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Itogi Nauki i Tekhniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory, 2018, Volume 152, Pages 125–142
(Mi into357)
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This article is cited in 2 scientific papers (total in 2 papers)
Asymptotic Problem for Second-Order Ordinary Differential Equation with Nonlinearity Corresponding to Butterfly Catastrophe
O. Yu. Khachay Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
Abstract:
For the second-order nonlinear ordinary differential equation ${u''_{xx}=u^5-tu^3-x}$, we prove the existence and uniqueness of a strictly increasing solution, which satisfies an initial condition and a limit condition at infinity and whose graph lies between the zero equation and the continuous graph of the root of the nondifferential equation ${u^5-tu^3-x=0}$. For this solution, we find an asymptotics, which is uniform on the ray ${t\in(-\infty,-M^t)}$ as $x\to+\infty$; separately, we construct asymptotics on the ray ${s>M^s}$ and on the segment ${0\leq s\leq M^s}$, where ${s=|t|^{-5/2}x}$ is the variable compressed with respect to $x$. Using the method of matching asymptotic expansions, we construct a composite asymptotic expansion of the solution to the Cauchy problem whose initial conditions are found from the theorem on the existence of solutions to the original problem. Finally, we construct a uniform asymptotic expansion under the restriction ${t\leq 0}$ as ${x^2+t^2\to\infty}$.
Keywords:
matching asymptotic expansions, nonlinear ordinary differential equation, nonlinear equation of mathematical physics, butterfly catastrophe.
Citation:
O. Yu. Khachay, “Asymptotic Problem for Second-Order Ordinary Differential Equation with Nonlinearity Corresponding to Butterfly Catastrophe”, Mathematical physics, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 152, VINITI, Moscow, 2018, 125–142; J. Math. Sci. (N. Y.), 252:2 (2021), 247–265
Linking options:
https://www.mathnet.ru/eng/into357 https://www.mathnet.ru/eng/into/v152/p125
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Abstract page: | 97 | Full-text PDF : | 34 | References: | 8 | First page: | 3 |
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