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Reconstruction of the nonlocal Sturm-Liouville operator with boundary conditions depending on the parameter

Year 2024, Volume: 53 Issue: 2, 314 - 320, 23.04.2024
https://doi.org/10.15672/hujms.1244992

Abstract

In the present paper, we consider the Sturm--Liouville equation with nonlocal boundary conditions depending polynomially on the parameter. We obtain a result and give an algorithm for the reconstruction of the coefficients of the problem using asymptotics of the nodal points.

References

  • [1] S. Akbarpoor, H. Koyunbakan and A.Dabbaghian, Solving inverse nodal problem with spectral parameter in boundary conditions, Inv. Prob. Sci. Eng. 27 (12), 1790–1801, 2019.
  • [2] S. Albeverio, R.O. Hryniv and L.P. Nizhnik, Inverse spectral problems for non-local Sturm–Liouville operators, Inverse Probl. 23 (2), 523, 2007.
  • [3] A.V. Bitsadze and A.A. Samarskii, Some elementary generalizations of linear elliptic boundary value problems, Dokl. Akad. Nauk SSSR 185, 739–740, 1969.
  • [4] S.A. Buterin and C.T. Shieh, Inverse nodal problem for differential pencils, Appl. Math. Lett. 22, 1240–1247, 2009.
  • [5] Y.H. Cheng, C.K. Law and J. Tsay, Remarks on a new inverse nodal problem, J. Math. Anal. Appl. 248, 145–155, 2000.
  • [6] Y.H. Cheng, Reconstruction of the Sturm-Liouville operator an a p-star graph with nodal data, Rocky Mt. J. Math. 42 (5), 1431–1446, 2012.
  • [7] S. Currie and B.A. Watson, Inverse nodal problems for Sturm-Liouville equations on graphs, Inverse Problems, 23, 2029–2040, 2007.
  • [8] Y. Çakmak and B. Keskin, Inverse nodal problem for the quadratic pencil of the Sturm-Liouville equations with parameter-dependent nonlocal boundary condition, Turkish J. Math. 47 (1), Article 26. 2023.
  • [9] N. Gordeziani, On some non-local problems of the theory of elasticity, Bulletin of TICMI, 4, 43–46, 2000.
  • [10] N. J. Guliyev, Inverse square singularities and eigenparameter-dependent boundary conditions are two sides of the same coin, Q. J. Math. haad004, 2023.
  • [11] Y. Guo and G. Wei, Inverse problems: dense nodal subset on an interior subinterval, J. Differ. Equ. 255 (7), 2002–2017, 2013.
  • [12] O.H. Hald and L.R. McLaughlin, Solutions of inverse nodal problems, Inv. Probl. 5, 307–347, 1989.
  • [13] Y.T. Hu , C.F. Yang and X.C. Xu, Inverse nodal problems for the Sturm–Liouville operator with nonlocal integral conditions, J. Inv. Ill-Posed Probl. 25 (6), 799–806, 2017.
  • [14] B. Keskin and A.S. Ozkan, Inverse nodal problems for impulsive Sturm-Liouville equation with boundary conditions depending on the parameter, Adv. Anal. 2 (3), 151-156, 2017.
  • [15] H. Koyunbakan and S. Mosazadeh, Inverse nodal problem for discontinuous Sturm– Liouville operator by new Prüfer Substitutions, Math. Sci., 1–8, 2021.
  • [16] H. Koyunbakan and E. Yilmaz, Reconstruction of the potential function and its derivatives for the diffusion operator, Z. Naturforschung A, 63 (3-4), 127–130, 2008.
  • [17] C.K. Law and C.F. Yang, Reconstructing the potential function and its derivatives using nodal data, Inverse Probl. 14 299–312, 1998.
  • [18] L.I. Mammadova and I.M. Nabiev, Uniqueness of recovery of the Sturm-Liouville operator with a spectral parameter quadratically entering the boundary condition, Vestn. Tomsk. Gos. Univ. Mat. Mech. (79), 14–24, 2022.
