Araştırma Makalesi
BibTex RIS Kaynak Göster

The Transmittal-Characteristic Function of Three-Interval Periodic Sturm-Liouville Problem with Transmission Conditions

Yıl 2021, Sayı: 37, 26 - 34, 31.12.2021
https://doi.org/10.53570/jnt.1018600

Öz

In this paper, we study the periodic Sturm-Liouville problem, defined on three non-intersecting intervals with four supplementary conditions which are imposed at two internal points of interaction, the so-called
transmission conditions. We first prove that the eigenvalues are real and the system of eigenfunctions is an orthogonal system. Secondly, some auxiliary initial-value problems are defined and transmittal-characteristic function is constructed in terms of solutions of these initial-value problems. Finally, we
establish that the eigenvalues of the considered problem are the zeros of the transmittal-characteristic function.

Destekleyen Kurum

Amasya University

Proje Numarası

FMB-BAP 20-0440

Teşekkür

This work was supported by the Office of Scientific Research Projects Coordination at Amasya University, Grant number: FMB-BAP 20-0440.

Kaynakça

  • İ. Çelik, G. Gökmen, Approximate Solution of Periodic Sturm Liouville Problems with Chebyshev Collocation Method, Applied Mathematics and Computation 170(1) (2005) 285-295.
  • X. Haoa, L. Liu, Y. Wu, Existence and Multiplicity Results for Nonlinear Periodic Boundary Value Problems, Nonlinear Analysis: Theory, Methods and Applications 72(9-10) (2010) 3635-3642.
  • J. Li, J. J. Nieto, J. Shen, Impulsive Periodic Boundary Value Problems of First-Order Differential Equations, Journal of Mathematical Analysis and Applications 325(1) (2007) 226-236.
  • J. Qiu, Positive Solutions for a Nonlinear Periodic Boundary-Value Problem with a Parameter, Electronic Journal of Differential Equations 2012(133) (2012) 1-10.
  • G. V. Berghe, M. V. Daele, H. D. Meyer, A Modified Difference Scheme for Periodic and Semiperiodic Sturm-Liouville Problems, Applied Numerical Mathematics 18(1-3) (1995) 69-78.
  • P. A. Binding, P. B. Rynne, Half-Eigenvalues of Periodic Sturm-Liouville Problems, Journal of Differential Equations 206(2) (2004) 280-305.
  • P. Binding, H. Volkmer, A rüfer Angle Approach to the Periodic Sturm-Liouville Problem, The American Mathematical Monthly 119(6) (2012) 477-484.
  • A. Boumenir, Eigenvalues of Periodic Sturm-Liouville Problems by the Shannon-Whittaker Sampling Theorem, Mathematics of Computation of the American Mathematical Society 68(227) (1999) 1057-1066.
  • F. Geng, Y. Xu, D. Zhu, Periodic Boundary Value Problems for First-Order Impulsive Dynamic Equations on Time Scales, Nonlinear Analysis: Theory, Methods and Applications 69(11) (2008) 4074-4087.
  • K. V. Khmelnytskaya, H. Rosu, C. A. Gonzalez, Periodic Sturm-Liouville Problems Related to Two Riccati Equations of Constant Coefficients, Annals of Physics 325(3) (2010) 596-606.
  • V. A. Sadovnichii, Y. T. Sultanaev, A. M. Akhtyamov, {\em Inverse Sturm-Liouville Problem with Generalized Periodic Boundary Conditions, Differential Equations 45(4) (2009) 526-538.
  • A. A. Shkalikov, O. A. Veliev, On the Riesz Basis Property of the Eigen-and Associated Functions of Periodic and Antiperiodic Sturm-Liouville Problems, Mathematical Notes 85(5-6) (2009) 647-660.
