eISSN: 2618-6446
Conferences Latest Issue Archive Future Issues About Us JOURNALS

SETSCI - Volume 3 (2018)
ISAS2018-Winter - 2nd International Symposium on Innovative Approaches in Scientific Studies, Samsun, Turkey, Nov 30, 2018

Completeness of the Weak Eigenfunctions of one Boundary-Value-Transmission Problem (ISAS2018-Winter_223)
Hayati Olgar1*, O. Sh.   Mukhtarov  2, F. S. Muhtarov3, Kadriye Aydemir4
1Gaziosmanpaşa University, Tokat, Turkey
2AzerbaijanNational Academy of Sciences  , Bakü, Azerbaijan
3AzerbaijanNational Academy of Sciences  , Bakü, Azerbaijan
4Amasya University, Amasya, Turkey
* Corresponding author: holgar@gmail.com
Published Date: 2019-01-14   |   Page (s): 1168-1172   |    5     3

ABSTRACT Recently, the basis properties and eigenfunction expansions in various function spaces of the eigenfunction of the regular
boundary value problems with spectral parameter in the boundary conditions have been investigated by many mathematicans. However in different areas of applied mathematics and physics many problems arise in the form of singular boundary value problems
involving transmission conditions at the interior singular points. Such problems are called boundary-value-transmission problems.
For example, some boundary value problems with transmission conditions arise in heat and mass transfer problems, in vibrating string problems when the string loaded with additional point masses, in diffraction problems. It is not clear how the extend
the classical methods to a problem with additional transmission conditions. Another major diffuculty lies in the completeness of
the eigenfunctions, since the eigenvalues of a Sturm-Liouville problems with transmission conditions may not have an asymptotic expansion. This study devoted to the investigation of the Sturm-Liouville type boundary-value problems with supplementary
transmission conditions. We introduce a new concept, so-called generalized eigenfunctions and study the completeness and basis
properties of systems of generalized eigenfunctions for the problem under consideration.  
KEYWORDS Boundary value problems, boundary conditions, weak eigenfunctions, eigenvalue, completeness
REFERENCES [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] K. Aydemir, H. Olgar, O. Sh. Mukhtarov and F. S. Muhtarov, Di ˘ fferential operator equations with interface conditions in modified direct sum spaces, Filomat, 32:3 (2018), 921-931.
[3] B. P. Belinskiy and J. P. Dauer, On a regular Sturm - Liouville problem on a finite interval with the eigenvalue parameter appearing linearly in the boundary conditions, Spectral theory and computational methods of SturmLiouville problem. Eds. D. Hinton and P. W. Schaefer, 1997.
[4] B. P. Belinskiy and J. P. Dauer, Eigenoscillations of mechanical systems with boundary conditions containing the frequency, Quarterly of Applied Math. 56 (1998), 521-541.
[5] P. A. Binding, P. J. Browne and B. A. Watson, Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter, II, J. Comput. Appl. Math. 148 (2002), 147168.
[6] 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.
[7] I. C. Gohberg and M. G. Krein, Introduction to The Theory of Linear Non-Selfadjoint Operators, Translation of Mathematical Monographs, vol. 18, Amer. Math. Soc., Providence, Rhode Island, 1969.
[8] M. Kandemir and O. Sh. Mukhtarov, Nonlocal Sturm-Liouville Problems with Integral Terms in the Boundary Conditions, Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 11, pp. 1-12.
[9] M. V. Keldysh, On the eigenvalues and eigenfunctions of certain classes of non-self-adjoint equations, Dokl Akad. Nauk SSSR (in Russian) 77 (1951), 11-14; English transl. in this volume.
[10] N. B. Kerimov and R. G. Poladov, Basis properties of the system of eigenfunctions in the Sturm-Liouville problem with a spectral parameter in the boundary conditions, ISSN 1064-5624, Doklady Mathematics, 2012, Vol. 85, No. 1, pp. 8-13.
[11] O. A. Ladyzhenskaia, The Boundary Value Problems of Mathematical Physics, Springer-Verlag, New York 1985.
[12] A. V. Likov and Y. A. Mikhalilov, The Theory of Heat and Mass Transfer, Qosenergaizdat, 1963. (In Russian).
[13] O.Sh. Mukhtarov, H. Olgar and K. Aydemir, Resolvent Operator and Spectrum of New Type Boundary Value ˇ
Problems, Filomat, 29:7 (2015), 1671-1680.
[14] O. Sh. Mukhtarov and S. Yakubov, Problems for differential equations with transmission conditions, Applicable Anal. 81 (2002), 10331064
[15] O. Sh. Mukhtarov, Discontinuous boundary-value problem with spectral parameter in boundary conditions.
Turkish J. Math. 18 (1994), 183192.
[16] H. Olgar and O. Sh. Mukhtarov, Weak Eigenfunctions Of Two-Interval Sturm-Liouville Problems Together ˇ With Interaction Conditions, Journal of Mathematical Physics, 58, 042201 (2017) DOI: 10.1063/1.4979615.
[17] H. Olgar, O. Sh. Mukhtarov and K. Aydemir, Some properties of eigenvalues and generalized eigenvectors ˇ of one boundary value problem, Filomat, 32:3 (2018), 911-920.
[18] A. M. Sarsenbi and A. A. Tengaeva, On the basis properties of root functions of two generalized eigenvalue problems, ISSN 0012-2661, Differential Equations, 2012, Vol. 48, No. 2, pp. 306-308.
[19] I. Titeux and Y. Yakubov, Completeness of root functions for thermal conduction in a strip with piecewise continuous coefficients, Math. Models Methods Appl. Sc. 7 (1997), 10351050.
[20] J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z., 133(1973), 301-312.
[21] A. Wang, J. Sun, X. Hao and S. Yao, Completeness of eigenfunctions of Sturm - Liouville problems with transmission conditions, Methods and Application of Analysis, 16(3) (2009), 299-312.
[22] A. Zettl, Adjoint and self-adjoint boundary value problems with interface conditions, SIAM J. Appl. Math. 16 (1968), 851859.

SET Technology - Turkey

eISSN  : 2618-6446

E-mail : info@set-science.com
+90 533 2245325

Tokat Technology Development Zone Gaziosmanpaşa University Taşlıçiftlik Campus, 60240 TOKAT-TURKEY
©2018 SET Technology