Sturm-Liouville Problems with Discontinuities at Two Interior Points
In this paper, we study the Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at two interior points. We establish a new operator A associated with the problem, prove the operator A is self-adjoint in an appropriate space H, construct the basic solutions and investigate some properties of the eigenvalues and corresponding eigenfunctions, then obtain asymptotic formulas for the eigenvalues and eigenfunctions, its Green function and the resolvent operator are also involved.
- E. C. Titchmarsh, Eigenfunctions Expansion Associated with Second Order Dierential Equations, part 1, Oxford University Press, London, UK, 1962.
- M. A. Naimark, Linear Dierential Operators, part 2, Harrap:London, 1968.
- Zhijiang Cao, Ordinary Dierential Operator, Shanghai, 1986 (in Chinese).
- Z. Akdogan, M. Demirci, O. Mukhtarov, Green function of discontinuous boundart-value problem with transmission conditions, Mathematical Models and Methods in Applied Sciences, 30:14(2007), 1719-1738.
- A. V. Likov, Yu. A. Mikhailov, The Theory of Heat and Mass Transfer, Qosenerqoizdat, 1963( in Russian).
- M. Mukhtarov Osh, S. Yakubov, Problems for ordinary dierential equations with transmission conditions, Applicable Analysis, 81:5(2002), 1033-1064.
- M. Kadakal and M. Mukhtarov Osh, Eigenvalues and eigenfunctions of Sturm-Liouville problem with two-point discontinuities containing eigenparameter-dependent boundary conditions, Scientia Magna, 6:4(2010), 33-48.
- M. Kadakal and M, Mukhtarov Osh, Discontinuous Sturm-Liouville problems containing eigenparameter in the boundary conditions, Acta Mathematica Sinica,22:5(2006), 15191528.
- M. Mukhtarov Osh and M. Kadakal, On a Sturm-Liouville type problem with discontinuous in two-points, Far East Journal of Applied Mathematics, 19:3(2005), 337-352.
- C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions,Proceedings of the Royal Society of Edinburgh A, 77(1977),293-308.
- M, Mukhtarov Osh and M. Kadakal, Some spectral properties of one Sturm-Liouville type problem with discontinuous weightSiberian Mathematical Journal, 46:4(2005),681694.
- Q. Yang and W. Wang, Asymptotic behavior of a dierential operator with discontinuities at two points, Mathematical Methods in the Applied Sciences, 34:4(2011),373-383.
- E.Tunc, M. Mukhtarov Osh, Fundamental solutions and eigenvalues of one boundaryvalue problem with transmission conditions, Applied mathematics and Computation, 157:2(2004), 347-355.
- Q. Yang and W. Wang, Spectral properties of Sturm-Liouville operators with discontinuities at nite points, Mathematical Sciences, 6(2012), no. 34, 9 pages.
- Demirci, M. Akdogan, Z, Mukhtarov. Asymptotic behavior of eigenvalues and eigenfunctions of one discontinuous boundary-value problem. Int. J. Comput. Cognition, 2:3(2004), 101-113.
- M. Mukhtarov Osh and M. Kadakal. and Muhtarov, F. S. Eigenvalues and normalized eigenfunctions of discontinuous Sturm-Liouville problem with transmission conditions, Reports on mathematical Physics, 54:1(2004), 41-56.
- J. Walter,Regular Eigenvalue Problems with Eigenvalue Parameter in the Boundary Conditions, Math. Z., 133: 4(1973), 301-312.
- E. C.Titchmarsh, Eigenfunction Expansion Associated with Second Order Dierential Equations, 2nd edition, Oxford University Press, London, 1962.
- Erdogan sen, A Sturm-Liouville problem with a discontinuous coecient and containing an eigenparameter in the boundary condition, Physics Research International, 2013 (2013), 9 pages, 2013.
- M. Kadakal and M. Mukhtarov Osh, Sturm-Liouville problems with discontinuities at two points, Computers and Mathematics with Applications, 54(2007), 1367-1379.
- M. Mukhtarov Osh, M. Kadakal, Altusik N, Eigenvalues and eigenfunctions of discontinuous Sturm-Liouville problems with eigenparameter in the boundary conditions, Indian Journal of Pure and Applied Mathematics, 102(2004), 159-193.
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