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Computation of Trace and Nodal Points of Eigenfunctions for a Sturm-Liouville Problem with Retarded Argument

Year 2018, Volume: 39 Issue: 3, 597 - 607, 30.09.2018

Erdoğan Şen

https://doi.org/10.17776/csj.416602
Cited By: 1

Abstract

In this study, a formula for regularized sums of eigenvalues and nodal
points of eigenfunctions for a discontinuous Sturm-Liouville problem with a
constant retarded argument. Contrary to standart problems the spectral
parameter appears not only in the differential equation, but also in one of the
boundary conditions. Thus, we see whether the nodal points of eigenfunctions
and the trace change or not.

Keywords

Differential equation with retarded argument transmission conditions spectrum nodal points regularized trace

References

  • [1]. Norkin S.B., Differential equations of the second order with retarded argument, Translations of Mathematical Monographs, AMS, Providence, RI, 1972.
  • [2]. Mukhtarov O.Sh., Kadakal M., Muhtarov F.S., On discontinuous Sturm-Liouville problems with transmission conditions, J. Math. Kyoto Univ., 44-4 (2004) 779-798.
  • [3]. Mukhtarov O.Sh., Tunç E., Eigenvalue problems for Sturm--Liouville equations with transmission conditions, Israel J. Math. 144 (2004) 367--380.
  • [4]. Aydemir K., Mukhtarov O.Sh., Class of Sturm-Liouville problems with eigenparameter dependent transmission conditions, Numer. Funct. Anal. Optim., 38-10 (2017) 1260-1275.
  • [5]. Çakmak Y., Keskin B., Uniqueness theorems for Sturm-Liouville operator with parameter dependent boundary conditions and finite number of transmission conditions, Cumhuriyet Sci. J., 38-3 (2017) 535-543.
  • [6]. Kandemir M., Asymptotic distribution of eigenvalues for fourth-order boundary-value problem with discontinuous coefficients and transmission conditions, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 66-1 (2017) 133-152.
  • [7]. Li K., Sun J., Hao X., Weyl function of Sturm-Liouville problems with transmission conditions at finite interior points, Mediter. J. Math., 14-189 (2017) 1-15.
  • [8]. Uğurlu E., Taş K., A new method for dissipative dynamic operator with transmission conditions, Complex Anal. Oper. Theory, 12 (2018) 1027-1055.
  • [9]. Zettl A., Adjoint and self-adjoint boundary value problems with interface conditions, SIAM J. Appl. Math., 16 (1968) 851-859.
  • [10]. Buschmann D., Stolz G., Weidmann J., One-dimensional Schrödinger operators with local point interactions, J. Reine Angew. Math., 467 (1995) 169-186.
  • [11]. Gelfand I.M., Levitan B.M., On a formula for eigenvalues of a differential operator of second order, Doklady Akademii Nauk SSSR, 88 (1953) 593--596 (Russian).
  • [12]. Dikii L.A., Trace formulas for Sturm-Liouville differential equations, Uspekhi Mat. Nauk, 12-3 (1958) 111-143.
  • [13]. Maksudov F.G., Bayramoglu M., Adıguzelov E., On a regularized traces of the Sturm-Liouville operator on a finite interval with the unbounded operator coefficient, Doklady Akademii Nauk SSSR, 277-4 (1984); English translation: Soviet Math. Dokl., 30 (1984) 169-173.
  • [14]. Bayramoglu M., Sahinturk H., Higher order regularized trace formula for the regular Sturm-Liouville equation contained spectral parameter in the boundary condition, Appl. Math. Comput., 186 (2007) 1591-1599.
  • [15]. Gül E., On the regularized trace of a second order differential operator, Appl. Math. Comput., 198 (2008) 471-480.
  • [16]. Yang C-F., Trace and inverse problem of a discontinuous Sturm-Liouville operator with retarded argument, J. Math. Anal. Appl., 395 (2012) 30-41.
  • [17]. Bayramoglu M., Bayramov A., Şen E., A regularized trace formula for a discontinuous Sturm-Liouville operator with delayed argument, Electronic Journal of Differential Equations, 2017-104 (2017) 1-12.
  • [18]. Hira F., A trace formula for the Sturm-Liouville type equation with retarded argument, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 66-1 (2017) 124-132.
  • [19]. Keskin B., Ozkan A.S., Inverse nodal problems for impulsive Sturm-Liouville equation with boundary conditions depending on the parameter, Advances in Analysis, 2-3 (2017) 151-156.
  • [20]. Keskin B., Reconstruction of the Volterra-type integro-differential operator from nodal points, Boundary Value Problems, 2018-47 (2018) 1-8.
  • [21]. Keskin B., Ozkan A.S., Inverse nodal problems for Dirac-type integro-differential operators, J. Differ. Eq., 263 (2017) 8838-8847.
  • [22]. Bayramov A., Çalışkan S., Uslu S., Computation of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument, Appl. Math. Comput., 191 (2007) 592-600.
  • [23]. Şen E., Bayramov A., Calculation of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument which contains a spectral parameter in the boundary condition, Math. Comput. Model., 54-(11-12) (2011) 3090-3097.
  • [24]. Yurko V.A., Inverse Spectral Problems for Differential Operators and Their Applications, Gordon and Breach, Amsterdam, 2000.

