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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| Format: | Article |
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MDPI AG
2021-11-01
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| Series: | Mathematics |
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| author | Raul Argun Alexandr Gorbachev Dmitry Lukyanenko Maxim Shishlenin |
| author_facet | Raul Argun Alexandr Gorbachev Dmitry Lukyanenko Maxim Shishlenin |
| author_sort | Raul Argun |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-03-10T05:19:11Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T05:19:11Z |
| publishDate | 2021-11-01 |
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| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-d5df7a9d82ea4ce185fc38cecedb04b52023-11-23T00:14:42ZengMDPI AGMathematics2227-73902021-11-01922289410.3390/math9222894On 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 FrontRaul Argun0Alexandr Gorbachev1Dmitry Lukyanenko2Maxim Shishlenin3Department of Mathematics, Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, RussiaDepartment of Mathematics, Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, RussiaDepartment of Mathematics, Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, RussiaInstitute of Computational Mathematics and Mathematical Geophysics of SB RAS, 630090 Novosibirsk, RussiaThe 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.https://www.mdpi.com/2227-7390/9/22/2894coefficient inverse problemreaction–diffusion–advection equationreaction–diffusion equationsingularly perturbed probleminverse problem with data on the position of a reaction frontblow-up |
| spellingShingle | 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 Mathematics 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 |
| 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_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_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_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 |
| 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 |
| url | https://www.mdpi.com/2227-7390/9/22/2894 |
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