Finally, we establish that the eigenvalues of the considered problem are the zeros of the transmittal-characteristic function.
📌 3 Key Takeaways
🎯
Objective & Hypothesis: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.
📊
Key Findings & Data:We first prove that the eigenvalues are real and the system of eigenfunctions is an orthogonal system.
💡
Conclusion & Impact:Finally, we establish that the eigenvalues of the considered problem are the zeros of the transmittal-characteristic function.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Periodic transmittal Sturm-Liouville problem · characteristic function · spectrum
Abstract
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.
MUKHTAROV, O., AYDEMİR, K. (2022). The Transmittal-Characteristic Function of Three-Interval Periodic Sturm-Liouville Problem with Transmission Conditions. Journal of New Theory, Cilt: 2021 - Sayı: 37, 26–34. https://ajindex.com/articles/565749
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