Geciken Argümanlı Bir Sturm-Liouville Probleminin İzinin ve Özfonksiyonlarının Düğüm Noktalarının Hesaplanması

Year 2018, Volume: 39 Issue: 3, 597 - 607, 30.09.2018

Erdoğan Şen

https://doi.org/10.17776/csj.416602
Cited By: 1

Abstract

Biz bu
çalışmada sabit geciken argümanlı süreksiz bir Sturm-Liouville probleminin
özdeğerleri için bir iz formülü ve özfonksiyonları için düğüm noktaları elde
ettik. Standart problemlerin aksine spektral parametre sadece diferansiyel denklemde
değil ayrıca sınır koşullarının birinde de yer alır. Böylece bu durumun
özfonksiyonların düğüm noktalarını ve izini değiştirip değiştirmediğini görmüş
oluyoruz.

Keywords

Geciken argümanlı diferansiyel denklem geçiş koşulları düğüm noktaları düzenli iz spectrum

References

  • [1]. Norkin S.B., Differential equations of the second order with retarded argument, Translations of Mathematical Monographs, AMS, Providence, RI, 1972.
  • [2]. Mukhtarov O.Sh., Kadakal M., Muhtarov F.S., On discontinuous Sturm-Liouville problems with transmission conditions, J. Math. Kyoto Univ., 44-4 (2004) 779-798.
  • [3]. Mukhtarov O.Sh., Tunç E., Eigenvalue problems for Sturm--Liouville equations with transmission conditions, Israel J. Math. 144 (2004) 367--380.
  • [4]. Aydemir K., Mukhtarov O.Sh., Class of Sturm-Liouville problems with eigenparameter dependent transmission conditions, Numer. Funct. Anal. Optim., 38-10 (2017) 1260-1275.
  • [5]. Çakmak Y., Keskin B., Uniqueness theorems for Sturm-Liouville operator with parameter dependent boundary conditions and finite number of transmission conditions, Cumhuriyet Sci. J., 38-3 (2017) 535-543.
  • [6]. Kandemir M., Asymptotic distribution of eigenvalues for fourth-order boundary-value problem with discontinuous coefficients and transmission conditions, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 66-1 (2017) 133-152.
  • [7]. Li K., Sun J., Hao X., Weyl function of Sturm-Liouville problems with transmission conditions at finite interior points, Mediter. J. Math., 14-189 (2017) 1-15.
  • [8]. Uğurlu E., Taş K., A new method for dissipative dynamic operator with transmission conditions, Complex Anal. Oper. Theory, 12 (2018) 1027-1055.
  • [9]. Zettl A., Adjoint and self-adjoint boundary value problems with interface conditions, SIAM J. Appl. Math., 16 (1968) 851-859.
  • [10]. Buschmann D., Stolz G., Weidmann J., One-dimensional Schrödinger operators with local point interactions, J. Reine Angew. Math., 467 (1995) 169-186.
  • [11]. Gelfand I.M., Levitan B.M., On a formula for eigenvalues of a differential operator of second order, Doklady Akademii Nauk SSSR, 88 (1953) 593--596 (Russian).
  • [12]. Dikii L.A., Trace formulas for Sturm-Liouville differential equations, Uspekhi Mat. Nauk, 12-3 (1958) 111-143.
