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

The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of...

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Autores principales:	Raul Argun, Alexandr Gorbachev, Dmitry Lukyanenko, Maxim Shishlenin
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	coefficient inverse problem reaction–diffusion–advection equation reaction–diffusion equation singularly perturbed problem inverse problem with data on the position of a reaction front blow-up Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/d5df7a9d82ea4ce185fc38cecedb04b5
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spelling	oai:doaj.org-article:d5df7a9d82ea4ce185fc38cecedb04b52021-11-25T18:16:56ZOn 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 Front10.3390/math92228942227-7390https://doaj.org/article/d5df7a9d82ea4ce185fc38cecedb04b52021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2894https://doaj.org/toc/2227-7390The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of the numerical solving process. One of the approaches to solving inverse problems of the class under consideration is the use of methods of asymptotic analysis. These methods, under certain conditions, make it possible to construct the so-called reduced formulation of the inverse problem. Usually, a differential equation in this formulation has a lower dimension/order with respect to the differential equation, which is included in the full statement of the inverse problem. In this paper, we consider an example that leads to a reduced formulation of the problem, the solving of which is no less a time-consuming procedure in comparison with the numerical solving of the problem in the full statement. In particular, to obtain an approximate numerical solution, one has to use the methods of the numerical diagnostics of the solution’s blow-up. Thus, it is demonstrated that the possibility of constructing a reduced formulation of the inverse problem does not guarantee its more efficient solving. Moreover, the possibility of constructing a reduced formulation of the problem does not guarantee the existence of an approximate solution that is qualitatively comparable to the true one. In previous works of the authors, it was shown that an acceptable approximate solution can be obtained only for sufficiently small values of the singular parameter included in the full statement of the problem. However, the question of how to proceed if the singular parameter is not small enough remains open. The work also gives an answer to this question.Raul ArgunAlexandr GorbachevDmitry LukyanenkoMaxim ShishleninMDPI AGarticlecoefficient inverse problemreaction–diffusion–advection equationreaction–diffusion equationsingularly perturbed probleminverse problem with data on the position of a reaction frontblow-upMathematicsQA1-939ENMathematics, Vol 9, Iss 2894, p 2894 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	coefficient inverse problem reaction–diffusion–advection equation reaction–diffusion equation singularly perturbed problem inverse problem with data on the position of a reaction front blow-up Mathematics QA1-939
spellingShingle	coefficient inverse problem reaction–diffusion–advection equation reaction–diffusion equation singularly perturbed problem inverse problem with data on the position of a reaction front blow-up Mathematics QA1-939 Raul Argun Alexandr Gorbachev Dmitry Lukyanenko Maxim Shishlenin 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
description	The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of the numerical solving process. One of the approaches to solving inverse problems of the class under consideration is the use of methods of asymptotic analysis. These methods, under certain conditions, make it possible to construct the so-called reduced formulation of the inverse problem. Usually, a differential equation in this formulation has a lower dimension/order with respect to the differential equation, which is included in the full statement of the inverse problem. In this paper, we consider an example that leads to a reduced formulation of the problem, the solving of which is no less a time-consuming procedure in comparison with the numerical solving of the problem in the full statement. In particular, to obtain an approximate numerical solution, one has to use the methods of the numerical diagnostics of the solution’s blow-up. Thus, it is demonstrated that the possibility of constructing a reduced formulation of the inverse problem does not guarantee its more efficient solving. Moreover, the possibility of constructing a reduced formulation of the problem does not guarantee the existence of an approximate solution that is qualitatively comparable to the true one. In previous works of the authors, it was shown that an acceptable approximate solution can be obtained only for sufficiently small values of the singular parameter included in the full statement of the problem. However, the question of how to proceed if the singular parameter is not small enough remains open. The work also gives an answer to this question.
format	article
author	Raul Argun Alexandr Gorbachev Dmitry Lukyanenko Maxim Shishlenin
author_facet	Raul Argun Alexandr Gorbachev Dmitry Lukyanenko Maxim Shishlenin
author_sort	Raul Argun
title	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
title_short	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
title_full	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
title_fullStr	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
title_full_unstemmed	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
title_sort	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
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/d5df7a9d82ea4ce185fc38cecedb04b5
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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

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