  • S. Somali, V. Oger, Improvement of Eigenvalues of Sturm-Liouville Problem with t-Periodic Boundary Conditions, Journal of Computational and Applied Mathematics 180(2) (2005) 433-441.
  • Y. Yuan, J. Sun, A. Zettl, Eigenvalues of Periodic Sturm Liouville Problems, Linear Algebra and Its Applications 517 (2017) 148-166.
  • Y. Wang, J. Li, Z. Cai. Positive Solutions of Periodic Boundary Value Problems for the Second-Order Differential Equation with a Parameter, Boundary Value Problems Article Number 49 2017(1) (2017) 1-11.
  • Y. Zhao, H. Chen, B. Qin, Periodic Boundary Value Problems for Second-Order Functional Differential Equations with Impulse, Advances in Difference Equations Article Number 134 2014(1) (2014) 1-12.
  • I. Akbarfam, A. Jodayree, Resolvent Operator and Self-Adjointness of Sturm-Liouville Operators with a Finite Number of Transmission Conditions, Mediterranean Journal Of Mathematics 11(2) (2014) 447-462.
  • B. P. Allahverdiev, E. Bairamov, E. Ugurlu, Eigenparameter Dependent Sturm-Liouville Problems in Boundary Conditions with Transmission Conditions, Journal of Mathematical Analysis and Applications 401(1) (2013) 388-396.
  • B. P. Allahverdiev, H. Tuna, On the Resolvent of Singular Sturm-Liouville Operators with Transmission Conditions, Mathematical Methods in The Applied Sciences 43(7) (2020) 388-396.
  • J. Ao, J. Sun, Eigenvalues of a Class of Fourth-Order Boundary Value Problems with Transmission Conditions Using Matrix Theory, Linear and Multilinear Algebra 69(9) (2021) 1610-1624.
  • K. Aydemir, H. Olğar, O. Sh. Mukhtarov, Differential Operator Equations with Interface Conditions in Modified Direct Sum Spaces, Filomat 32(3) (2018) 921-931.
  • E. Bairamov, E. Uğurlu, On the Characteristic Values of the Real Component of a Dissipative Boundary Value Transmission Problem, Applied Mathematics and Computation 218(19) (2012) 9657-9663.
  • H. Olğar, O. Sh. Mukhtarov, Weak Eigenfunctions Of Two-Interval Sturm-Liouville Problems Together With Interaction Conditions, Journal of Mathematical Physics 58(042201) (2017) DOI: 10.1063/1.4979615.
  • H. Olğar, F. Muhtarov, The Basis Property of the System of Weak Eigenfunctions of a Discontinuous Sturm-Liouville Problem, Mediterranean Journal of Mathematics Article Number 114 14(3) (2017) 1-13.
  • O. Sh. Mukhtarov, K. Aydemir, Discontinuous Sturm-Liouville Problems Involving an Abstract Linear Operator, Journal of Applied Analysis and Computation 10(4) (2020) 1545-1560.
  • O. Sh. Mukhtarov, H. Olğar, K. Aydemir, I. Sh. Jabbarov, Operator-Pencil Realization of One Sturm-Liouville Problem with Transmission Conditions, Applied and Computational Mathematics 17(2) (2018) 284-294.
  • O. Sh. Mukhtarov, M. Yücel, K. Aydemir, A New Generalization of the Differential Transform Method for Solving Boundary Value Problems, Journal of New Results in Science 10(2) (2021) 49-58.
  • E. Şen, Sturm-Liouville Problems With Retarded Argument and A Finite Number of Transmission Conditions, Electronic Journal Of Differential Equations 2017(310) (2017) 1-8.
  • E. C. Titchmarsh, Eigenfunctions Expansion Associated with Second Order Differential Equations I, Second edn. Oxford University Press, London, 1962.
Yıl 2021, Sayı: 37, 26 - 34, 31.12.2021
https://doi.org/10.53570/jnt.1018600

