Articles

THE REPRESENTATION OF THE SOLUTION OF STURM-LIOUVILLE EQUATION WITH DISCONTINUITY CONDITIONS

  • Ozge AKCAY
Expand
  • Department of Mathematics, Mersin University, Mersin 33343, Turkey

Received date: 2017-02-13

  Revised date: 2017-09-30

  Online published: 2018-08-25

Supported by

This work is supported by the Scientific and Technological Research Council of Turkey (TÜB?TAK).

Abstract

The aim of this paper is to construct the integral representation of the solution of Sturm-Liouville equation with eigenparameter-dependent discontinuity conditions at an interior point of the finite interval. Moreover, we examine the properties of the kernel function of this integral representation and obtain the partial differential equation provided by this kernel function.

Cite this article

Ozge AKCAY . THE REPRESENTATION OF THE SOLUTION OF STURM-LIOUVILLE EQUATION WITH DISCONTINUITY CONDITIONS[J]. Acta mathematica scientia, Series B, 2018 , 38(4) : 1195 -1213 . DOI: 10.1016/S0252-9602(18)30808-7

References

[1] Akhmedova E N, Huseynov H M. On solution of the inverse Sturm-Liouville problem with discontinuous coefficients. Trans Natl Acad Sci Azerb, 2007, 27:33-44
[2] Amirov R Kh. On Sturm-Liouville operators with discontinuity conditions inside an interval. J Math Anal Appl, 2006, 317:163-176
[3] Amirov R Kh, Ozkan A S, Keskin B. Inverse problems for impulsive Sturm-Liouville operator with spectral parameter linearly contained in boundary conditions. Integral Transforms and Special Functions, 2009, 20:607-618
[4] Anderssen R S. The effect of discontinuous in density and shear velocity on the asymptotic overtone structure of torsional eigenfrequencies of the earth. Geophys J R Astr Soc, 1997, 50:303-309
[5] Aydemir K, Mukhtarov O Sh. Completeness of one two-interval boundary value problem with transmission conditions. Miskolc Mathematical Notes, 2014, 15:293-303
[6] Aydemir K, Mukhtarov O Sh. Class of Sturm-Liouville problems with eigenparameter dependent transmission conditions. Numerical Functional Analysis and Optimization, 2017, 38:1260-1275
[7] Freiling G, Yurko V A. Inverse Sturm-Liouville Problems and Their Applications. Huntington, NY:Nova Science Publishers Inc, 2001
[8] Gomilko A, Pivovarchik V. On basis properties of a part of eigenfunctions of the problem of vibrations of a smooth inhomogeneous string damped at the midpoint. Math Nachr, 2002, 245:72-93
[9] Guseinov I M, Mammadova L I. Reconstruction of the diffusion equation with singular coefficients for two spectra. Doklady Mathematics, 2014, 90:401-404
[10] Hald O H. Discontinuous inverse eigenvalue problems. Comm Pure Appl Math, 1984, 37:539-577
[11] Huseynov H M, Dostuyev F Z. On determination of Sturm-Liouville operator with discontinuity conditions with respect to spectral data. Proceedings of the Institute of Mathematics and Mechanics, 2016, 42:143-153
[12] Kadakal M, Mukhtarov O Sh. Discontinuous Sturm-Liouville problems containing eigenparameter in the boundary conditions. Acta Mathematica Sinica, English Series, 2006, 22:1519-1528
[13] Kruger R J. Inverse problems for nonabsorbing media with discontinuous material properties. J Math Phys, 1982, 23:396-404
[14] Lapwood F R, Usami T. Free Oscillations of the Earth. Cambridge:Cambridge Univ Press, 1981
[15] Levitan B M. Inverse Sturm-Liouville Problems. Utrecht:VNU Sci Press, 1987
[16] Lykov A V, Mikhailov Y A. The Theory of Heat and Mass Transfer. Moscow:Qosenergaizdat, 1963(Russian)
[17] Mamedov Kh R, Akcay O. Inverse eigenvalue problem for a class of Dirac operators with discontinuous coefficient. Boundary Value Problems, 2014, 2014:110, DOI:10.1186/1687-2770-2014-110
[18] Mamedov Kh R, Akcay O. Necessary and sufficient conditions for the solvability of inverse problem for a class of Dirac operators. Miskolc Mathematical Notes, 2015, 16:257-275
[19] Mammadova L I. Representation of the solution of Sturm-Liouville equation with discontinuity conditions interior to interval. Proceedings of IMM of NAS of Azerbaijan, 2010, 33:127-136
[20] Marchenko V A. Sturm-Liouville Operators and Applications. Providence, Rhode Island:AMS Chelsea Publishing, 2011
[21] Mochizuki K, Trooshin I. Inverse problem for interior spectral data of Sturm-Liouville operator. J Inverse Ill-Posed Probl, 2001, 9:425-433
[22] Muhtarov F, Kandemir M, Mukhtarov O Sh. Multi-point transmission problems for Sturm-Liouville equation with an abstract linear operator. AIP Conference Proceedings, 2017, 1833:020022, DOI:10.1063/1.4981670
[23] Mukhtarov O Sh, Aydemir K. Eigenfunction expansion for Sturm-Liouville problems with transmission conditions at one interior point. Acta Mathematica Scientia, 2015, 35B:639-649
[24] Mukhtarov O Sh, Kandemir M. Asymptotic behaviour of eigenvalues for the discontinuous boundary value problem with functional-transmission conditions. Acta Mathematica Scientia, 2002, 22B:335-345
[25] Ozkan A S, Keskin B. Spectral problems for SturmLiouville operator with boundary and jump conditions linearly dependent on the eigenparameter. Inverse Problems in Science and Engineering, 2012, 20:799-808
[26] Olgar H, Mukhtarov O S, Aydemir K. Some properties of eigenvalues and generalized eigenvectors of one boundary-value problem. AIP Conference Proceedings, 2016, 1759:020060, DOI:10.1063/1.4959674
[27] Olgar H, Muhtarov F S. The basis property of the system of weak eigenfunctions of a discontinuous SturmLiouville problem. Mediterr J Math, 2017, 14:114, DOI:10.1007/s00009-017-0915-9
[28] Pöschel J, Trubowitz E. Inverse Spectral Theory. New York:Academic Press, 1987
[29] Şen E. A class of second-order differential operators with eigenparameter-dependent boundary and transmission conditions. Math Methods Appl Sci, 2014, 37:2952-2961
[30] Şen E, Mukhtarov O Sh. Spectral properties of discontinuous Sturm-Liouville problems with a finite number of transmission conditions. Mediterr J Math, 2014, DOI:10.1007/s00009-014-0487-x
[31] Shepelsky D G. The inverse problem of reconstruction of medium's conductivity in a class of discontinuous and increasing functions. Adv Soviet Math, 1994, 19:209-231
[32] Tikhonov A N, Samarskii A A. Equations of Mathematical Physics. Dover, New York:Dover Books on Physics and Chemistry, 1990
[33] Yang C -F, Yang X -P. An interior inverse problem for the Sturm-Liouville operator with discontinuous conditions. Appl Math Lett, 2009, 22:1315-1319
[34] Yang C -F. An interior spectral problem for discontinuous boundary value problems. Integral Equations and Operator Theory, 2009, 65:593-604.
[35] Willis C. Inverse problems for torsional modes. Geophys J R Astr Soc, 1984, 78:847-853
Options
Outlines

/