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Russia; Vietnam Moscow; Tuyen Quang
Section Mathematics
Title Pseudospectral method for second-order autonomous nonlinear differential equations
Author(-s) Nhat L.A.ab
Affiliations Peoples' Friendship University of Russia (RUDN University)a, Tan Trao Universityb
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
UDC 519.624
MSC 34B15, 65D25
DOI 10.20537/vm190106
Received 25 February 2019
Language English
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