On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems

Paine, John; Hoog, Frank de; Anderssen, R. S.

doi:10.1007/bf02241779

Cited by 108 publications

(109 citation statements)

References 3 publications

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“…in his PhD Thesis (1979). Then it is followed up in various papers [13,14,16,19,20,26]. The correction terms can be added to the numerical eigenvalues in order to obtain better eigenvalues of the problem.…”

Section: Correction Termsmentioning

confidence: 99%

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Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four

Rattana¹,

Böckmann²

2013

Journal of Computational and Applied Mathematics

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Bibliografische Information der Deutschen NationalbibliothekDie Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.de abrufbar. Universitätsverlag AbstractThis paper examines and develops matrix methods to approximate the eigenvalues of a fourth order Sturm-Liouville problem subjected to a kind of fixed boundary conditions, furthermore, it extends the matrix methods for a kind of general boundary conditions. The idea of the methods comes from finite difference and Numerov's method as well as boundary value methods for second order regular Sturm-Liouville problems. Moreover, the determination of the correction term formulas of the matrix methods are investigated in order to obtain better approximations of the problem with fixed boundary conditions since the exact eigenvalues for q = 0 are known in this case. Finally, some numerical examples are illustrated.

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Section: Correction Termsmentioning

confidence: 99%

“…However, the two most common matrix methods are finite different method (FDM) and Numerov's method [12]. Correction terms of the methods for second order SLP have been described in [12,13,14,15,16,17,18,19,20] to obtain better numerical eigenvalues of the problem.…”

Section: Introductionmentioning

confidence: 99%

Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four

Rattana¹,

Böckmann²

2013

Journal of Computational and Applied Mathematics

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“…Also, although the theory of asymptotic correction [11,22] concerns equations in the Liouville normal form (1), recent numerical results [13,16] show that it can also improve results for equations in the impedance form (ay ) + λay = 0 .…”

Section: C128mentioning

confidence: 99%

“…In many important cases [4,8,9], the correction is known in closed form. Originally proposed [22] for centered finite difference solution of the direct problem (1-2) with c 1 = c 2 = 0 , asymptotic correction was soon extended to higher order methods [11,25] and to more general differential operators [4,6,25].…”

Section: U3mentioning

confidence: 99%

Solving inverse Sturm-Liouville problems: theory and practice

Andrew

2017

ANZIAMJ

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Theoretical results on the solution of inverse Sturm-Liouville problems generally consider only idealized problems requiring much more data than is available in real applications. Typical theorems describe problems where infinitely many eigenvalues are known exactly, but in most applications we know only approximations of a finite, and usually small, number of eigenvalues. This paper considers how idealized theoretical results may assist practical numerical computation. It also reviews recent progress on a class of numerical methods for inverse Sturm-Liouville problems, it discusses some open questions, and it announces a new convergence result.

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“…In [21] a kind of successive approximation method is developed. In [23,24], the Sturm-Liouville problem is replaced by its finite-dimensional approximation and a regularization procedure is developed based on the known asymptotic representation of the spectrum as the potential is small.…”

mentioning

confidence: 99%

Inverse problem on the semi-axis: local approach

Avdonin

Belinskiy

Matthews³

2011

Tamkang J. Math.

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Abstract. We consider the problem of reconstruction of the potential for the wave equation on the semi-axis. We use the local versions of the Gelfand-Levitan and Krein equations, and the linear version of Simon's approach. For all methods, we reduce the problem of reconstruction to a second kind Fredholm integral equation, the kernel and the right-hand-side of which arise from an auxiliary second kind Volterra integral equation. A second-order accurate numerical method for the equations is described and implemented. Then several numerical examples verify that the algorithms can be used to reconstruct an unknown potential accurately. The practicality of each approach is briefly discussed. Accurate data preparation is described and implemented.

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On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems

Cited by 108 publications

References 3 publications

Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four

Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four

Solving inverse Sturm-Liouville problems: theory and practice

Inverse problem on the semi-axis: local approach

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