Upper and lower bounds to eigenvalues by weighting function approximations (Q789930)

A method is developed making use of variational principles and Rayleigh's quotient which yield lower bounds to eigenvalues. The method is the counterpart of the Rayleigh-Ritz method in the sense that the results obtained from both methods will improve, i.e. approach to the exact value, as more and more terms are considered, both rely on variational principles, they are similar systematically and conceptually but this method yields lower bounds to eigenvalues which cannot be obtained from the Rayleigh-Ritz method. For the flexural vibrations of clamped square plates \textit{L. de Vito}, \textit{G. Fichera}, \textit{A. Fusciardi} and \textit{M. Schaerf} [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 40, 725-733 (1966; Zbl 0143.17206)] and \textit{N. W. Bazley}, \textit{D. W. Fox} and \textit{J. T. Stadter} [Z. Angew. Math. Mech. 47, 251-260 (1967; Zbl 0161.442)] have presented analysis for evaluation of both lower and upper bounds and have reported accurate numerical results for a large number of eigenvalues. Recently, \textit{K. Vijayakumar} and \textit{G. K. Ramaiah} [J. Sound Vib. 56, 127-135 (1978; Zbl 0379.70024) and ibid. 59, 335-347 (1978; Zbl 0381.70027)] have also reported very accurate lower and upper bounds for a large number of eigenfrequencies for clamped rectangular orthotropic plates. The reviewer would like the authors to extend the method suggested in the present paper for the vibrations of clamped rectangular orthotropic plates and compare the values thus obtained with the results reported in the above mentioned references.