Eigenvalue Problems with Interface Conditions

Abstract

Sturm-Liouville type boundary value problems arise a result of using the Fourier’s method of separation of variables to solve the classical partial differential equations of mathematical physics, such as the Laplace’s equation, the heat equation and the wave equation. A large class of physical problems require the investigation of the Sturm-Liouville type problems with the eigen-parameter in the boundary conditions. Also, many physical processes, such as the vibration of loaded strings, the interaction of atomic particles, electrodynamics of complex medium, aerodynamics, polymer rheology or the earth’s free oscillations yield. Sturm-Liouville eigenvalue problems( see, for example, [1, 6, 12, 13]). On the other hand, the Sturm-Liouville problems with transmission conditions (such conditions are known by various names including transmission conditions, interface conditions, jump conditions and discontinuous conditions) arise in problems of heat and mass transfer, various physical transfer problems [8], radio science [7], and geophysics [9]. In this work we shall investigate some spectral properties of a regular Sturm-Liouville problem on a finite interval with the transmission conditions at a point of interaction. We prove that the set of eigenfunctions for the problem under consideration forms a basis in the corresponding Hilbert space.

Keywords

• Sturm-Liouville type boundary value problems
• transmission conditions
• transmission

References

1. [1] K. Aydemir and O. Mukhtarov, Variational principles for spectral analysis of one Sturm-Liouville problem with transmission conditions, Advances in Difference Equations, 2016:76 (2016)
2. [2] E. Bairamov and E. Ugurlu, On the characteristic values of the real component of a dissipative boundary value transmission problem, Appl. Math. and Comp. 218, 9657-9663(2012).
3. [3] P. A. Binding and P. J. Browne, Oscillation theory for indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 1123-1136.
4. [4] C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edin. 77A(1977), P. 293-308.
5. [5] B. Harmsen and A. Li, Discrete Sturm-Liouville problems with parameter in the boundary conditions, Journal of Difference Equations and Applications, 8(11)(2002), 969-981.
6. [6] Q. Kong, H. Wu and A. Zettl, Geometric aspects of Sturm-Liouville problems I. Structures on spaces of boundary conditions, Proceedings of the Royal Society of Edinburgh Section A Mathematics, 130(3), 2000, 561-589.
7. [7] F. R. Lapwood and T. Usami, Free Oscillations of the Earth, Cambridge University Press, Cambridge, 1981.
8. [8] A. V. Likov and Yu. A. Mikhailov, The Theory of Heat and Mass Transfer, Translated from Russian by I. Shechtman, Israel Program for Scientific Translations, Jerusalem, 1965.

9. [9] O. N. Litvinenko and V. I. Soshnikov, The Theory of Heteregenous Lines and their Applications in Radio Engineering (in Russian), Radio, Moscow, 1964.
10. [10] O. Sh. Mukhtarov and K. Aydemir, The Eigenvalue Problem with Interaction Conditions at One Interior Singular Point, Filomat 31:17 (2017), 5411-5420.
11. [11] O. Sh. Mukhtarov, H. Olgar and K. Aydemir, Resolvent Operator and Spectrum of New Type Boundary Value Problems. Filomat, 29(7) (2015), 1671-1680.
12. [12] J. D. Pryce, Numerical solution of Sturm-Liouville Problems, Oxford University Press, 1993.
13. [13] M. Rotenberg, Theory and application of Sturmian functions. Adv. Atomic. and Mol. Phys. 6(1970).
14. [14] A. N. Tikhonov and A. A. Samarskii, Equations Of Mathematical Physics, Oxford and New York, Pergamon (1963).
15. [15] J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z., 133(1973), 301-312.
16. [16] N. N. Voitovich, B. Z. Katsenelbaum and A. N. Sivov, Generalized Method Of Eigen-Vibration In The Theory Of Diffraction, Nakua, Moskow, 1997 (Russian).