Numerical Scheme for the Pseudoparabolic Singularly Perturbed Initial-boundary Problem with Interior Transitional Layer
Evolution equations are derived for the contrasting-structure-type solution of the gen-eralized Kolmogorov–Petrovskii–Piskunov (GKPP) equation with the small parameter with high order derivatives. The GKPP equation is a pseudoparabolic equation with third order derivatives. This equation describes n...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2016-06-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/340 |
| _version_ | 1826558996519059456 |
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| author | A. A. Bykov |
| author_facet | A. A. Bykov |
| author_sort | A. A. Bykov |
| collection | DOAJ |
| description | Evolution equations are derived for the contrasting-structure-type solution of the gen-eralized Kolmogorov–Petrovskii–Piskunov (GKPP) equation with the small parameter with high order derivatives. The GKPP equation is a pseudoparabolic equation with third order derivatives. This equation describes numerous processes in physics, chemistry, biology, for example, magnetic field generation in a turbulent medium and the moving front for the carriers in semiconductors. The profile of the moving internal transitional layer (ITL) is found, and an expression for drift speed of the ITL is derived. An adaptive mesh (AM) algorithm for the numerical solution of the initial-boundary value problem for the GKPP equation is developed and rigorously substantiated. AM algorithm for the special point of the first kind is developed, in which drift speed of the ITL in the first order of the asymptotic expansion turns to zero. Sufficient conditions for ITL transitioning through the special point within finite time are formulated. AM algorithm for the special point of the second kind is developed, in which drift speed of the ITL in the first order formally turns to infinity. Substantiation of the AM method is given based on the method of differential inequalities. Upper and lower solutions are derived. The results of the numerical algorithm are presented. |
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| format | Article |
| id | doaj.art-e0495e7df5d047798ada131db46873f1 |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2025-03-14T08:53:22Z |
| publishDate | 2016-06-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-e0495e7df5d047798ada131db46873f12025-03-02T12:46:58ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172016-06-0123325928210.18255/1818-1015-2016-3-259-282297Numerical Scheme for the Pseudoparabolic Singularly Perturbed Initial-boundary Problem with Interior Transitional LayerA. A. Bykov0Faculty of Physics, Lomonosov Moscow State UniversityEvolution equations are derived for the contrasting-structure-type solution of the gen-eralized Kolmogorov–Petrovskii–Piskunov (GKPP) equation with the small parameter with high order derivatives. The GKPP equation is a pseudoparabolic equation with third order derivatives. This equation describes numerous processes in physics, chemistry, biology, for example, magnetic field generation in a turbulent medium and the moving front for the carriers in semiconductors. The profile of the moving internal transitional layer (ITL) is found, and an expression for drift speed of the ITL is derived. An adaptive mesh (AM) algorithm for the numerical solution of the initial-boundary value problem for the GKPP equation is developed and rigorously substantiated. AM algorithm for the special point of the first kind is developed, in which drift speed of the ITL in the first order of the asymptotic expansion turns to zero. Sufficient conditions for ITL transitioning through the special point within finite time are formulated. AM algorithm for the special point of the second kind is developed, in which drift speed of the ITL in the first order formally turns to infinity. Substantiation of the AM method is given based on the method of differential inequalities. Upper and lower solutions are derived. The results of the numerical algorithm are presented.https://www.mais-journal.ru/jour/article/view/340singularly perturbed equationinterior transitional layerfinite difference methodasymptotic expansion |
| spellingShingle | A. A. Bykov Numerical Scheme for the Pseudoparabolic Singularly Perturbed Initial-boundary Problem with Interior Transitional Layer Моделирование и анализ информационных систем singularly perturbed equation interior transitional layer finite difference method asymptotic expansion |
| title | Numerical Scheme for the Pseudoparabolic Singularly Perturbed Initial-boundary Problem with Interior Transitional Layer |
| title_full | Numerical Scheme for the Pseudoparabolic Singularly Perturbed Initial-boundary Problem with Interior Transitional Layer |
| title_fullStr | Numerical Scheme for the Pseudoparabolic Singularly Perturbed Initial-boundary Problem with Interior Transitional Layer |
| title_full_unstemmed | Numerical Scheme for the Pseudoparabolic Singularly Perturbed Initial-boundary Problem with Interior Transitional Layer |
| title_short | Numerical Scheme for the Pseudoparabolic Singularly Perturbed Initial-boundary Problem with Interior Transitional Layer |
| title_sort | numerical scheme for the pseudoparabolic singularly perturbed initial boundary problem with interior transitional layer |
| topic | singularly perturbed equation interior transitional layer finite difference method asymptotic expansion |
| url | https://www.mais-journal.ru/jour/article/view/340 |
| work_keys_str_mv | AT aabykov numericalschemeforthepseudoparabolicsingularlyperturbedinitialboundaryproblemwithinteriortransitionallayer |