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

Yıl 2024, Cilt: 53 Sayı: 2, 314 - 320, 23.04.2024

İbrahim Adalar Ahmet Sinan Özkan

https://doi.org/10.15672/hujms.1244992

Öz

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.

Anahtar Kelimeler

nodal points Sturm-Liouville equation nonlocal boundary conditions inverse problem

Kaynakça

  • [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.

Yıl 2024, Cilt: 53 Sayı: 2, 314 - 320, 23.04.2024

İbrahim Adalar Ahmet Sinan Özkan

https://doi.org/10.15672/hujms.1244992

Öz

Anahtar Kelimeler

Nodal points Sturm-Liouville equation Nonlocal boundary conditions Inverse problem

Kaynakça

  • [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.
Toplam 39 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

İbrahim Adalar 0000-0002-4224-0972

Ahmet Sinan Özkan 0000-0002-9703-8982

Erken Görünüm Tarihi 15 Ağustos 2023
Yayımlanma Tarihi 23 Nisan 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 53 Sayı: 2

Kaynak Göster

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. Nisan 2024;53(2):314-320. doi:10.15672/hujms.1244992
Chicago Adalar, İbrahim, ve Ahmet Sinan Özkan. “Reconstruction of the Nonlocal Sturm-Liouville Operator With Boundary Conditions Depending on the Parameter”. Hacettepe Journal of Mathematics and Statistics 53, sy. 2 (Nisan 2024): 314-20. https://doi.org/10.15672/hujms.1244992.
EndNote Adalar İ, Özkan AS (01 Nisan 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 ve A. S. Özkan, “Reconstruction of the nonlocal Sturm-Liouville operator with boundary conditions depending on the parameter”, Hacettepe Journal of Mathematics and Statistics, c. 53, sy. 2, ss. 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 (Nisan 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 ve Ahmet Sinan Özkan. “Reconstruction of the Nonlocal Sturm-Liouville Operator With Boundary Conditions Depending on the Parameter”. Hacettepe Journal of Mathematics and Statistics, c. 53, sy. 2, 2024, ss. 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.

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