Lower bounds for eigenvalues of Sturm-Liouville problems with discontinuous coefficients: integral equation methods
Authors:
C. O. Horgan, J. P. Spence and A. N. Andry
Journal:
Quart. Appl. Math. 39 (1982), 455-465
MSC:
Primary 34B25
DOI:
https://doi.org/10.1090/qam/644100
MathSciNet review:
644100
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Abstract: This paper is concerned with application of the theory of Fredholm integral equations to Sturm-Liouville problems with discontinuous coefficients. Such problems occur naturally in many areas of application involving mechanics of heterogeneous media. Due to the nonsmoothness of the coefficients, the eigenvalue spectrum may exhibit severe irregularities. Lower bounds for eigenvalues are obtained which reflect this behavior. Numerical results are presented for example problems previously treated using other methods.
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S. Nemat-Nasser and S. Minagawa, Harmonic waves in layered composites: comparison among several schemes, J. Appl. Mech. 42, 699–704 (1975)
C. O. Horgan, K.-W. Lang and S. Nemat-Nasser, Harmonic waves in layered composites: new bounds on eigenfrequencies, J. Appl. Mech. 45, 829–833 (1978)
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S. Nemat-Nasser and S. Minagawa, Harmonic waves in layered composites: comparison among several schemes, J. Appl. Mech. 42, 699–704 (1975)
C. O. Horgan, K.-W. Lang and S. Nemat-Nasser, Harmonic waves in layered composites: new bounds on eigenfrequencies, J. Appl. Mech. 45, 829–833 (1978)
R. S. Anderssen and J. R. Cleary, Asymptotic structure in torsional free oscillations of the earth I—overtone structure, Geophys. J. R. Astr. Soc. 39, 241–268 (1974)
A. McNabb, R. S. Anderssen and E. R. Lapwood, Asymptotic behavior of the eigenvalues of a Sturm-Liouville system with discontinuous coefficients, J. Math. Anal. Appl. 54, 741–751 (1976)
S. Nemat-Nasser and K.-W. Lang, Eigenvalue problems for heat conduction in composite materials, Iranian J. Sci. Technology 7, 243–260 (1979)
C. O. Horgan and S. Nemat-Nasser, Bounds on eigenvalues of Sturm-Liouville problems with discontinuous coefficients. J. of Appl. Math. and Phys. (ZAMP), 30, 77–86 (1979)
C. O. Horgan and S. Nemat-Nasser, Variational methods for eigenvalue problems in composites, Proceedings of the IUTAM Symposium on Variational Methods in the Mechanics of Solids, Northwestern University, Sept. 1978 (S. Nemat-Nasser, ed.), Pergamon Press, New York (1980), 52–58
K.-W. Lang and S. Nemat-Nasser, Vibration and buckling of composite beams, J. Struct. Mech. 5, 395–419 (1977)
E. H. Lee and W. H. Yang, On waves in composite materials with periodic structure, SIAM J. Appl. Math. 25, 492–499 (1973)
G. H. Golub, L. Jenning and W. H. Yang, Waves in periodically structured media, J. of Computational Phys. 17, 349–357 (1975)
D. H. Hodges, Direct solution for Sturm-Liouville systems with discontinuous coefficients, AIAA Journal 17, 924–926 (1979)
W. B. Bickford, Lower bounds to eigenvalues of piecewise continuous elastic systems, J. of Appl. Math. and Phys. (ZAMP), 30, 65–75 (1979)
B. E. Goodwin and W. E. Boyce, The vibrations of a random elastic string: the method of integral equations, Quart. Appl. Math. 22, 261–266 (1964)
I. Babuška and J. E. Osborn, Numerical treatment of eigenvalue problems for differential equations with discontinuous coefficients, Math. of Computation 32, 991–1023 (1978)
L. Collatz, The numerical treatment of differential equations, (3rd ed.), Springer-Verlag, Berlin, 1960
J. A. Cochran, Analysis of linear integral equations, McGraw-Hill, New York, 1972
M. G. Krein, On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability, AMS Translations Ser. 2, 1, 163–187 (1955)
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Article copyright:
© Copyright 1982
American Mathematical Society
