Analytical Solutions of a Nonlinear Convection-Diffusion Equation With Polynomial Sources
Nonlinear convection–diffusion equations are widely used for the description of various processes and phenomena in physics, mechanics and biology. In this work we consider a family of nonlinear ordinary differential equations which is a traveling wave reduction of a nonlinear convection–diffusion equat...
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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 |
| Subjects: | |
| Online Access: | https://www.mais-journal.ru/jour/article/view/344 |
| _version_ | 1797877855733940224 |
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| author | N. A. Kudryashov D. I. Sinelshchikov |
| author_facet | N. A. Kudryashov D. I. Sinelshchikov |
| author_sort | N. A. Kudryashov |
| collection | DOAJ |
| description | Nonlinear convection–diffusion equations are widely used for the description of various processes and phenomena in physics, mechanics and biology. In this work we consider a family of nonlinear ordinary differential equations which is a traveling wave reduction of a nonlinear convection–diffusion equation with a polynomial source. We study a question about integrability of this family of nonlinear ordinary differential equations. We consider both stationary and non–stationary cases of this equation with and without convection. In order to construct general analytical solutions of equations from this family we use an approach based on nonlocal transformations which generalize the Sundman transformations. We show that in the stationary case without convection the general analytical solution of the considered family of equations can be constructed without any constraints on its parameters and can be expressed via the Weierstrass elliptic function. Since in the general case this solution has a cumbersome form we find some correlations on the parameters which allow us to construct the general solution in the explicit form. We show that in the non–stationary case both with and without convection we can find a general analytical solution of the considered equation only imposing some correlation on the parameters. To this aim we use criteria for the integrability of the Lienard equation which have recently been obtained. We find explicit expressions in terms of exponential and elliptic functions for the corresponding analytical solutions. |
| first_indexed | 2024-04-10T02:23:48Z |
| format | Article |
| id | doaj.art-1c87ab02e33b4802818a4c5c0ef5ec20 |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2024-04-10T02:23:48Z |
| publishDate | 2016-06-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-1c87ab02e33b4802818a4c5c0ef5ec202023-03-13T08:07:34ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172016-06-0123330931610.18255/1818-1015-2016-3-309-316301Analytical Solutions of a Nonlinear Convection-Diffusion Equation With Polynomial SourcesN. A. Kudryashov0D. I. Sinelshchikov1Национальный исследовательский ядерный университет МИФИНациональный исследовательский ядерный университет МИФИNonlinear convection–diffusion equations are widely used for the description of various processes and phenomena in physics, mechanics and biology. In this work we consider a family of nonlinear ordinary differential equations which is a traveling wave reduction of a nonlinear convection–diffusion equation with a polynomial source. We study a question about integrability of this family of nonlinear ordinary differential equations. We consider both stationary and non–stationary cases of this equation with and without convection. In order to construct general analytical solutions of equations from this family we use an approach based on nonlocal transformations which generalize the Sundman transformations. We show that in the stationary case without convection the general analytical solution of the considered family of equations can be constructed without any constraints on its parameters and can be expressed via the Weierstrass elliptic function. Since in the general case this solution has a cumbersome form we find some correlations on the parameters which allow us to construct the general solution in the explicit form. We show that in the non–stationary case both with and without convection we can find a general analytical solution of the considered equation only imposing some correlation on the parameters. To this aim we use criteria for the integrability of the Lienard equation which have recently been obtained. We find explicit expressions in terms of exponential and elliptic functions for the corresponding analytical solutions.https://www.mais-journal.ru/jour/article/view/344аналитические решенияэллиптические функциинелокальные преобразованияуравнения льенара |
| spellingShingle | N. A. Kudryashov D. I. Sinelshchikov Analytical Solutions of a Nonlinear Convection-Diffusion Equation With Polynomial Sources Моделирование и анализ информационных систем аналитические решения эллиптические функции нелокальные преобразования уравнения льенара |
| title | Analytical Solutions of a Nonlinear Convection-Diffusion Equation With Polynomial Sources |
| title_full | Analytical Solutions of a Nonlinear Convection-Diffusion Equation With Polynomial Sources |
| title_fullStr | Analytical Solutions of a Nonlinear Convection-Diffusion Equation With Polynomial Sources |
| title_full_unstemmed | Analytical Solutions of a Nonlinear Convection-Diffusion Equation With Polynomial Sources |
| title_short | Analytical Solutions of a Nonlinear Convection-Diffusion Equation With Polynomial Sources |
| title_sort | analytical solutions of a nonlinear convection diffusion equation with polynomial sources |
| topic | аналитические решения эллиптические функции нелокальные преобразования уравнения льенара |
| url | https://www.mais-journal.ru/jour/article/view/344 |
| work_keys_str_mv | AT nakudryashov analyticalsolutionsofanonlinearconvectiondiffusionequationwithpolynomialsources AT disinelshchikov analyticalsolutionsofanonlinearconvectiondiffusionequationwithpolynomialsources |