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Invariant imbedding and the calculation of eigenvalues for Sturm-Liouville systems

Invariante Einbettung und die Berechnung von Eigenwerten für Sturm-Liouville-Systeme

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Summary

A new technique based upon an invariant imbedding-Ricatti transformation approach is presented for the calculation of eigenvalues ofSturm-Liouville type systems of differential equations.

A very simple numerical procedure is developed which is easily programmed and which uses reliable subroutines. The method is capable of handling a large class of problems. Included among these are problems in which the eigenvalues appears in a non-linear fashion, cases in which the eigenvalue occurs in the boundary condition, and equations which have singularities.

The numerical computations are generally well-conditioned and very accurate results were obtained.

Zusammenfassung

Es wird eine neue Technik, basierend auf einer Methode, welche invariante Einbettung (Invariant Imbedding) mit derRicatti-Transformation kombiniert, dargestellt zum Zwecke der Berechnung der Eigenwerte von Differentialgleichungssystemen, welche vomSturm-Liouville-Typ sind.

Eine sehr einfache numerische Prozedur ist entwickelt worden, die leicht zu programmieren ist und verläßliche Unterprogramme benutzt. Dieses Verfahren vermag eine umfangreiche Klasse von Problemen zu handhaben. Unter ihnen sind solche inbegriffen, bei denen der Eigenwert in einer nichtlinearen Form auftritt; auch Fälle, wo der Eigenwert in der Randbedingung vorkommt; und schließlich auch Gleichungen, die Singularitäten aufweisen.

Es sei betont, daß die hier von uns behandelten numerischen Berechnungen im allgemeinen stabil sind und sehr genaue Ergebnisse hervorzubringen vermögen.

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Authors and Affiliations

  1. Sandia Laboratory, 87115, Albuquerque, New Mexico, USA

    M. R. Scott

  2. Department of Mathematics and Statistics, The University of New Mexico, 87106, Albuquerque, New Mexico, USA

    L. F. Shampine & G. M. Wing

Authors
  1. M. R. Scott
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  2. L. F. Shampine
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  3. G. M. Wing
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With 3 Figures

Work was supported by the United States Atomic Energy Commission and supported in part under NSF Grant GP 5967.

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Scott, M.R., Shampine, L.F. & Wing, G.M. Invariant imbedding and the calculation of eigenvalues for Sturm-Liouville systems. Computing 4, 10–23 (1969). https://doi.org/10.1007/BF02236538

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  • DOI: https://doi.org/10.1007/BF02236538

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