  • [19] L.I. Mammadova, I.M. Nabiev and Ch. H. Rzayeva, Uniqueness of the solution of the inverse problem for differential operator with semiseparated boundary conditions, Baku Math. J. 1 (1), 47–52, 2022.
  • [20] J.R. McLaughlin, Inverse spectral theory using nodal points as data– a uniqueness result, J. Differ. Equ. 73, 354–362, 1988.
  • [21] S. Mosazadeh, The uniqueness theorem for inverse nodal problems with a chemical potential, Iran. J. Math. Chem. 8 (4), 403–411, 2017.
  • [22] I.M. Nabiev, Reconstruction of the Differential Operator with Spectral Parameter in the Boundary Condition, Mediterr. J. Math. 19, 124, 2022.
  • [23] A.M. Nakhushev, Equations of Mathematical Biology, Moscow: Vysshaya Shkola, 1995. (in Russian)
  • [24] L. Nizhnik, Inverse nonlocal Sturm–Liouville problem, Inverse Probl. 26 (12), 125006, 2010.
  • [25] A.S. Ozkan and I. Adalar, Inverse nodal problems for Sturm-Liouville equation with nonlocal boundary conditions, J. Math. Anal. Appl. 520 (1), 126907, 2023.
  • [26] A.S. Ozkan and I. Adalar, Inverse nodal problem for Dirac operator with integral type nonlocal boundary conditions, Math. Meth. Appl. Sci. 46 (1), 986–993, 2023.
  • [27] A.S. Ozkan and B. Keskin, Inverse nodal problems for Sturm–Liouville equation with eigenparameter-dependent boundary and jump conditions, Inv. Probl. Sci. Eng. 23 (8), 1306–1312, 2015.
  • [28] C.T. Shieh and V.A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl. 347 266–272, 2008.
  • [29] F. Sun, K. Li and J. Cai, Bounds on the non-real eigenvalues of nonlocal indefinite Sturm–Liouville problems with coupled boundary conditions, Complex Anal. Oper. Theory, 16 (30), 2022.
  • [30] Y.P. Wang and C.T. Shieh, Inverse problems for Sturm–Liouville operators on a compact equilateral graph by partial nodal data, Math. Models Meth. Appl. Sci. 44 (1), 693–704, 2021.
  • [31] Y.P. Wang, E. Yılmaz and S. Akbarpoor, The numerical solution of inverse nodal problem for integro-differential operator by Legendre wavelet method, Int. J. Comput. Math. 100 (1), 219–232, 2023.
  • [32] Y.P. Wang and V.A. Yurko, On the inverse nodal problems for discontinuous Sturm– Liouville operators, J. Differ. Equ. 260 (5), 4086–4109, 2016.
  • [33] X.J. Xu and C.F. Yang, Inverse nodal problem for nonlocal differential operators, Tamkang J. Math. 50 (3), 337–347, 2019.
  • [34] C.F. Yang, Inverse nodal problem for a class of nonlocal Sturm-Liouville operator, Math. Model. Anal. 15 (3), 383–392, 2010.
  • [35] X.F. Yang, A solution of the nodal problem, Inverse Probl. 13 203–213, 1997.
  • [36] X.F. Yang, A new inverse nodal problem, J. Differ. Equ. 169, 633–653, 2001.
  • [37] C.F. Yang and X.P. Yang, Inverse nodal problems for the Sturm-Liouville equation with polynomially dependent on the eigenparameter, Inv. Probl. Sci. Eng. 19 (7), 951-961, 2011.
  • [38] E. Yılmaz and H. Koyunbakan, Reconstruction of potential function and its derivatives for Sturm–Liouville problem with eigenvalues in boundary condition, Inv. Prob. Sci. Eng. 18 (7), 935-944, 2010.
  • [39] V.A. Yurko, Inverse Spectral Problems for Differential Operators and Their Applications, Gordon and Breach, Amsterdam, 2000.
Year 2024, Volume: 53 Issue: 2, 314 - 320, 23.04.2024
https://doi.org/10.15672/hujms.1244992

