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Green's Functions for Two-Interval Sturm-Liouville Problems in Direct Sum Space

Year 2018, Volume: 10, 117 - 120, 29.12.2018

Abstract

The theme of the development of the theory and applications of
Green's Functions is skilfully used to motivate and connect clear
accounts of the theory of distributions, Fourier series and
transforms, Hilbert spaces, linear integral equations. In this work
we analyze the Green's functions of boundary value problems defined
on two interval and associated with Schrodinger operators with
interaction conditions. We have constructed some special
eigensolutions of this problem and presented a formula and the
existence condition of Green's function in terms of the general
solution of a corresponding homogeneous equation. We have obtained
the relation between two Green's functions of two nonhomogeneous
problems. It allows us to find Green's function for the same
equation but with different additional conditions. These problems
include the cases in which the boundary has two, one or none
vertices. In each case, the Green's functions, the eigenvalues and
the eigenfunctions are given in terms of asymptotic formulas. A
preliminary study of two-point regular boundary value problems with
additional transmission conditions was developed by the authors of
this study  under the denomination of two-point transmission
boundary value problems. In each case, it is essential to describe
the solutions of the Schrodinger equation on the interior nodes of
the path. As an consequence of this property, we can characterize
those boundary value problems that are regular and then we obtain
their corresponding Green's function, as well as the eigenvalues and
the eigenfunctions for the regular case.

References

  • Allahverdiev, B. P., Bairamov, E. and E. Ugurlu, Eigenparameter dependent Sturm-Liouville problems in boudary conditions with transmissionconditions, J. Math. Anal. Appl. 401, 88-396 (2013)
  • Amirov, R.Kh., Ozkan, A.S. and Keskin, B., Inverse problems for impulsive Sturm-Liouville operator with spectral parameter linearly containedin boundary conditions, Integral Transforms and Special Functions, 20(8)(2009), 607-618.
  • Ao J. and Sun J. Matrix representations of SturmˆuLiouville problems with coupled eigenparameter-dependent boundary conditions, AppliedMathematics and Computation 244, 142ˆu148(2014).
  • Aydemir, K., Mukhtarov, O., A Class of Sturm-Liouville Problems with Eigenparameter Dependent Transmission Conditions, NumericalFunctional Analysis and Optimization, (2017), Doi: 10.1080/01630563.2017.1316995.
  • Binding, P. A. and Browne, P. J., Oscillation theory for indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions,Proc. Roy. Soc. Edinburgh Sect. A 127.
  • Kandemir, M. and Mukhtarov, O. Sh., Nonlocal Sturm-Liouville Problems with Integral Terms in the Boundary Conditions, Electronic Journalof Di erential Equations, Vol. 2017 (2017), No. 11, pp. 1-12.
  • Markus, A. S. and Matsayev, V. I. Comparison theorems for spectra of linear Operators and spectral asymptotics, Tr. Mosk. Mat. Obshch.,(1982), vol.45, pp. 133- 181; English transl., Trans. Moscow Math. Soc., 1(1984), 139-188.
  • Mukhtarov, O.Sh., Olˇgar, H. and Aydemir, K., Resolvent Operator and Spectrum of New Type Boundary Value Problems, Filomat, 29:7 (2015),1671-1680.
  • Mukhtarov, O. Sh. and Kandemir, M., Asymptotic behaviour of eigenvalues for the discontinuous boundary- value problem with Functional-Transmissin conditions, Acta Mathematica Scientia vol 22 B(3), (2002) pp.335-345.
  • Olğar, H., Mukhtarov, O. Sh. and Aydemir, K., Some properties of eigenvalues and generalized eigenvectors of one boundary value problem,Filomat, 32:3, 911-920(2018).
  • Panakhov, E. S. and Gulsen, T., On discontinuous Dirac systems with eigenvalue dependent boundary conditions, AIP Conference Proceeding,1648, 260003 (2015), 1-4; https://doi.org/10.1063/1.4912520.
Year 2018, Volume: 10, 117 - 120, 29.12.2018

