Equiconvergence with Fourier Series for Non-Classical Sturm-Liouville Problems

Year 2019, Volume: 11 Issue: 2, 107 - 111, 31.12.2019

Oktay Mukhtarov Kadriye Aydemir Hayati Olğar

Abstract

Sturm-Liouville type boundary-value problems arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of chemistry, aerodynamics, electrodynamics of complex medium or polymer rheology. For example the vibration of a homogeneous loaded strings, the earth's free oscillations,  the interaction of atomic particles, sound, surface waves, heat transfer in a rod with heat capacity concentrated at the ends, electromagnetic waves and gravitational waves can be solved using the Sturmian theory. A large class of physical problems require the investigation of the Sturm-Liouville type problems with discontinuities. Examples are vibration problems under various loads such as a vibrating string with a tip mass or heat conduction through a liquid solid interface.

In this study we shall investigate some properties of the eigenfunctions of one discontinuous Sturm-Liouville Problem. We shall prove some preliminary results related to the basic solutions, Green's function, resolvent operator and selfadjointness of the considered problem. Particularly we shall present a new approach for constructing the Green's function which is not standard one generally found in textbooks. The obtained results are implemented to the investigation of the basis properties of the system of eigenfunctions in modified Hilbert spaces. Finally, we shall show that the eigenfunction expansion series regarding the convergence behaves in the same way as an ordinary Fourier series.

Keywords

Discontinuous Sturm-Liouville Problems, Green-function, jump conditions

References

• Aliyev, Z.S., Dun'yamalieva, A.A., {\em Basis properties of root functions of the Sturm-Liouville problem with a spectral parameter in the boundary conditions}, Doklady Mathematics, \textbf{88/1}(2013), 441--445.
• Allahverdiev, P.B., Bairamov, E., U\u{g}urlu,, E., {\em Eigenparameter dependent Sturm-Liouville problems in boundary conditions with transmission conditions}, J. Math. Anal. Appl., \textbf{401}(2013), 388--396.
• Ao, J., Sun, J., {\em Matrix representations of Sturm-Liouville problems with coupled eigenparameter-dependent boundary conditions}, Applied Mathematics and Computation, \textbf{244}(2014), 142--148.
• Bairamov, E., U\u{g}urlu,, E., {\em Krein's Theorems for a Dissipative Boundary Value Transmission Problem}, Complex Anal. Oper. Theory, \textbf{7}(2013), 831--842.
• Binding, P.A., Browne, P.J., Patrick J., {\em Oscillation theory for indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions}, Proc. Roy. Soc. Edinburg Sect., \textbf{127/6}(1997), 1123--1136.
• Buterin, S.A., {\em On half inverse problem for differential pencils with the spectral parameter in boundary conditions}, Tamkang J. Math., \textbf{42/3}(2011), 355--364.
• Fulton, C.T., {\em Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions}, Proc. Roy. Soc. Edin., \textbf{77A}(1977), 293--308.
• Kandemir, M., Mukhtarov, O.Sh, Yakubov, Y.Y., {\em Irregular boundary value problems with discontinuous coefficients and the eigenvalue parameter}, Mediterr, J. Math., \textbf{6}(2009), 317--338.
• Li, K., Sun, J., Hao, X., Bao, Q., {\em Spectral analysis for discontinuous non-self-adjoint singular Dirac operators with eigenparameter dependent boundary condition}, J. Math. Anal. Appl., \textbf{453}(2017), 304--316.
• Mukhtarov, O.Sh., Ol\v{g}ar, H., Aydemir, K., {\em Resolvent operator and spectrum of new type boundary value problems}, Filomat, \textbf{29:7}(2015), 1671--1680.
• Ol\v{g}ar, H., Mukhtarov, O.Sh., {\em Weak eigenfunctions of two-interval Sturm-Liouville problems together with interaction conditions}, Journal of Mathematical Physics, \textbf{58}(2017), 042201.
• Panakhov E.S., Sat, M., {\em Reconstruction of potential function for Sturm-Liouville operator with coulomb potential}, Boundary Value Problems, \textbf{49}(2013), pp.2013.
• Schneider, A., {\em A note on eigenvalue problems with eigenvalue parameter in the boundary condition}, Math. Z., \textbf{136}(1794), 163--167.
• Sherstyuk, A.I., Problems of Theoretical Physics, Gos. Univ., Leningrad Vol. 3., 1988.
• Shkalikov, A.A., {\em Boundary value problems for ordinary differential equations with a parameter in boundary condition}, Trudy Sem. Imeny I.G. Petrowsgo, \textbf{9}(1983), 190--229.
• Walter, J., {\em Regular eigenvalue problems with eigenvalue parameter in the boundary conditions}, Math. Z., \textbf{133}(1973), 301--312.
• Wang A., Zettl, A., {\em Self-adjoint Sturm-Liouville problems with discontinuous boundary conditions}, Methods And Applications of Analysis, \textbf{22(1)}(2015), 037--066.
• Yakubov, S., Yakubov, Y., \textit{Abel basis of root functions of regular boundary value problems}, Math. Nachr., \textbf{19}7(1999), 157--187.
• Zhang, X., Sun, J., {\em The determinants of fourth order dissipative operators with transmission conditions}, J. Math. Anal. Appl., \textbf{410/1}(2014), 55--69.