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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2019, Volume 29, Issue 1, Pages 61–72 (Mi vuu666)  

MATHEMATICS

Pseudospectral method for second-order autonomous nonlinear differential equations

L. A. Nhatab

a Tan Trao University, Tuyen Quang, 22227, Vietnam
b Peoples' Friendship University of Russia (RUDN University), ul. Miklukho-Maklaya, 6, Moscow, 117198, Russia

Abstract: Autonomous nonlinear differential equations constituted a system of ordinary differential equations, which often applied in different areas of mechanics, quantum physics, chemical engineering science, physical science, and applied mathematics. It is assumed that the second-order autonomous nonlinear differential equations have the types ${u}"({x}) - {u}'({x}) = {f}[{u}({x})]$ and ${u}"({x}) + {f}[{u}({x})]{u}'({x}) + {u}({x}) = 0$ on the range $[-1, 1]$ with the boundary values ${u}[-1]$ and ${u}[1]$ provided. We use the pseudospectral method based on the Chebyshev differentiation matrix with Chebyshev–Gauss–Lobatto points to solve these problems. Moreover, we build two new iterative procedures to find the approximate solutions. In this paper, we use the programming language Mathematica version 10.4 to represent the algorithms, numerical results and figures. In the numerical results, we apply the well-known Van der Pol oscillator equation and gave good results. Therefore, they will be able to be applied to other nonlinear systems such as the Rayleigh equations, the Lienard equations, and the Emden–Fowler equations.

Keywords: pseudospectral method, Chebyshev differentiation matrix, Chebyshev polynomial, autonomous equations, nonlinear differential equations, Van der Pol oscillator.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation
The publication has been prepared with the support of the “RUDN University Program 5-100”.


DOI: https://doi.org/10.20537/vm190106

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Bibliographic databases:

UDC: 519.624
MSC: 34B15, 65D25
Received: 25.02.2019
Language:

Citation: L. A. Nhat, “Pseudospectral method for second-order autonomous nonlinear differential equations”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 29:1 (2019), 61–72

Citation in format AMSBIB
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\by L.~A.~Nhat
\paper Pseudospectral method for second-order autonomous nonlinear differential equations
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2019
\vol 29
\issue 1
\pages 61--72
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\crossref{https://doi.org/10.20537/vm190106}
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  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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