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
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.
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
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.