Approximate solution of some perturbed boundary value problems
type: Item
instance of: scholarly article
title: Approximate solution of some perturbed boundary value problems
zbMATH Open document ID: 0634.73020
DOI: 10.1016/0021-8928(86)90098-5
author: L. D. Akulenko
author: A. S. Shamaev
published in: Journal of Applied Mathematics and Mechanics
publication date: 1986-01-01T00:00:00Z
review text: An analysis of elastic oscillatory systems with distributed and slightly variable parameters is often reduced to the solution of the linear boundary value problem of Sturm-Liouville type \(((1+\epsilon \sigma (x))X')'+\lambda\) \(2(1+\epsilon \delta (x))X=0\), \(X(0)=X(1)=0\), \(x\in [0,1]\), \(\epsilon\in [0,\epsilon]\), \(\lambda\) \(2\geq 0\), where \(\epsilon\) is a small parameter characterizing the nonhomogeneity. The authors analyze some reasons for the appearance of secular terms in asymptotic expansions for eigenvalues and eigenfunctions in (A) and construct uniform (with respect to the index number n) expansions enabling to avoid them. They reduce (A) to a system in ``amplitude-phase'' variables and develop an asymptotic procedure based on a correction of the phase \(\psi_ n\) which is in general similar to conventional methods in the theory of nonlinear oscillations. An approximation of functions \(f(y)\in C_ 0^{(2)}\) using the approximate basis of the problem (A) is also studied. Afterwards this technique is applied to Dirichlet's problem describing the oscillations of nonhomogeneous elastic systems with clamped ends.
Mathematics Subject Classification ID: 74H45
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34L99
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 34B05
zbMATH DE Number: 4033207
zbMATH Keywords: elastic oscillatory systems
zbMATH Keywords: distributed
zbMATH Keywords: slightly variable parameters
zbMATH Keywords: linear boundary value problem
zbMATH Keywords: Sturm-Liouville type
zbMATH Keywords: nonhomogeneity
zbMATH Keywords: secular terms in asymptotic expansions
zbMATH Keywords: system in ``amplitude-phase'' variables
zbMATH Keywords: correction of the phase
zbMATH Keywords: Dirichlet's problem
MaRDI profile type: Publication
full work available at URL: 0021-8928(86)90098-5
OpenAlex ID: W2095378992
cites work: Q3279399
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description: scientific article; zbMATH DE number 4033207