Abstract

References

  • [1] S. Akbarpoor, H. Koyunbakan and A.Dabbaghian, Solving inverse nodal problem with spectral parameter in boundary conditions, Inv. Prob. Sci. Eng. 27 (12), 1790–1801, 2019.
  • [2] S. Albeverio, R.O. Hryniv and L.P. Nizhnik, Inverse spectral problems for non-local Sturm–Liouville operators, Inverse Probl. 23 (2), 523, 2007.
  • [3] A.V. Bitsadze and A.A. Samarskii, Some elementary generalizations of linear elliptic boundary value problems, Dokl. Akad. Nauk SSSR 185, 739–740, 1969.
  • [4] S.A. Buterin and C.T. Shieh, Inverse nodal problem for differential pencils, Appl. Math. Lett. 22, 1240–1247, 2009.
  • [5] Y.H. Cheng, C.K. Law and J. Tsay, Remarks on a new inverse nodal problem, J. Math. Anal. Appl. 248, 145–155, 2000.
  • [6] Y.H. Cheng, Reconstruction of the Sturm-Liouville operator an a p-star graph with nodal data, Rocky Mt. J. Math. 42 (5), 1431–1446, 2012.
  • [7] S. Currie and B.A. Watson, Inverse nodal problems for Sturm-Liouville equations on graphs, Inverse Problems, 23, 2029–2040, 2007.
  • [8] Y. Çakmak and B. Keskin, Inverse nodal problem for the quadratic pencil of the Sturm-Liouville equations with parameter-dependent nonlocal boundary condition, Turkish J. Math. 47 (1), Article 26. 2023.
  • [9] N. Gordeziani, On some non-local problems of the theory of elasticity, Bulletin of TICMI, 4, 43–46, 2000.
  • [10] N. J. Guliyev, Inverse square singularities and eigenparameter-dependent boundary conditions are two sides of the same coin, Q. J. Math. haad004, 2023.
  • [11] Y. Guo and G. Wei, Inverse problems: dense nodal subset on an interior subinterval, J. Differ. Equ. 255 (7), 2002–2017, 2013.
  • [12] O.H. Hald and L.R. McLaughlin, Solutions of inverse nodal problems, Inv. Probl. 5, 307–347, 1989.
  • [13] Y.T. Hu , C.F. Yang and X.C. Xu, Inverse nodal problems for the Sturm–Liouville operator with nonlocal integral conditions, J. Inv. Ill-Posed Probl. 25 (6), 799–806, 2017.
  • [14] B. Keskin and A.S. Ozkan, Inverse nodal problems for impulsive Sturm-Liouville equation with boundary conditions depending on the parameter, Adv. Anal. 2 (3), 151-156, 2017.
  • [15] H. Koyunbakan and S. Mosazadeh, Inverse nodal problem for discontinuous Sturm– Liouville operator by new Prüfer Substitutions, Math. Sci., 1–8, 2021.
  • [16] H. Koyunbakan and E. Yilmaz, Reconstruction of the potential function and its derivatives for the diffusion operator, Z. Naturforschung A, 63 (3-4), 127–130, 2008.
  • [17] C.K. Law and C.F. Yang, Reconstructing the potential function and its derivatives using nodal data, Inverse Probl. 14 299–312, 1998.
  • [18] L.I. Mammadova and I.M. Nabiev, Uniqueness of recovery of the Sturm-Liouville operator with a spectral parameter quadratically entering the boundary condition, Vestn. Tomsk. Gos. Univ. Mat. Mech. (79), 14–24, 2022.
  • [19] L.I. Mammadova, I.M. Nabiev and Ch. H. Rzayeva, Uniqueness of the solution of the inverse problem for differential operator with semiseparated boundary conditions, Baku Math. J. 1 (1), 47–52, 2022.
  • [20] J.R. McLaughlin, Inverse spectral theory using nodal points as data– a uniqueness result, J. Differ. Equ. 73, 354–362, 1988.
  • [21] S. Mosazadeh, The uniqueness theorem for inverse nodal problems with a chemical potential, Iran. J. Math. Chem. 8 (4), 403–411, 2017.
  • [22] I.M. Nabiev, Reconstruction of the Differential Operator with Spectral Parameter in the Boundary Condition, Mediterr. J. Math. 19, 124, 2022.
  • [23] A.M. Nakhushev, Equations of Mathematical Biology, Moscow: Vysshaya Shkola, 1995. (in Russian)
  • [24] L. Nizhnik, Inverse nonlocal Sturm–Liouville problem, Inverse Probl. 26 (12), 125006, 2010.
  • [25] A.S. Ozkan and I. Adalar, Inverse nodal problems for Sturm-Liouville equation with nonlocal boundary conditions, J. Math. Anal. Appl. 520 (1), 126907, 2023.
  • [26] A.S. Ozkan and I. Adalar, Inverse nodal problem for Dirac operator with integral type nonlocal boundary conditions, Math. Meth. Appl. Sci. 46 (1), 986–993, 2023.
  • [27] A.S. Ozkan and B. Keskin, Inverse nodal problems for Sturm–Liouville equation with eigenparameter-dependent boundary and jump conditions, Inv. Probl. Sci. Eng. 23 (8), 1306–1312, 2015.
  • [28] C.T. Shieh and V.A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl. 347 266–272, 2008.
  • [29] F. Sun, K. Li and J. Cai, Bounds on the non-real eigenvalues of nonlocal indefinite Sturm–Liouville problems with coupled boundary conditions, Complex Anal. Oper. Theory, 16 (30), 2022.
  • [30] Y.P. Wang and C.T. Shieh, Inverse problems for Sturm–Liouville operators on a compact equilateral graph by partial nodal data, Math. Models Meth. Appl. Sci. 44 (1), 693–704, 2021.
  • [31] Y.P. Wang, E. Yılmaz and S. Akbarpoor, The numerical solution of inverse nodal problem for integro-differential operator by Legendre wavelet method, Int. J. Comput. Math. 100 (1), 219–232, 2023.
  • [32] Y.P. Wang and V.A. Yurko, On the inverse nodal problems for discontinuous Sturm– Liouville operators, J. Differ. Equ. 260 (5), 4086–4109, 2016.
  • [33] X.J. Xu and C.F. Yang, Inverse nodal problem for nonlocal differential operators, Tamkang J. Math. 50 (3), 337–347, 2019.
  • [34] C.F. Yang, Inverse nodal problem for a class of nonlocal Sturm-Liouville operator, Math. Model. Anal. 15 (3), 383–392, 2010.
  • [35] X.F. Yang, A solution of the nodal problem, Inverse Probl. 13 203–213, 1997.
  • [36] X.F. Yang, A new inverse nodal problem, J. Differ. Equ. 169, 633–653, 2001.
  • [37] C.F. Yang and X.P. Yang, Inverse nodal problems for the Sturm-Liouville equation with polynomially dependent on the eigenparameter, Inv. Probl. Sci. Eng. 19 (7), 951-961, 2011.
  • [38] E. Yılmaz and H. Koyunbakan, Reconstruction of potential function and its derivatives for Sturm–Liouville problem with eigenvalues in boundary condition, Inv. Prob. Sci. Eng. 18 (7), 935-944, 2010.
  • [39] V.A. Yurko, Inverse Spectral Problems for Differential Operators and Their Applications, Gordon and Breach, Amsterdam, 2000.
There are 39 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

