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.

**[1]**E. H. Lee,*A survey of variational methods for elastic wave propagation analysis in composites with periodic structures*, in*Dynamics of Composites*(E. H. Lee, ed.) ASME, New York (1972), 122-138**[2]**S. Nemat-Nasser and S. Minagawa,*Harmonic waves in layered composites: comparison among several schemes*, J. Appl. Mech.**42**, 699-704 (1975)**[3]**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)**[4]**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)**[5]**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**(1976), no. 3, 741–751. MR**0404752**, https://doi.org/10.1016/0022-247X(76)90193-1**[6]**S. Nemat-Nasser and K.-W. Lang,*Eigenvalue problems for heat conduction in composite materials*, Iranian J. Sci. Technology**7**, 243-260 (1979)**[7]**C. O. Horgan and S. Nemat-Nasser,*Bounds on eigenvalues of Sturm-Liouville problems with discontinuous coefficients*, Z. Angew. Math. Phys.**30**(1979), no. 1, 77–86 (English, with French summary). MR**526211**, https://doi.org/10.1007/BF01597482**[8]**S. Nemat-Nasser and C. O. Horgan,*Variational methods for eigenvalue problems with discontinuous coefficients*, Mechanics today, Vol. 5, Pergamon, Oxford-New York, 1980, pp. 365–376. MR**591514****[9]**K.-W. Lang and S. Nemat-Nasser,*Vibration and buckling of composite beams*, J. Struct. Mech.**5**, 395-419 (1977)**[10]**E. H. Lee and W. H. Yang,*On waves in composite materials with periodic structure*, SIAM J. Appl. Math.**25**, 492-499 (1973)**[11]**G. H. Golub, L. Jenning and W. H. Yang,*Waves in periodically structured media*, J. of Computational Phys.**17**, 349-357 (1975)**[12]**D. H. Hodges,*Direct solution for Sturm-Liouville systems with discontinuous coefficients*, AIAA Journal**17**, 924-926 (1979)**[13]**William B. Bickford,*Lower bounds to eigenvalues of piecewise continuous elastic systems*, Z. Angew. Math. Phys.**30**(1979), no. 1, 65–75 (English, with German summary). MR**526210**, https://doi.org/10.1007/BF01597481**[14]**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)**[15]**I. Babuška and J. E. Osborn,*Numerical treatment of eigenvalue problems for differential equations with discontinuous coefficients*, Math. Comp.**32**(1978), no. 144, 991–1023. MR**0501962**, https://doi.org/10.1090/S0025-5718-1978-0501962-0**[16]**Lothar Collatz,*The numerical treatment of differential equations. 3d ed*, Translated from a supplemented version of the 2d German edition by P. G. Williams. Die Grundlehren der mathematischen Wissenschaften, Bd. 60, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR**0109436****[17]**James Alan Cochran,*The analysis of linear integral equations*, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1972. McGraw-Hill Series in Modern Applied Mathematics. MR**0447991****[18]**M. G. Krein,*On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability*, Amer. Math. Soc. Transl. (2)**1**(1955), 163–187. MR**0073776**

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DOI:
https://doi.org/10.1090/qam/644100

Article copyright:
© Copyright 1982
American Mathematical Society