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Application of the Ritz method to non-standard eigenvalue problems

Published online by Cambridge University Press:  09 April 2009

A. L. Andrew
Affiliation:
La trobe University Melbourne
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There is an extensive literature on application of the Ritz method to eigenvalue problems of the type where L1, L2 are positive definite linear operators in a Hilbert space (see for example [1]). The classical theory concerns the case in which there exists a minimum (or maximum) eigenvalue, and subsequent eigenvalues can be located by a well-known mini-max principle [2; p. 405]. This paper considers the possibility of application of the Ritz method to eigenvalue problems of the type (1) where the linear operators L1L2 are not necessarily positive definite and a minimum (or maximum) eigenvalue may not exist. The special cases considered may be written with the eigenvalue occurring in a non-linear manner.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Mikhlin, S. G., Variational methods in mathematical physics (Pergamon, 1964).Google Scholar
[2]Courant, R. and Hilbert, D., Methods of mathematical physics. Vol. 1 (Interscience, 1953).Google Scholar
[3]Andrew, A. L., ‘Higher modes of non-radial oscillations of stars by variational methods’, Aust. J. Phys. 20 (1967), 363368.CrossRefGoogle Scholar
[4]Chandrasekhar, S., ‘A general variational principle governing the radial and the non-radial oscillations of gaseous masses’, Astrophys. J. 139 (1964), 664674.CrossRefGoogle Scholar
[5]Ledoux, P. and Walraven, Th., ‘Variable stars’, in Flügge, S., Handbuch der Physik, Vol. 51, (Springer, Berlin, 1958), 353604.Google Scholar
[6]Lebovitz, N. R., ‘On Schwarzschild's criterion for the stability of gaseous masses’, Astrophys. J. 142 (1965), 229242.Google Scholar
[7]Cowling, T. G., ‘The non-radial oscillations of polytropic stars’, Monthly Notices Roy. Astronom. Soc. 101 (1941) 367375.Google Scholar
[8]Ledoux, P. and Smeyers, P., ‘Sur le spectre des oscillations non radiales d'un modèle stellaireC. R. Acad. Sci. Paris. Sér. B 262 (1966), 841844.Google Scholar
[9]Owen, J. W., ‘The non-radial oscillations of centrally condensed stars’, Monthly Notices Roy. Astronom. Soc. 117 (1957), 384392.CrossRefGoogle Scholar
[10]Van der Borght, R., ‘The evolution of massive stars initially composed of pure hydrogen’, Aust. J. Phys. 17 (1964), 165174.CrossRefGoogle Scholar
[11]Wan, Fook Sun and Van der Borght, R., ‘Note on Cowling's method in the theory of non-radial oscillations of massive stars’, Aust. J. Phys. 19 (1966), 467470.Google Scholar
[12]Andrew, A. L., ‘A note on the Ritz method with an application to overtone stellar pulsation theory’, J. Aust. Math. Soc. 8 (1968), 275286.CrossRefGoogle Scholar
[13]Chandrasekhar, S. and Lebovitz, N. R., ‘Non-radial oscillations of gaseous masses’, Astrophys. J. 140 (1964), 15171528.CrossRefGoogle Scholar
[14]Robe, H. and Brandt, L., ‘Sur les oscillations non radiales des polytropes’, Ann. Astrophys. 29 (1966), 517524.Google Scholar
[15]Tassoul, J. L., ‘Sur l'instabilité convective d'une masse gazeuse inhomogàne’, Ann. Astrophys. 30 (1967), 363370.Google Scholar
[16]Robe, H., ‘Sur les oscillations non radiales d'une sphère’, Acad. Roy. Belg. Bull. Cl. Sci. 51 (1965), 598603.Google Scholar
[17]Marcus, M., ‘Basic theorems in matrix theory’, Nat. Bur. Standards Appl. Math. Ser. 57 (1960).Google Scholar
[18]Mihlin, S. G. (Mikhlin), ‘The stability of the Ritz method’, Soviet Math. Dokl. 1 (1960), 12301233.Google Scholar
[19]Lebovitz, N. R., ‘On the onset of convective instability’, Astrophys. J. 142 (1965), 12571260.CrossRefGoogle Scholar
[20]Andrew, A. L., ‘Non-radial oscillations of massive stars by variational methods’, Acad. Roy. Belg. Bull. Cl. Sci. 54 (1968), (10461055).Google Scholar