Research Article
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Sturm-Liouville Problems with Polynomially Eigenparameter Dependent Boundary Conditions

Year 2023, Volume: 27 Issue: 6, 1235 - 1242, 18.12.2023
https://doi.org/10.16984/saufenbilder.1304365

Abstract

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 𝑳𝟏- function on the interval (𝟎, 𝟏). Besides, the asymptotic formulae are given for the derivatives of the solutions.

References

  • [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.
Year 2023, Volume: 27 Issue: 6, 1235 - 1242, 18.12.2023
https://doi.org/10.16984/saufenbilder.1304365

Abstract

References

  • [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.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Ayşe KABATAŞ 0000-0003-3273-3666

Early Pub Date December 1, 2023
Publication Date December 18, 2023
Submission Date May 28, 2023
Acceptance Date July 31, 2023
Published in Issue Year 2023 Volume: 27 Issue: 6

Cite

APA KABATAŞ, A. (2023). Sturm-Liouville Problems with Polynomially Eigenparameter Dependent Boundary Conditions. Sakarya University Journal of Science, 27(6), 1235-1242. https://doi.org/10.16984/saufenbilder.1304365
AMA KABATAŞ A. Sturm-Liouville Problems with Polynomially Eigenparameter Dependent Boundary Conditions. SAUJS. December 2023;27(6):1235-1242. doi:10.16984/saufenbilder.1304365
Chicago KABATAŞ, Ayşe. “Sturm-Liouville Problems With Polynomially Eigenparameter Dependent Boundary Conditions”. Sakarya University Journal of Science 27, no. 6 (December 2023): 1235-42. https://doi.org/10.16984/saufenbilder.1304365.
EndNote KABATAŞ A (December 1, 2023) Sturm-Liouville Problems with Polynomially Eigenparameter Dependent Boundary Conditions. Sakarya University Journal of Science 27 6 1235–1242.
IEEE A. KABATAŞ, “Sturm-Liouville Problems with Polynomially Eigenparameter Dependent Boundary Conditions”, SAUJS, vol. 27, no. 6, pp. 1235–1242, 2023, doi: 10.16984/saufenbilder.1304365.
ISNAD KABATAŞ, Ayşe. “Sturm-Liouville Problems With Polynomially Eigenparameter Dependent Boundary Conditions”. Sakarya University Journal of Science 27/6 (December 2023), 1235-1242. https://doi.org/10.16984/saufenbilder.1304365.
JAMA KABATAŞ A. Sturm-Liouville Problems with Polynomially Eigenparameter Dependent Boundary Conditions. SAUJS. 2023;27:1235–1242.
MLA KABATAŞ, Ayşe. “Sturm-Liouville Problems With Polynomially Eigenparameter Dependent Boundary Conditions”. Sakarya University Journal of Science, vol. 27, no. 6, 2023, pp. 1235-42, doi:10.16984/saufenbilder.1304365.
Vancouver KABATAŞ A. Sturm-Liouville Problems with Polynomially Eigenparameter Dependent Boundary Conditions. SAUJS. 2023;27(6):1235-42.

Sakarya University Journal of Science (SAUJS)