Öz

Proje Numarası

FMB-BAP 20-0440

Kaynakça

  • İ. Çelik, G. Gökmen, Approximate Solution of Periodic Sturm Liouville Problems with Chebyshev Collocation Method, Applied Mathematics and Computation 170(1) (2005) 285-295.
  • X. Haoa, L. Liu, Y. Wu, Existence and Multiplicity Results for Nonlinear Periodic Boundary Value Problems, Nonlinear Analysis: Theory, Methods and Applications 72(9-10) (2010) 3635-3642.
  • J. Li, J. J. Nieto, J. Shen, Impulsive Periodic Boundary Value Problems of First-Order Differential Equations, Journal of Mathematical Analysis and Applications 325(1) (2007) 226-236.
  • J. Qiu, Positive Solutions for a Nonlinear Periodic Boundary-Value Problem with a Parameter, Electronic Journal of Differential Equations 2012(133) (2012) 1-10.
  • G. V. Berghe, M. V. Daele, H. D. Meyer, A Modified Difference Scheme for Periodic and Semiperiodic Sturm-Liouville Problems, Applied Numerical Mathematics 18(1-3) (1995) 69-78.
  • P. A. Binding, P. B. Rynne, Half-Eigenvalues of Periodic Sturm-Liouville Problems, Journal of Differential Equations 206(2) (2004) 280-305.
  • P. Binding, H. Volkmer, A rüfer Angle Approach to the Periodic Sturm-Liouville Problem, The American Mathematical Monthly 119(6) (2012) 477-484.
  • A. Boumenir, Eigenvalues of Periodic Sturm-Liouville Problems by the Shannon-Whittaker Sampling Theorem, Mathematics of Computation of the American Mathematical Society 68(227) (1999) 1057-1066.
  • F. Geng, Y. Xu, D. Zhu, Periodic Boundary Value Problems for First-Order Impulsive Dynamic Equations on Time Scales, Nonlinear Analysis: Theory, Methods and Applications 69(11) (2008) 4074-4087.
  • K. V. Khmelnytskaya, H. Rosu, C. A. Gonzalez, Periodic Sturm-Liouville Problems Related to Two Riccati Equations of Constant Coefficients, Annals of Physics 325(3) (2010) 596-606.
  • V. A. Sadovnichii, Y. T. Sultanaev, A. M. Akhtyamov, {\em Inverse Sturm-Liouville Problem with Generalized Periodic Boundary Conditions, Differential Equations 45(4) (2009) 526-538.
  • A. A. Shkalikov, O. A. Veliev, On the Riesz Basis Property of the Eigen-and Associated Functions of Periodic and Antiperiodic Sturm-Liouville Problems, Mathematical Notes 85(5-6) (2009) 647-660.
  • S. Somali, V. Oger, Improvement of Eigenvalues of Sturm-Liouville Problem with t-Periodic Boundary Conditions, Journal of Computational and Applied Mathematics 180(2) (2005) 433-441.
  • Y. Yuan, J. Sun, A. Zettl, Eigenvalues of Periodic Sturm Liouville Problems, Linear Algebra and Its Applications 517 (2017) 148-166.
  • Y. Wang, J. Li, Z. Cai. Positive Solutions of Periodic Boundary Value Problems for the Second-Order Differential Equation with a Parameter, Boundary Value Problems Article Number 49 2017(1) (2017) 1-11.
  • Y. Zhao, H. Chen, B. Qin, Periodic Boundary Value Problems for Second-Order Functional Differential Equations with Impulse, Advances in Difference Equations Article Number 134 2014(1) (2014) 1-12.
  • I. Akbarfam, A. Jodayree, Resolvent Operator and Self-Adjointness of Sturm-Liouville Operators with a Finite Number of Transmission Conditions, Mediterranean Journal Of Mathematics 11(2) (2014) 447-462.
  • B. P. Allahverdiev, E. Bairamov, E. Ugurlu, Eigenparameter Dependent Sturm-Liouville Problems in Boundary Conditions with Transmission Conditions, Journal of Mathematical Analysis and Applications 401(1) (2013) 388-396.
  • B. P. Allahverdiev, H. Tuna, On the Resolvent of Singular Sturm-Liouville Operators with Transmission Conditions, Mathematical Methods in The Applied Sciences 43(7) (2020) 388-396.
  • J. Ao, J. Sun, Eigenvalues of a Class of Fourth-Order Boundary Value Problems with Transmission Conditions Using Matrix Theory, Linear and Multilinear Algebra 69(9) (2021) 1610-1624.
  • K. Aydemir, H. Olğar, O. Sh. Mukhtarov, Differential Operator Equations with Interface Conditions in Modified Direct Sum Spaces, Filomat 32(3) (2018) 921-931.
  • E. Bairamov, E. Uğurlu, On the Characteristic Values of the Real Component of a Dissipative Boundary Value Transmission Problem, Applied Mathematics and Computation 218(19) (2012) 9657-9663.
  • H. Olğar, O. Sh. Mukhtarov, Weak Eigenfunctions Of Two-Interval Sturm-Liouville Problems Together With Interaction Conditions, Journal of Mathematical Physics 58(042201) (2017) DOI: 10.1063/1.4979615.
  • H. Olğar, F. Muhtarov, The Basis Property of the System of Weak Eigenfunctions of a Discontinuous Sturm-Liouville Problem, Mediterranean Journal of Mathematics Article Number 114 14(3) (2017) 1-13.
  • O. Sh. Mukhtarov, K. Aydemir, Discontinuous Sturm-Liouville Problems Involving an Abstract Linear Operator, Journal of Applied Analysis and Computation 10(4) (2020) 1545-1560.
  • O. Sh. Mukhtarov, H. Olğar, K. Aydemir, I. Sh. Jabbarov, Operator-Pencil Realization of One Sturm-Liouville Problem with Transmission Conditions, Applied and Computational Mathematics 17(2) (2018) 284-294.
  • O. Sh. Mukhtarov, M. Yücel, K. Aydemir, A New Generalization of the Differential Transform Method for Solving Boundary Value Problems, Journal of New Results in Science 10(2) (2021) 49-58.
  • E. Şen, Sturm-Liouville Problems With Retarded Argument and A Finite Number of Transmission Conditions, Electronic Journal Of Differential Equations 2017(310) (2017) 1-8.
  • E. C. Titchmarsh, Eigenfunctions Expansion Associated with Second Order Differential Equations I, Second edn. Oxford University Press, London, 1962.
Toplam 29 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik, Uygulamalı Matematik
Bölüm Araştırma Makalesi
Yazarlar

