Interpolation perturbation method for solving nonlinear problems
type: Item
instance of: scholarly article
title: Interpolation perturbation method for solving nonlinear problems
zbMATH Open document ID: 0911.65060
DOI: 10.1007/BF00132806
published in: Applied Mathematics and Mechanics. (English Edition)
publication date: 1999-05-16T00:00:00Z
review text: The interpolation perturbation method [cf. \textit{Yiwu Yuan}, Mechanics and Practice, Vol. 12, No. 1 (1990); in Chinese] is designed to produce \(O(\varepsilon^2)\) approximations to the solution of first-order regularly perturbed ordinary differential equations, with better global properties than does the usual perturbation method. Given a system (i) \(V'(t)= f(\varepsilon, t,V(t))\), (ii) \(V(t_0)= c\), \(0\leq\varepsilon< \varepsilon_0\), suppose that \(V_0(t)\) is the solution of (i), (ii) when \(\varepsilon= 0\). The author introduces an interpolation function \(y(t)\) which satisfies (iii) \((1- y)= (y- g)/g= K\), \(g= f(\varepsilon, t,V)/f(0,t,V)\). Replacing \(V\) by \(V_0\) in (iii) one solves for \(K\), \(y\) and truncates these solutions to obtain \(K_*\), \(y_*\) of first order in \(\varepsilon\). Next, one solves (iii) for \(V_*= V+ O(\varepsilon^2)\) in terms of \(K_*\), \(y_*\). The author presents three test problems in which the new method is applied and compared with the exact solution and solutions using the regular perturbation method or the method of scales. The new method is shown to produce better accuracy over a wider range of the \(\varepsilon\) parameter than do the usual methods.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 1159632
zbMATH Keywords: interpolation perturbation method
zbMATH Keywords: test problems
zbMATH Keywords: method of scales
author: Yi-Wu Yuan
reviewed by: John B. Butler jun.
MaRDI profile type: Publication
cites work: The calculation of semi-circular corrugatd tubes - An application of the general solution for ring shells
cites work: Surface waves in closed basins under principal and autoparametric resonances
full work available at URL: https://doi.org/10.1007/bf00132806
OpenAlex ID: W1993279101
description: scientific article; zbMATH DE number 1159632