A method for constructing a complete bifurcation picture of a boundary value problem for nonlinear partial differential equations: application of the Kolmogorov-Arnold theorem

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Abstract

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.  

About the authors

Vasily Alexandrovich Gromov

National Research University "Higher School of Economics"

ORCID iD: 0000-0001-5891-6597
Scopus Author ID: 35228959300
ResearcherId: M-6614-2015
ul. Myasnitskaya 20, Moscow, 101000, Russia

Korney Kirillovich Tomashchuk

National Research University "Higher School of Economics"

ul. Myasnitskaya 20, Moscow, 101000, Russia

Yury Nikolaevich Beschastnov

National Research University "Higher School of Economics"

ORCID iD: 0000-0001-6511-5894
ul. Myasnitskaya 20, Moscow, 101000, Russia

Artem Aleksandrovich Sidorenko

National Research University "Higher School of Economics"

ul. Myasnitskaya 20, Moscow, 101000, Russia

Vasily Vladimirovich Kakurin

National Research University "Higher School of Economics"

ORCID iD: 0009-0004-3660-871X
ul. Myasnitskaya 20, Moscow, 101000, Russia

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