Algorithm to construct bifurcation structure of non-linear boundary problem for von Karman equations

Authors

  • Vasilii A. Gromov The Department of Computational Mathematics and Mathematical Cybernetics of the Faculty of applied mathematics of Oles Honchar Dnipro National University, Dnipro http://orcid.org/0000-0001-5891-6597

DOI:

https://doi.org/10.20535/SRIT.2308-8893.2017.1.0.08

Keywords:

von Karman equations, solution branching of non-linear boundary problems for partial differential equations, primary and secondary bifurcation paths

Abstract

In the frameworks of the generalized Kantorovich method, a novel approach to detect and analyze singular points of a non-linear boundary problem for von Karman equations is proposed: an algorithm suggests that a sequence of single-dimensional boundary problems is constructed in order to solve the two-dimensional boundary problem in question. The aforesaid single-dimensional boundary problems are reduced to the equivalent Cauchy problems. In doing so, one calculates the Frechet matrix, whose degeneracy is necessary and sufficient conditions of branching. The simulation reveals the bifurcation structure for von Karman equations with the constant right term. In that case, the structure includes primary and secondary bifurcation paths.

Author Biography

Vasilii A. Gromov, The Department of Computational Mathematics and Mathematical Cybernetics of the Faculty of applied mathematics of Oles Honchar Dnipro National University, Dnipro

Vasilii A. Gromov,

Ph. D., a senior researcher and associate professor of the Department of Computational Mathematics and Mathematical Cybernetics of the Faculty of applied mathematics of Oles Honchar Dnipro National University.

Research interests: the direct and inverse bifurcation problems for PDEs, non-linear computational mathematics, chaotic time series prediction.

 

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Published

2017-03-21

Issue

Section

Mathematical methods, models, problems and technologies for complex systems research