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Bounds on eigenvalues of Sturm-Liouville problems with discontinuous coefficients

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Abstract

This paper is concerned with obtaining upper and lower bounds for the eigenvalues of Sturm-Liouville problems with discontinuous coefficients. Such problems occur naturally in many areas of composite material mechanics.

The problem is first transformed by using an analog of the classical Liouville transformation. Upper bounds are obtained by application of a Rayleigh-Ritz technique to the transformed problem. Explicit lower bounds in terms of the coefficients are established. Numerical examples illustrate the accuracy of the results.

Résumé

Dans cet article les bornes supérieures et inférieures sont détermineés pour les valeurs caractéristiques des problèmes de Sturm-Liouville avec des coefficients discontinus. De tels problèmes se trouvent naturellement dans la mécanique des materiaux composites.

Après avoir transformé ce problème en utilisant un analogue de la transformation classique de Liouville, les bornes supérieures sont obtenues par l'application d'une technique de Rayleigh-Ritz au problème transformé. Les bornes inférieurs sont determinées en fonction des coefficients sous une forme explicite. Quelques exemples numériques montrent l'exactitude des résultats.

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Author notes

  1. C. O. Horgan

    Present address: College of Engineering, Michigan State University, East Lansing, Michigan, U.S.A.

Authors and Affiliations

  1. The Technological Institute, Northwestern University, Evanston, Illinois, USA

    C. O. Horgan & S. Nemat-Nasser

Authors
  1. C. O. Horgan
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  2. S. Nemat-Nasser
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Additional information

This work was supported by the U.S. Army Research Office under Grants DAH C04-75-G-0059, DAAG 29-76-G-0063 and DAAG 29-77-G-0034.

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Horgan, C.O., Nemat-Nasser, S. Bounds on eigenvalues of Sturm-Liouville problems with discontinuous coefficients. Journal of Applied Mathematics and Physics (ZAMP) 30, 77–86 (1979). https://doi.org/10.1007/BF01597482

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  • DOI: https://doi.org/10.1007/BF01597482

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