Moving Front Solution of the Reaction-Diffusion Problem
In this paper, we study the moving front solution of the reaction-diffusion initialboundary value problem with a small diffusion coefficient. Problems in such statements can be used to model physical processes associated with the propagation of autowave fronts, in particular, in biophys...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Yaroslavl State University
2017-06-01
|
| Series: | Моделирование и анализ информационных систем |
| Subjects: | |
| Online Access: | https://www.mais-journal.ru/jour/article/view/517 |
| _version_ | 1797878026566893568 |
|---|---|
| author | Evgeny A. Antipov Vladimir T. Volkov Natalia T. Levashova Nikolay N. Nefedov |
| author_facet | Evgeny A. Antipov Vladimir T. Volkov Natalia T. Levashova Nikolay N. Nefedov |
| author_sort | Evgeny A. Antipov |
| collection | DOAJ |
| description | In this paper, we study the moving front solution of the reaction-diffusion initialboundary value problem with a small diffusion coefficient. Problems in such statements can be used to model physical processes associated with the propagation of autowave fronts, in particular, in biophysics or in combustion. The moving front solution is a function the distinctive feature of which is the presence in the domain of its definition of a subdomain where the function has a large gradient. This subdomain is called an internal transition layer. In the nonstationary case, the position of the transition layer varies with time which, as it is well known, complicates the numerical solution of the problem as well as the justification of the correctness of numerical calculations. In this case the analytical method is an essential component of the study. In the paper, asymptotic methods are applied for analytical investigation of the solution of the problem posed. In particular, an asymptotic approximation of the solution as an expansion in powers of a small parameter is constructed by the use of the Vasil’eva algorithm and the existence theorem is carried out using the asymptotic method of differential inequalities. The methods used also make it possible to obtain an equation describing the motion of the front. For this purpose a transition to local coordinates takes place in the region of the front localization. In the present paper, in comparison with earlier publications dealing with two-dimensional problems with internal transition layers the transition to local coordinates in the vicinity of the front has been modified, that led to the simplification of the algorithm of determining the equation of the curve motion. |
| first_indexed | 2024-04-10T02:26:11Z |
| format | Article |
| id | doaj.art-89270ea56533480cbd2a9daa9aaf9a07 |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2024-04-10T02:26:11Z |
| publishDate | 2017-06-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-89270ea56533480cbd2a9daa9aaf9a072023-03-13T08:07:29ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172017-06-0124325927910.18255/1818-1015-2017-3-259-279367Moving Front Solution of the Reaction-Diffusion ProblemEvgeny A. Antipov0Vladimir T. Volkov1Natalia T. Levashova2Nikolay N. Nefedov3Московский государственный университет имени М.В. ЛомоносоваМосковский государственный университет имени М.В. ЛомоносоваМосковский государственный университет имени М.В. ЛомоносоваМосковский государственный университет имени М.В. ЛомоносоваIn this paper, we study the moving front solution of the reaction-diffusion initialboundary value problem with a small diffusion coefficient. Problems in such statements can be used to model physical processes associated with the propagation of autowave fronts, in particular, in biophysics or in combustion. The moving front solution is a function the distinctive feature of which is the presence in the domain of its definition of a subdomain where the function has a large gradient. This subdomain is called an internal transition layer. In the nonstationary case, the position of the transition layer varies with time which, as it is well known, complicates the numerical solution of the problem as well as the justification of the correctness of numerical calculations. In this case the analytical method is an essential component of the study. In the paper, asymptotic methods are applied for analytical investigation of the solution of the problem posed. In particular, an asymptotic approximation of the solution as an expansion in powers of a small parameter is constructed by the use of the Vasil’eva algorithm and the existence theorem is carried out using the asymptotic method of differential inequalities. The methods used also make it possible to obtain an equation describing the motion of the front. For this purpose a transition to local coordinates takes place in the region of the front localization. In the present paper, in comparison with earlier publications dealing with two-dimensional problems with internal transition layers the transition to local coordinates in the vicinity of the front has been modified, that led to the simplification of the algorithm of determining the equation of the curve motion.https://www.mais-journal.ru/jour/article/view/517задача реакция-диффузиядвумерный движущийся фронтасимптотическое представлениемалый параметрасимптотический метод дифференциальных неравенств |
| spellingShingle | Evgeny A. Antipov Vladimir T. Volkov Natalia T. Levashova Nikolay N. Nefedov Moving Front Solution of the Reaction-Diffusion Problem Моделирование и анализ информационных систем задача реакция-диффузия двумерный движущийся фронт асимптотическое представление малый параметр асимптотический метод дифференциальных неравенств |
| title | Moving Front Solution of the Reaction-Diffusion Problem |
| title_full | Moving Front Solution of the Reaction-Diffusion Problem |
| title_fullStr | Moving Front Solution of the Reaction-Diffusion Problem |
| title_full_unstemmed | Moving Front Solution of the Reaction-Diffusion Problem |
| title_short | Moving Front Solution of the Reaction-Diffusion Problem |
| title_sort | moving front solution of the reaction diffusion problem |
| topic | задача реакция-диффузия двумерный движущийся фронт асимптотическое представление малый параметр асимптотический метод дифференциальных неравенств |
| url | https://www.mais-journal.ru/jour/article/view/517 |
| work_keys_str_mv | AT evgenyaantipov movingfrontsolutionofthereactiondiffusionproblem AT vladimirtvolkov movingfrontsolutionofthereactiondiffusionproblem AT nataliatlevashova movingfrontsolutionofthereactiondiffusionproblem AT nikolaynnefedov movingfrontsolutionofthereactiondiffusionproblem |