The Principal Eigenvalue and The Principal Eigenfunction of A Boundary-Value-Transmission Problem

Year 2019, Volume: 11 Issue: 2, 97 - 100, 31.12.2019

Kadriye Aydemir , Hayati Olğar , Oktay Mukhtarov

Abstract

 It is well-know that the Sturm-Liouville theory justifies the "separation of variables"n method for voluminous partial differential equation problems. For Sturm-Liouville problems the Rayleigh quotient is  the basis of an important approximation method that is used in physics, as well as in engineering. Although any eigenvalue can be related to its eigenfunction by the Rayleigh quotient, this quotient cannot be used to determine the exact value of the eigenvalue since eigenfunction is unknown. However, interesting and significant results can be obtained from

the Rayleigh quotient without solving the differential equation(i.e. even in the case when the eigenfunction is not known). For example, Rayleigh quotient can be quite useful in estimating the eigenvalue.  

It is the purpose of this paper to extend and generalize such important spectral properties as  eigenfunction expansion and Parseval equality for Sturm-Liouville problems with interior

singularities. We shall investigate certain spectral problems arising in the theory of the convergence of the eigenfunction expansion. Particularly, by modifying the Green's function method we

shall extend and generalize such important spectral properties as Parseval's equality, Rayleigh quotient and Rayleigh-Ritz  formula for the considered problem.

Keywords

Boundary-value problem, Rayleigh quotient

References

• Aydemir, K., Mukhtarov, O.Sh., Ol\v{g}ar, H., {\em Differential operator equations with interface conditions in modified direct sum spaces}, AIP Conference Proceeding, \textbf{1759}(2016), Doi: 10.1063/1.4959642.
• Courant, R., Hilbert, D., Methods of Mathematical Physics, vol. 1, Interscience, New York, 1953.
• Gesztesy, F., Kirsch, W., {\em One-dimensional Schrodinger operators with interactions singular on a discrete set}, J. Reine Angew. Math., \textbf{362}(1985), 27--50.
• Ismailov, Z., Ipek, P., {\em Selfadjoint singular differential operators of first order and their spectrum}, Electronic Journal of Differential Equations, (2016), 1--9
• Kandemir, M., Yakubov, Y., {\em Regular boundary value problems with a discontinuous coefficient, functional-multipoint conditions, and a linear spectral parameter}, Israel Journal of Mathematics, \textbf{180}(2010), 255--270.
• Likov, A.V., Mikhalilov, Y.A., Theory of Heat and Mass Transfer, Qosenergaizdat, (In Russian), 1963.
• Ol\v{g}ar, H., Mukhtarov, O.Sh., Aydemir, K., {\em Some properties of eigenvalues and generalized eigenvectors of one boundary value problem}, Filomat, \textbf{32(3)}(2018), 911--920.
• Pham Huy, H., Sanchez-Palencia, E., {\em Phenomenes des transmission a travers des couches minces de conductivite elevee} J. Math. Anal. Appl., \textbf{47}(1974), 284--309.
• Titeux, I., Yakubov, Ya., {\em Completeness of root functions for thermal conduction in a strip with piecewise continuous coefficients}, Math. Models Methods Appl. Sc., \textbf{7(7)}(1997), 1035--1050.
• Yakubov, S., Completeness of Root Functions of Regular Differential Operators, Pitman Monographs and Surveys in Pure and Applied Mathematics, 71. Longman Scientific and Technical, Harlow; Copublished in the United States with John Wiley and Sons, Inc., New York, 1994.