On a shooting algorithm for Sturm-Liouville eigenvalue problems with periodic and semi-periodic boundary conditions
type: Item
instance of: scholarly article
title: On a shooting algorithm for Sturm-Liouville eigenvalue problems with periodic and semi-periodic boundary conditions
zbMATH Open document ID: 0804.65085
author: Xingzhi Ji
published in: Journal of Computational Physics
publication date: 1995-01-15T00:00:00Z
review text: The eigenvalues of a regular Sturm-Liouville system on \((0,a)\) with periodic and semi-periodic boundary conditions satisfy an equation (i) \(\varphi_ 1 (a, \lambda) + \varphi_ 2' (a, \lambda) = D (\lambda) = \pm 2\), where \(\varphi_ 1\), \(\varphi_ 2\) are basic solutions of the differential equation satisfying initial conditions \(\varphi_ 1 (0, \lambda) = 1\), \(\varphi_ 1' (0, \lambda) = 0\), \(\varphi_ 2 (0, \lambda) = 0\), \(\varphi_ 2' (0, \lambda) = 1\). The author applies the shooting method to generate a sequence \(\lambda^{(k)}\), \(\mu^{(k)}\) of numbers and functions \(\varphi_ i (x, \lambda^{(k)})\), \(\varphi_ i (x, \mu^{(k)})\), \(i=1,2\), such that \(\lambda^{(k)}\), \(\mu^{(k)}\) converge to eigenvalues \(\lambda, \mu\) of the periodic and semi-periodic boundary value problems respectively. Starting values for \(\lambda^{(k)}\), \(\mu^{(k)}\) are based on the known behavior of \(D(\lambda)\). The new algorithm is applied to three test problems, one of which is the Mathieu equation. Calculations made on a CYBER 860 computer for the first ten pairs of eigenvalues are accurate to \(10^{-4}\).
Mathematics Subject Classification ID: 65L15
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34L15
zbMATH DE Number: 569522
zbMATH Keywords: convergence
zbMATH Keywords: eigenvalue
zbMATH Keywords: Sturm-Liouville system
zbMATH Keywords: semi-periodic boundary conditions
zbMATH Keywords: shooting method
zbMATH Keywords: algorithm
zbMATH Keywords: test problems
zbMATH Keywords: Mathieu equation
reviewed by: John B. Butler jun.
MaRDI profile type: Publication
full work available at URL: jcph.1994.1045
OpenAlex ID: W2085339468
DOI: 10.1006/JCPH.1994.1045
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description: scientific article; zbMATH DE number 569522