  • [13]. Maksudov F.G., Bayramoglu M., Adıguzelov E., On a regularized traces of the Sturm-Liouville operator on a finite interval with the unbounded operator coefficient, Doklady Akademii Nauk SSSR, 277-4 (1984); English translation: Soviet Math. Dokl., 30 (1984) 169-173.
  • [14]. Bayramoglu M., Sahinturk H., Higher order regularized trace formula for the regular Sturm-Liouville equation contained spectral parameter in the boundary condition, Appl. Math. Comput., 186 (2007) 1591-1599.
  • [15]. Gül E., On the regularized trace of a second order differential operator, Appl. Math. Comput., 198 (2008) 471-480.
  • [16]. Yang C-F., Trace and inverse problem of a discontinuous Sturm-Liouville operator with retarded argument, J. Math. Anal. Appl., 395 (2012) 30-41.
  • [17]. Bayramoglu M., Bayramov A., Şen E., A regularized trace formula for a discontinuous Sturm-Liouville operator with delayed argument, Electronic Journal of Differential Equations, 2017-104 (2017) 1-12.
  • [18]. Hira F., A trace formula for the Sturm-Liouville type equation with retarded argument, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 66-1 (2017) 124-132.
  • [19]. Keskin B., Ozkan A.S., Inverse nodal problems for impulsive Sturm-Liouville equation with boundary conditions depending on the parameter, Advances in Analysis, 2-3 (2017) 151-156.
  • [20]. Keskin B., Reconstruction of the Volterra-type integro-differential operator from nodal points, Boundary Value Problems, 2018-47 (2018) 1-8.
  • [21]. Keskin B., Ozkan A.S., Inverse nodal problems for Dirac-type integro-differential operators, J. Differ. Eq., 263 (2017) 8838-8847.
  • [22]. Bayramov A., Çalışkan S., Uslu S., Computation of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument, Appl. Math. Comput., 191 (2007) 592-600.
  • [23]. Şen E., Bayramov A., Calculation of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument which contains a spectral parameter in the boundary condition, Math. Comput. Model., 54-(11-12) (2011) 3090-3097.
  • [24]. Yurko V.A., Inverse Spectral Problems for Differential Operators and Their Applications, Gordon and Breach, Amsterdam, 2000.
There are 24 citations in total.

Details

Primary Language English
Journal Section Natural Sciences
Authors

Erdoğan Şen 0000-0001-6603-2652

Publication Date September 30, 2018
Submission Date April 18, 2018
Acceptance Date September 28, 2018
Published in Issue Year 2018Volume: 39 Issue: 3

Cite

APA Şen, E. (2018). Computation of Trace and Nodal Points of Eigenfunctions for a Sturm-Liouville Problem with Retarded Argument. Cumhuriyet Science Journal, 39(3), 597-607. https://doi.org/10.17776/csj.416602

Cited By

Sınır Koşulunda Öz Parametre Bulunduran Bir Sturm-Liouville Operatörü İçin Ters Nodal Problem
Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi
Sertaç GÖKTAŞ
https://doi.org/10.17714/gumusfenbil.672966