An inverse problem for a Sturm-Liouville operator
type: Item
instance of: scholarly article
title: An inverse problem for a Sturm-Liouville operator
zbMATH Open document ID: 0798.34015
author: Bruce D. Lowe
author: William Rundell
published in: Journal of Mathematical Analysis and Applications
publication date: 1994-05-10T00:00:00Z
review text: Consider an identification problem for the coefficient \(q(x)\) in the differential equation (1) \(-u''(x) + q(x) u(x) = \psi (x)\), \(0<x<1\), \(u(0) = 0\), \(u(1) = 0\). It is assumed throughout that 0 is not a Dirichlet eigenvalue of \(-d^ 2/dx^ 2 + q(x)\) on \((0,1)\), so the forward solution to (1) for a given \(\psi (x)\) is unique. The authors prove the following result: Let \(u_ j (x) = u_ j (x;q)\) denote the solution of (1) corresponding to the function \(\psi_ j (x)\) in the basis \(\{\psi_ j (x)\}^ \infty_{j=1}\) of \(L^ 2(0,1)\). Then \(q(x) \in L^ 2 (0,1)\) is uniquely determined by the measurements \(\{u_ j'(0)\}^ \infty_{j=1}\). In Section 2 the special case when \(\psi_ j(x) = \sin \pi jx\) is investigated. In Section 3, a Newton algorithm is used to construct approximations of \(q(x) \approx 0\) from the finite data \(\{u_ j'(0)\}^ N_{j=1}\).
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 34B24
zbMATH DE Number: 567032
zbMATH Keywords: Sturm-Liouville operator
zbMATH Keywords: inverse problem
zbMATH Keywords: identification
zbMATH Keywords: eigenvalue
reviewed by: G. Dimitriu
MaRDI profile type: Publication
full work available at URL: jmaa.1994.1013
OpenAlex ID: W2078455689
DOI: 10.1006/JMAA.1994.1013
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description: scientific article; zbMATH DE number 567032