Research Article
Year 2019,
Volume: 11 Issue: 2, 97 - 100, 31.12.2019
Abstract
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.
Year 2019,
Volume: 11 Issue: 2, 97 - 100, 31.12.2019
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
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.
There are 10 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Articles |
| Authors | |
| Publication Date | December 31, 2019 |
| Published in Issue | Year 2019 Volume: 11 Issue: 2 |