstract
The paper considers the features of numerical reconstruction of the advection coefficient when solving the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type. Information on the position of a reaction front is used as data of the inverse problem. An important question arises: is it possible to obtain a mathematical connection between the unknown coefficient and the data of the inverse problem? The methods of asymptotic analysis of the direct problem help to solve this question. But the reduced statement of the inverse problem obtained by the methods of asymptotic analysis contains a nonlinear integral equation for the unknown coefficient. The features of its solution are discussed. Numerical experiments demonstrate the possibility of solving problems of such class using the proposed methods.
Highlights
An important question arises: is it possible to obtain a mathematical connection between the unknown coefficient and the data of the inverse problem? The methods of asymptotic analysis of the direct problem help to solve this question
One of the main result of the paper is using the method of asymptotic analysis in order to obtain mathematical connection between the unknown coefficient and the data of the inverse problem
The second result is the algorithm of solving obtained nonlinear integral equation
Summary
Problems for nonlinear singularly perturbed reaction-diffusion-advection equations arise in gas dynamics [1], combustion theory [2], chemical kinetics [3,4,5,6,7,8,9,10], nonlinear wave theory [11], biophysics [12,13,14,15,16], medicine [17,18,19,20], ecology [21,22,23,24,25], finance [26] and other fields of science [27]. 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 to observe in an experiments (the front is an distinguishable contrast structure). We consider the question of the possibility of recovering the advection coefficient in the generalized Burgers equation [42] from the data on the dynamics of the reaction front. To effectively solve such inverse problems the methods of asymptotic analysis can be used [3,43].
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