Sturm-Liouville Problems with Polynomially Eigenparameter Dependent Boundary Conditions
Sturm-Liouville Problems with Polynomially Eigenparameter Dependent Boundary Conditions
Sturm-Liouville equation on a finite interval together with boundary conditions arises from the infinitesimal, vertical vibrations of a string with the ends subject to various constraints. The coefficient (also called potential) function in the differential equation is in a close relationship with the density of the string. In this sense, the computation of solutions plays a rather important role in both mathematical and physical fields. In this study, asymptotic behaviors of the solutions for Sturm-Liouville problems associated with polynomially eigenparameter dependent boundary conditions are obtained when the potential function is real valued ??- function on the interval (?, ?). Besides, the asymptotic formulae are given for the derivatives of the solutions.
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- [1] E. Başkaya, "On the gaps of Neumann eigenvalues for Hill’s equation with symmetric double well potential," Tbillisi Mathematical Journal, vol. 8, pp. 139-145, 2021.
- [2] E. Başkaya, "Periodic and semiperiodic eigenvalues of Hill’s equation with symmetric double well potential," TWMS Journal of Applied and Engineering Mathematics, vol. 10, no. 2, pp. 346-352, 2020.
- [3] H. Coşkun, E. Başkaya, A. Kabataş, "Instability intervals for Hill’s equation with symmetric single well potential," Ukrainian Mathematical Journal, vol. 71, no. 6, pp. 977-983, 2019.
- [4] G. Freiling, V. A. Yurko, "Inverse problems for Sturm-Liouville equations with boundary conditions polynomially dependent on the spectral parameter," Inverse problems, vol. 26, no. 6, 055003, 2010.
- [5] A. Kabataş, "Eigenfunction and Green’s function asymptotics for Hill’s equation with symmetric single well potential," Ukrainian Mathematical Journal, vol. 74, no. 2, pp. 191-203, 2022.
- [6] A. Kabataş, "On eigenfunctions of Hill’s equation with symmetric double well potential," Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, vol. 71, no. 3, pp. 634-649, 2022.
- [7] W. A. Woldegerima, "The Sturm- Liouville boundary value problems and their applications," LAP Lambert Academic Publishing, Germany, 2011.
- [8] R. E. Kraft, R. W. Wells, "Adjointness properties for differential systems with eigenvalue-dependent boundary conditions, with application to flowduct acoustics," Journal of the Acoustical Society of America, vol. 61, pp. 913-922, 1977.
- [9] T. V. Levitina, E. J. Brandas, "Computational techniques for prolate spheroidal wave functions in signal processing," Journal of Computational Methods in Sciences and Engineering, vol. 1, pp. 287-313, 2001.
- [10] E. Başkaya, "Asymptotic eigenvalues of regular Sturm-Liouville problems with spectral parameter-dependent boundary conditions and symmetric single well potential," Turkish Journal of Mathematics and Computer Science, vol. 3, no. 1, pp. 44-50, 2021.
- [11] E. Başkaya, "Asymptotics of eigenvalues for Sturm-Liouville problem including eigenparameterdependent boundary conditions with integrable potential," New Trends in Mathematical Sciences, vol. 6, no. 3, pp. 39-47, 2018.
- [12] E. Başkaya, "Asymptotics of eigenvalues for Sturm-Liouville problem with eigenparameter dependent-boundary conditions," New Trends in Mathematical Sciences, vol. 6, no. 2, pp. 247-257, 2018.
- [13] H. Coşkun, E. Başkaya, "Asymptotics of eigenvalues for Sturm-Liouville problem with eigenvalue in the boundary condition for differentiable potential," Annals of Pure and Applied Mathematics, vol. 16, no. 1, pp. 7-19, 2018.
- [14] M. Zhang, K. Li, "Dependence of eigenvalues of Sturm-Liouville problems with eigenparameter dependent boundary conditions," Applied Mathematics and Computation, vol. 378, 125214, 2020.
- [15] E. Başkaya, "Asymptotics of eigenvalues for Sturm-Liouville problem including quadratic eigenvalue in the boundary condition," New Trends in Mathematical Sciences, vol. 6, no. 3, pp. 76-82, 2018.
- [16] P. A. Binding, P. J. Browne, B. A. Watson, "Equivalence of inverse Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparamete," Journal of Mathematical Analysis and Applications, vol. 291, pp. 246-261, 2004.
- [17] H. Coşkun, A. Kabataş, "Green’s function of regular Sturm-Liouville problem having eigenparameter in one boundary condition," Turkish Journal of Mathematics and Computer Science, vol. 4, pp. 1-9, 2016.
- [18] H. Coşkun, A. Kabataş, E. Başkaya, "On Green’s function for boundary value problem with eigenvalue dependent quadratic boundary condition," Boundary Value Problems, vol. 71, 2017.
- [19] A. Shkalikov, "Boundary problems for ordinary differential equations with parameter in the boundary conditions," Journal of Soviet Mathematics, vol. 33, pp. 1311-1342, 1986.
- [20] C. T. Fulton, S. A. Pruess, "Eigenvalue and eigenfunction asymptotics for regular Sturm-Liouville problems," Journal of Mathematical Analysis and Applications, vol. 188, pp. 297-340, 1994.
- [21] B. J. Harris, "The form of the spectral functions associated with Sturm- Liouville problems with continuous spectrum," Mathematika, vol. 44, pp. 149-161, 1997.
- [22] H. Coşkun, E. Başkaya, "Asymptotics of eigenvalues of regular Sturm- Liouville problems with eigenvalue parameter in the boundary condition for integrable potential," Mathematica Scandinavica, vol. 107, pp. 209-223, 2010.
- [23] H. Coşkun, A. Kabataş, "Asymptotic approximations of eigenfunctions for regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition for integrable potential," Mathematica Scandinavica, vol. 113, pp. 143-160, 2013.
- [24] Y. P. Wang, K. Y. Lien, C. T. Shieh, "On a uniqueness theorem of Sturm- Liouville equations with boundary conditions polynomially dependent on the spectral parameter," Boundary Value Problems, no. 28, 2018.