Abstract

References

  • Allahverdiev, B. P., Bairamov, E. and E. Ugurlu, Eigenparameter dependent Sturm-Liouville problems in boudary conditions with transmissionconditions, J. Math. Anal. Appl. 401, 88-396 (2013)
  • Amirov, R.Kh., Ozkan, A.S. and Keskin, B., Inverse problems for impulsive Sturm-Liouville operator with spectral parameter linearly containedin boundary conditions, Integral Transforms and Special Functions, 20(8)(2009), 607-618.
  • Ao J. and Sun J. Matrix representations of SturmˆuLiouville problems with coupled eigenparameter-dependent boundary conditions, AppliedMathematics and Computation 244, 142ˆu148(2014).
  • Aydemir, K., Mukhtarov, O., A Class of Sturm-Liouville Problems with Eigenparameter Dependent Transmission Conditions, NumericalFunctional Analysis and Optimization, (2017), Doi: 10.1080/01630563.2017.1316995.
  • Binding, P. A. and Browne, P. J., Oscillation theory for indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions,Proc. Roy. Soc. Edinburgh Sect. A 127.
  • Kandemir, M. and Mukhtarov, O. Sh., Nonlocal Sturm-Liouville Problems with Integral Terms in the Boundary Conditions, Electronic Journalof Di erential Equations, Vol. 2017 (2017), No. 11, pp. 1-12.
  • Markus, A. S. and Matsayev, V. I. Comparison theorems for spectra of linear Operators and spectral asymptotics, Tr. Mosk. Mat. Obshch.,(1982), vol.45, pp. 133- 181; English transl., Trans. Moscow Math. Soc., 1(1984), 139-188.
  • Mukhtarov, O.Sh., Olˇgar, H. and Aydemir, K., Resolvent Operator and Spectrum of New Type Boundary Value Problems, Filomat, 29:7 (2015),1671-1680.
  • Mukhtarov, O. Sh. and Kandemir, M., Asymptotic behaviour of eigenvalues for the discontinuous boundary- value problem with Functional-Transmissin conditions, Acta Mathematica Scientia vol 22 B(3), (2002) pp.335-345.
  • Olğar, H., Mukhtarov, O. Sh. and Aydemir, K., Some properties of eigenvalues and generalized eigenvectors of one boundary value problem,Filomat, 32:3, 911-920(2018).
  • Panakhov, E. S. and Gulsen, T., On discontinuous Dirac systems with eigenvalue dependent boundary conditions, AIP Conference Proceeding,1648, 260003 (2015), 1-4; https://doi.org/10.1063/1.4912520.
There are 11 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Oktay Mukhtarov

Kadriye Aydemir

Hayati Olğar

Publication Date December 29, 2018
Published in Issue Year 2018 Volume: 10

Cite

APA Mukhtarov, O., Aydemir, K., & Olğar, H. (2018). Green’s Functions for Two-Interval Sturm-Liouville Problems in Direct Sum Space. Turkish Journal of Mathematics and Computer Science, 10, 117-120.
AMA Mukhtarov O, Aydemir K, Olğar H. Green’s Functions for Two-Interval Sturm-Liouville Problems in Direct Sum Space. TJMCS. December 2018;10:117-120.
Chicago Mukhtarov, Oktay, Kadriye Aydemir, and Hayati Olğar. “Green’s Functions for Two-Interval Sturm-Liouville Problems in Direct Sum Space”. Turkish Journal of Mathematics and Computer Science 10, December (December 2018): 117-20.
EndNote Mukhtarov O, Aydemir K, Olğar H (December 1, 2018) Green’s Functions for Two-Interval Sturm-Liouville Problems in Direct Sum Space. Turkish Journal of Mathematics and Computer Science 10 117–120.
IEEE O. Mukhtarov, K. Aydemir, and H. Olğar, “Green’s Functions for Two-Interval Sturm-Liouville Problems in Direct Sum Space”, TJMCS, vol. 10, pp. 117–120, 2018.
ISNAD Mukhtarov, Oktay et al. “Green’s Functions for Two-Interval Sturm-Liouville Problems in Direct Sum Space”. Turkish Journal of Mathematics and Computer Science 10 (December 2018), 117-120.
JAMA Mukhtarov O, Aydemir K, Olğar H. Green’s Functions for Two-Interval Sturm-Liouville Problems in Direct Sum Space. TJMCS. 2018;10:117–120.
MLA Mukhtarov, Oktay et al. “Green’s Functions for Two-Interval Sturm-Liouville Problems in Direct Sum Space”. Turkish Journal of Mathematics and Computer Science, vol. 10, 2018, pp. 117-20.
Vancouver Mukhtarov O, Aydemir K, Olğar H. Green’s Functions for Two-Interval Sturm-Liouville Problems in Direct Sum Space. TJMCS. 2018;10:117-20.