A method for constructing a complete bifurcation picture of a boundary value problem for nonlinear partial differential equations: application of the Kolmogorov-Arnold theorem
The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partia...
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| Format: | Article |
| Language: | English |
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Saratov State University
2025-07-01
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| Series: | Известия высших учебных заведений: Прикладная нелинейная динамика |
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| Online Access: | https://andjournal.sgu.ru/sites/andjournal.sgu.ru/files/text-pdf/2025/07/and_2025-4_gromov_et-al_435-465.pdf |
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| author | Gromov, Vasily Alexandrovich Tomashchuk, Korney Kirillovich Beschastnov, Yury Nikolaevich Sidorenko, Artem Aleksandrovich Kakurin, Vasily Vladimirovich |
| author_facet | Gromov, Vasily Alexandrovich Tomashchuk, Korney Kirillovich Beschastnov, Yury Nikolaevich Sidorenko, Artem Aleksandrovich Kakurin, Vasily Vladimirovich |
| author_sort | Gromov, Vasily Alexandrovich |
| collection | DOAJ |
| description | The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations. Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions. Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations. |
| format | Article |
| id | doaj-art-8d6a46ae20664aa2bfd4e888b1efce99 |
| institution | Kabale University |
| issn | 0869-6632 2542-1905 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | Saratov State University |
| record_format | Article |
| series | Известия высших учебных заведений: Прикладная нелинейная динамика |
| spelling | doaj-art-8d6a46ae20664aa2bfd4e888b1efce992025-08-20T03:58:36ZengSaratov State UniversityИзвестия высших учебных заведений: Прикладная нелинейная динамика0869-66322542-19052025-07-0133443546510.18500/0869-6632-003160A method for constructing a complete bifurcation picture of a boundary value problem for nonlinear partial differential equations: application of the Kolmogorov-Arnold theoremGromov, Vasily Alexandrovich0Tomashchuk, Korney Kirillovich1Beschastnov, Yury Nikolaevich2Sidorenko, Artem Aleksandrovich3Kakurin, Vasily Vladimirovich4National Research University "Higher School of Economics", ul. Myasnitskaya 20, Moscow, 101000, RussiaNational Research University "Higher School of Economics", ul. Myasnitskaya 20, Moscow, 101000, RussiaNational Research University "Higher School of Economics", ul. Myasnitskaya 20, Moscow, 101000, RussiaNational Research University "Higher School of Economics", ul. Myasnitskaya 20, Moscow, 101000, RussiaNational Research University "Higher School of Economics", ul. Myasnitskaya 20, Moscow, 101000, RussiaThe purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations. Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions. Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations. https://andjournal.sgu.ru/sites/andjournal.sgu.ru/files/text-pdf/2025/07/and_2025-4_gromov_et-al_435-465.pdfbifurcation analysisnonlinear partial differential equationsboundary value problemskolmogorovarnold theorem |
| spellingShingle | Gromov, Vasily Alexandrovich Tomashchuk, Korney Kirillovich Beschastnov, Yury Nikolaevich Sidorenko, Artem Aleksandrovich Kakurin, Vasily Vladimirovich A method for constructing a complete bifurcation picture of a boundary value problem for nonlinear partial differential equations: application of the Kolmogorov-Arnold theorem Известия высших учебных заведений: Прикладная нелинейная динамика bifurcation analysis nonlinear partial differential equations boundary value problems kolmogorovarnold theorem |
| title | A method for constructing a complete bifurcation picture of a boundary value problem for nonlinear partial differential equations: application of the Kolmogorov-Arnold theorem |
| title_full | A method for constructing a complete bifurcation picture of a boundary value problem for nonlinear partial differential equations: application of the Kolmogorov-Arnold theorem |
| title_fullStr | A method for constructing a complete bifurcation picture of a boundary value problem for nonlinear partial differential equations: application of the Kolmogorov-Arnold theorem |
| title_full_unstemmed | A method for constructing a complete bifurcation picture of a boundary value problem for nonlinear partial differential equations: application of the Kolmogorov-Arnold theorem |
| title_short | A method for constructing a complete bifurcation picture of a boundary value problem for nonlinear partial differential equations: application of the Kolmogorov-Arnold theorem |
| title_sort | method for constructing a complete bifurcation picture of a boundary value problem for nonlinear partial differential equations application of the kolmogorov arnold theorem |
| topic | bifurcation analysis nonlinear partial differential equations boundary value problems kolmogorovarnold theorem |
| url | https://andjournal.sgu.ru/sites/andjournal.sgu.ru/files/text-pdf/2025/07/and_2025-4_gromov_et-al_435-465.pdf |
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