Solutions of the Second-order Nonlinear Parabolic System Modeling the Diffusion Wave Motion
The paper continues a long series of our research and considers a secondorder nonlinear evolutionary parabolic system. The system can be a model of various convective and diffusion processes in continuum mechanics, including mass transfer in a binary mixture. In hydrology, ecology, and mathematical...
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| Format: | Article |
| Language: | English |
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Irkutsk State University
2022-12-01
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| Series: | Известия Иркутского государственного университета: Серия "Математика" |
| Subjects: | |
| Online Access: | https://mathizv.isu.ru/en/article/file?id=1430 |
| _version_ | 1811189042417500160 |
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| author | A.L. Kazakov A.A. Lempert |
| author_facet | A.L. Kazakov A.A. Lempert |
| author_sort | A.L. Kazakov |
| collection | DOAJ |
| description | The paper continues a long series of our research and considers a secondorder nonlinear evolutionary parabolic system. The system can be a model of various convective and diffusion processes in continuum mechanics, including mass transfer in a binary mixture. In hydrology, ecology, and mathematical biology, it describes the propagation of pollutants in water and air, as well as population dynamics, including the interaction of two different biological species. We construct solutions that have the type of diffusion (heat) wave propagating over a zero background with a finite velocity. Note that the system degenerates on the line where the perturbed and zero (unperturbed) solutions are continuously joined. A new existence and uniqueness theorem is proved in the class of analytical functions. In this case, the solution has the desired type and is constructed in the form of characteristic series, the convergence of which is proved by the majorant method. We also present two new classes of exact solutions, the construction of which, due to ansatzes of a specific form, reduces to integrating systems of ordinary differential equations that inherit a singularity from the original formulation. The obtained results are expected to be helpful in modeling the evolution of the Baikal biota and the propagation of pollutants in the water of Lake Baikal near settlements |
| first_indexed | 2024-04-11T14:29:09Z |
| format | Article |
| id | doaj.art-d6df873612bc431397d758a5a9681bdc |
| institution | Directory Open Access Journal |
| issn | 1997-7670 2541-8785 |
| language | English |
| last_indexed | 2024-04-11T14:29:09Z |
| publishDate | 2022-12-01 |
| publisher | Irkutsk State University |
| record_format | Article |
| series | Известия Иркутского государственного университета: Серия "Математика" |
| spelling | doaj.art-d6df873612bc431397d758a5a9681bdc2022-12-22T04:18:44ZengIrkutsk State UniversityИзвестия Иркутского государственного университета: Серия "Математика"1997-76702541-87852022-12-014214358https://doi.org/10.26516/1997-7670.2022.42.43Solutions of the Second-order Nonlinear Parabolic System Modeling the Diffusion Wave MotionA.L. KazakovA.A. LempertThe paper continues a long series of our research and considers a secondorder nonlinear evolutionary parabolic system. The system can be a model of various convective and diffusion processes in continuum mechanics, including mass transfer in a binary mixture. In hydrology, ecology, and mathematical biology, it describes the propagation of pollutants in water and air, as well as population dynamics, including the interaction of two different biological species. We construct solutions that have the type of diffusion (heat) wave propagating over a zero background with a finite velocity. Note that the system degenerates on the line where the perturbed and zero (unperturbed) solutions are continuously joined. A new existence and uniqueness theorem is proved in the class of analytical functions. In this case, the solution has the desired type and is constructed in the form of characteristic series, the convergence of which is proved by the majorant method. We also present two new classes of exact solutions, the construction of which, due to ansatzes of a specific form, reduces to integrating systems of ordinary differential equations that inherit a singularity from the original formulation. The obtained results are expected to be helpful in modeling the evolution of the Baikal biota and the propagation of pollutants in the water of Lake Baikal near settlementshttps://mathizv.isu.ru/en/article/file?id=1430parabolic partial differential equationsanalytical solutiondiffusion waveexistence theoremexact solutionmathematical modeling |
| spellingShingle | A.L. Kazakov A.A. Lempert Solutions of the Second-order Nonlinear Parabolic System Modeling the Diffusion Wave Motion Известия Иркутского государственного университета: Серия "Математика" parabolic partial differential equations analytical solution diffusion wave existence theorem exact solution mathematical modeling |
| title | Solutions of the Second-order Nonlinear Parabolic System Modeling the Diffusion Wave Motion |
| title_full | Solutions of the Second-order Nonlinear Parabolic System Modeling the Diffusion Wave Motion |
| title_fullStr | Solutions of the Second-order Nonlinear Parabolic System Modeling the Diffusion Wave Motion |
| title_full_unstemmed | Solutions of the Second-order Nonlinear Parabolic System Modeling the Diffusion Wave Motion |
| title_short | Solutions of the Second-order Nonlinear Parabolic System Modeling the Diffusion Wave Motion |
| title_sort | solutions of the second order nonlinear parabolic system modeling the diffusion wave motion |
| topic | parabolic partial differential equations analytical solution diffusion wave existence theorem exact solution mathematical modeling |
| url | https://mathizv.isu.ru/en/article/file?id=1430 |
| work_keys_str_mv | AT alkazakov solutionsofthesecondordernonlinearparabolicsystemmodelingthediffusionwavemotion AT aalempert solutionsofthesecondordernonlinearparabolicsystemmodelingthediffusionwavemotion |