İbrahim Adalar 0000-0002-4224-0972

Ahmet Sinan Özkan 0000-0002-9703-8982

Early Pub Date August 15, 2023
Publication Date April 23, 2024
Published in Issue Year 2024 Volume: 53 Issue: 2

Cite

APA Adalar, İ., & Özkan, A. S. (2024). Reconstruction of the nonlocal Sturm-Liouville operator with boundary conditions depending on the parameter. Hacettepe Journal of Mathematics and Statistics, 53(2), 314-320. https://doi.org/10.15672/hujms.1244992
AMA Adalar İ, Özkan AS. Reconstruction of the nonlocal Sturm-Liouville operator with boundary conditions depending on the parameter. Hacettepe Journal of Mathematics and Statistics. April 2024;53(2):314-320. doi:10.15672/hujms.1244992
Chicago Adalar, İbrahim, and Ahmet Sinan Özkan. “Reconstruction of the Nonlocal Sturm-Liouville Operator With Boundary Conditions Depending on the Parameter”. Hacettepe Journal of Mathematics and Statistics 53, no. 2 (April 2024): 314-20. https://doi.org/10.15672/hujms.1244992.
EndNote Adalar İ, Özkan AS (April 1, 2024) Reconstruction of the nonlocal Sturm-Liouville operator with boundary conditions depending on the parameter. Hacettepe Journal of Mathematics and Statistics 53 2 314–320.
IEEE İ. Adalar and A. S. Özkan, “Reconstruction of the nonlocal Sturm-Liouville operator with boundary conditions depending on the parameter”, Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 2, pp. 314–320, 2024, doi: 10.15672/hujms.1244992.
ISNAD Adalar, İbrahim - Özkan, Ahmet Sinan. “Reconstruction of the Nonlocal Sturm-Liouville Operator With Boundary Conditions Depending on the Parameter”. Hacettepe Journal of Mathematics and Statistics 53/2 (April 2024), 314-320. https://doi.org/10.15672/hujms.1244992.
JAMA Adalar İ, Özkan AS. Reconstruction of the nonlocal Sturm-Liouville operator with boundary conditions depending on the parameter. Hacettepe Journal of Mathematics and Statistics. 2024;53:314–320.
MLA Adalar, İbrahim and Ahmet Sinan Özkan. “Reconstruction of the Nonlocal Sturm-Liouville Operator With Boundary Conditions Depending on the Parameter”. Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 2, 2024, pp. 314-20, doi:10.15672/hujms.1244992.
Vancouver Adalar İ, Özkan AS. Reconstruction of the nonlocal Sturm-Liouville operator with boundary conditions depending on the parameter. Hacettepe Journal of Mathematics and Statistics. 2024;53(2):314-20.