Sertaç GÖKTAŞ

Sınır Koşulunda Öz Parametre Bulunduran Bir Sturm-Liouville Operatörü İçin Ters Nodal Problem

Bu çalışmada, sınır koşulunda öz parametre ve ikinci derece diferensiyel denkleminde birden fazla potansiyel bulunduran bir Sturm- Liouville problemi ele alınmıştır. Prüfer dönüşümü yardımıyla bu problemin özdeğerlerinin ve nodal parametrelerinin asimptotik formülleri bulunmuştur. Ayrıca, potansiyel fonksiyonlar için bir yapılandırma formülü elde edilmiştir.

Anahtar Kelimeler:

Nodal Noktalar, Nodal Uzunluklar, Potansiyel Fonksiyon, Prüfer Dönüşümü

Inverse Nodal Problem for A Sturm-Liouville Operator with Eigenparameter in the Boundary Condition

In this study, Sturm-Liouville problem with eigenparameter in the boundary condition and more than one potential in the second order differential equation is considered. Asymptotic formulas of the eigenvalues and nodal parameters of this problem are found by Prüfer substitution. In addition, a reconstruction formula is obtained for potential functions.

Keywords:

Nodal Points, Nodal Lengths, Potential Function,

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Ambartsumyan, V.A., 1929. Über eine Frage der Eigenwerttheorie. Zeitschrift für Physik, 53, 690-695.
Birkhoff, G. ve Rota, G.C., 1989. Ordinary Differential Equations, 4 edition: Ginn, John Wiley & Sons, 416p.
Borg, G., 1946. Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Acta Mathematica, 78(1), 1-96.
Browne, P.J. ve Sleeman, B.D., 1996. Inverse Nodal Problems for Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions. Inverse Problems, 12, 377-381.
Chen, Y.T., Cheng, Y. H., Law, C.K. ve Tsay J., 2002. L1 Convergence of the Reconstruction Formula for the Potential Function. Proceedings of the American Mathematical Society, 130, 2319-2324.
Çakır, A., 2007. Kompleks Potansiyele Sahip Sturm-Liouville Operatörleri için Ters Saçılma Problemi ve Bazı Uygulamaları. Yüksek Lisans Tezi, Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü. Isparta, 58s.
Gel’fand, I.M. ve Levitan, B.M., 1951. On the Determination of a Differential Equation from Its Spectral Function. Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, 15(4), 309-360.
Goktas S., Koyunbakan H. ve Gulsen T., 2018. Inverse Nodal Problem for Polynomial Pencil of Sturm-Liouville Operator. Mathematical Methods in the Applied Sciences, 41, 7576–7582.
Guseinov, I.M., Nabiev A.A. ve Pashaev R.T., 2000. Transformation Operators and Asymptotic Formulas for the Eigenvalues of a Polynomial Pencil of Sturm-Liouville Operators. Sibirskii Matematicheskii Zhurnal, 41, 554-566.
Guseinov, I.M. ve Nabiev A.A., 2000. A Class of Inverse Problems for a Quadratic Pencil of Sturm-Liouville Operators. Differentsial'nye Uravneniya, 36(3), 418-420.
Hald, O.L. ve McLaughlin, J.R., 1989. Solutions of Inverse Nodal Problems. Inverse Problems, 5, 307-347.
Hoschtadt, H., 1973. The Inverse Sturm-Liouville Problem. Communications Pure and Applied Mathematics, 26, 715-729.
Kaplan, M., 2019. n-Potansiyel İçeren Sturm-Liouville Operatörü için Ters Nodal Problem. Yüksek Lisans Tezi, Fırat Üniversitesi Fen Bilimleri Enstitüsü. Elazığ, 25s.
Keskin, B. ve Özkan, A.S., 2017. Inverse Nodal Problems for Impulsive Sturm-Liouville Equation with Boundary Conditions Depending on the Parameter. Advances in Analysis, 2(3), 151-156.
Koyunbakan, H., 2009. Reconstruction of Potential Function for Diffusion Operator. Numerical Functional Analysis and Optimization, 30(1-2), 1-10.
Koyunbakan, H., 2011. Inverse Problem for a Quadratic Pencil of Sturm-Liouville Operator. Journal of Mathematical Analysis and Applications, 378, 549-554.
Koyunbakan, H. ve Panakhov, E.S., 2007. Half Inverse Problem for Diffusion Operators on the Finite İnterval. Journal of Mathematical Analysis and Applications, 326, 1024-1030.
Koyunbakan, H. ve Yılmaz, E., 2008. Reconstruction of the Potential Function and Its Derivatives for the Diffusion Operator. Verlag der Zeitschrift für Naturforch, 63a, 127-130.
McLaughlin, J.R., 1988. Inverse Spectral Theory Using Nodal Points as Data – A Uniqueness Result. Journal of Differential Equations, 73, 354–362.
Nabiev A.A., 2010. On a Fundemental System of Solutions of the matrix Schrödinger Equation with a Polynomial Energy-Dependent Potential. Mathematical Methods in the Applied Sciences, 33(11), 1372-1383.
Panakhov, E.S., Koyunbakan, H. ve Ic, U., 2010. Reconstruction Formula for the Potential Function of Sturm–Liouville Problem with Eigenparameter Boundary Condition. Inverse Problems in Science and Engineering, 18 (1), 173–180.
Pinasco, J.P. ve Scarola, C.A., 2015. Nodal Inverse Problem for Second Order Sturm-Liouville Operators with Indefinite Weights. Applied Mathematics and Computation, 256, 819-830.
Shukurov, A. Sh., 2009. The Inverse Problem for a Diffusion Operator. Proceeding of IMM of NAS of Azerbaijan, 30, 105-110.
Şen, E., 2017. A Regularized Trace Formula and Oscillation of Eigenfunctions of a Sturm-Liouville Operator with Retarded Argument at 2 Points of Discontinuity. Mathematical Methods in the Applied Sciences, 40(18), 7051-7061.
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.
Yang, C. F., 2014. Inverse Nodal Problems for the Sturm–Liouville Operator with a Constant Delay. Journal of Differential Equations, 257(4), 1288-1306.
Yılmaz E. ve Koyunbakan H., 2010. Reconstruction of Potential Function and Its Derivatives for Sturm–Liouville Problem with Eigenvalues in Boundary Condition. Inverse Problems in Science and Engineering, 18(7), 935–944.