Kadriye Aydemir 0000-0002-8378-3949

Oktay Mukhtarov 0000-0001-7480-6857

Proje Numarası FMB-BAP 20-0440
Yayımlanma Tarihi 31 Aralık 2021
Gönderilme Tarihi 3 Kasım 2021
Yayımlandığı Sayı Yıl 2021 Sayı: 37

Kaynak Göster

APA Aydemir, K., & Mukhtarov, O. (2021). The Transmittal-Characteristic Function of Three-Interval Periodic Sturm-Liouville Problem with Transmission Conditions. Journal of New Theory(37), 26-34. https://doi.org/10.53570/jnt.1018600
AMA Aydemir K, Mukhtarov O. The Transmittal-Characteristic Function of Three-Interval Periodic Sturm-Liouville Problem with Transmission Conditions. JNT. Aralık 2021;(37):26-34. doi:10.53570/jnt.1018600
Chicago Aydemir, Kadriye, ve Oktay Mukhtarov. “The Transmittal-Characteristic Function of Three-Interval Periodic Sturm-Liouville Problem With Transmission Conditions”. Journal of New Theory, sy. 37 (Aralık 2021): 26-34. https://doi.org/10.53570/jnt.1018600.
EndNote Aydemir K, Mukhtarov O (01 Aralık 2021) The Transmittal-Characteristic Function of Three-Interval Periodic Sturm-Liouville Problem with Transmission Conditions. Journal of New Theory 37 26–34.
IEEE K. Aydemir ve O. Mukhtarov, “The Transmittal-Characteristic Function of Three-Interval Periodic Sturm-Liouville Problem with Transmission Conditions”, JNT, sy. 37, ss. 26–34, Aralık 2021, doi: 10.53570/jnt.1018600.
ISNAD Aydemir, Kadriye - Mukhtarov, Oktay. “The Transmittal-Characteristic Function of Three-Interval Periodic Sturm-Liouville Problem With Transmission Conditions”. Journal of New Theory 37 (Aralık 2021), 26-34. https://doi.org/10.53570/jnt.1018600.
JAMA Aydemir K, Mukhtarov O. The Transmittal-Characteristic Function of Three-Interval Periodic Sturm-Liouville Problem with Transmission Conditions. JNT. 2021;:26–34.
MLA Aydemir, Kadriye ve Oktay Mukhtarov. “The Transmittal-Characteristic Function of Three-Interval Periodic Sturm-Liouville Problem With Transmission Conditions”. Journal of New Theory, sy. 37, 2021, ss. 26-34, doi:10.53570/jnt.1018600.
Vancouver Aydemir K, Mukhtarov O. The Transmittal-Characteristic Function of Three-Interval Periodic Sturm-Liouville Problem with Transmission Conditions. JNT. 2021(37):26-34.


TR Dizin 26024

Electronic Journals Library (EZB) 13651



Academindex 28993

SOBİAD 30256                                                   

Scilit 20865                                                  


29324 As of 2021, JNT is licensed under a Creative Commons Attribution-NonCommercial 4.0 International Licence (CC BY-NC).