Inverse nodal problem for nonlocal differential operators
Main Article Content
Abstract
Inverse nodal problem consists in constructing operators from the given zeros of their eigenfunctions. The problem of differential operators with nonlocal boundary condition appears, e.g., in scattering theory, diffusion processes and the other applicable fields. In this paper, we consider a class of differential operators with nonlocal boundary condition, and show that the potential function can be determined by nodal data.
Article Details
How to Cite
Xu, X.-J., & Yang, C.-F. . (2019). Inverse nodal problem for nonlocal differential operators. Tamkang Journal of Mathematics, 50(3), 337–347. https://doi.org/10.5556/j.tkjm.50.2019.3361
Issue
Section
Papers
References
[1] S. Albeverio, R. Hryniv and L. Nizhnik, Inverse spectral problems for nonlocal Sturm-Liouville operators [J], Inverse Problems, 23 (2007), 523-535.
[2] N. Bondarenko, S. A. Buterin, An inverse spectral problem for Sturm- Liouville operators with frozen argument [J], J. Math. Anal. Appl., 472 (2019), 1028-1041.
[3] S. A. Buterin, On an inverse spectral problem for a convolution integro- differential operator [J], Results in Mathematics, 50 (2007), 173-181.
[4] S. A. Buterin, C.-T. Shieh, Inverse nodal problem for differential pencils [J], Appl. Math. Lett., 22 (2009), 1240-1247.
[5] S. A. Buterin, C.-T. Shieh, Incomplete inverse spectral and nodal problems for differential pencils [J], Res. Math. 62 (2012), no.1-2, 167-179.
[6] S. A. Buterin, S. V. Vasiliev, On recovering a Sturm-Liouville-type operator with the frozen argument rationally proportioned to the interval length, J. Inv. Ill-posed Probl., (2019) https://doi.org/10.1515/jiip-2018-0047.
[7] B. Chanane, Computing the eigenvalues of a class of nonlocal Sturm-Liouville Problems [J], Mathematical and Computer Modelling, 50 (2009), 225-232.
[8] Y. H. Cheng, C. K. Law, J. Tsay, Remarks on a new inverse nodal problem [J], Journal of Mathematical Analysis and Applications, 248 (2000), 145-155.
[9] R. Ciegis, A. Stikonas, O. Stikoniene and O. Suboc, Stationary problems with nonlocal boundary conditions [J], Mathematical Modelling and Analysis, 6
(2001), 178-191.
[10] R. Ciegis, A. Stikonas, O. Stikoniene and O. Suboc, A monotonic finite differ-
ence scheme for a parabolic problem with nonlocal conditions [J], Differential
Equations, 38 (2002), 1027-1037.
[11] S. Currie, B. A. Watson, Inverse nodal problems for Sturm-Liouville equations
on graphs [J], Inverse Problems, 23 (2007), 2029-2040.
[12] W. Feller, The parabolic differential equations and the associated semigroups
of transformations [J], Ann.Math., 55 (1952), 468-519.
[13] Y Guo, G Wei, Inverse problem for differential pencils with incompletely
spectral information [J], Taiwanese J. Math., 19 (2015), 927-942.
[14] Y Guo, G Wei, Determination of differential pencils from dense nodal subset
in an interior subinterval [J], Israel J. Math., 206 (2015), no.1, 213-231.
[15] O. H. Hald, J. R. McLaughlin, Inverse problems using nodal position data- uniqueness results, algorithms and bounds [J], Proc. Centre Math. Anal. Aus-
tral. Nat. Univ., 17 (1988), 32-59.
[16] N. I. Ionkin, The solution of a certain boundary value problem of the theory of
heat conduction with a nonclassical boundary condition [J], Diff. Equations,
13 (1997), 294-304. (in Russian)
[17] H. Y. Liu, J. Zou, Zeros of the Bessel and spherical Bessel functions and their
applications for uniqueness in inverse acoustic obstacle scattering [J], IMA J.
Appl. Math., 72 (2007), no. 6, 817-831.
[18] H. Y. Liu, J. Zou, Uniqueness in an inverse acoustic obstacle scattering prob-
lem for both sound-hard and sound-soft polyhedral scatterers [J], Inverse
Problems, 22 (2006), no. 2, 515-524.
[19] H. Y. Liu, J. Zou, On unique determination of partially coated polyhedral
scatterers with far field measurements [J], Inverse Problems, 23 (2007), no. 1, 297-308.
[20] H. Y. Liu, On recovering polyhedral scatterers with acoustic far-field mea- surements [J], IMA J. Appl. Math., 74 (2009), no. 2, 264-272.
[21] H. Koyunbakan, A new inverse problem for the diffusion operator [J], Applied Mathematics Letters, 19 (2006), 995-999.
[22] K. V. Kravchenko, On differential operators with nonlocal boundary condi- tions [J], Differ. Uravn. 36 (2000), 464-469; English transl. in Differ. Equa- tions, 36 (2000), 517-523.
[23] J. R. McLaughlin, Inverse spectral theory using nadal points as data-a unique- ness result [J], Journal of Differential Equations, 73 (1988), 354-362.
[24] L. P. Nizhnik, Inverse eigenvalue problems for nonlocal Sturm-Liouville op- erators [J], Methods Funct. Anal. Topology, 15 (2009), 41-47.
[25] L. P. Nizhnik, Inverse nonlocal Sturm-Liouville problem [J], Inverse Problems, 26 (2010), 635-684.
[26] A. S. Ozkan, B. Keskin, Inverse nodal problems for Sturm-Liouville equation with eigenparameter-dependent boundary and jump conditions [J], Inverse Problems in Science and Engineering, 23 (2015), 1306-1312.
[27] A. D. Wentzell, On boundary conditions for multi-dimensional diffusion pro- cesses [J], Teor. Veroyatnost. Primenen., 4 (1959) 172-185.(in Russian)
[28] C. F. Yang, Inverse nodal problem for a class of nonlocal Sturm-Liouville operator [J], Mathematical Modelling and Analysis, 15 (2010), 383-392.
[29] C. F. Yang, Inverse nodal problems for the Sturm-Liouville operator with a
constant delay [J], Journal of Differential Equations, 257 (2014), 1288-1306.
[30] C. F. Yang, On the quasinodal map for the diffusion operator [J], Journal of
Functional Analysis, 266 (2014), 4236-4265.
[31] V. A. Yurko, Inverse nodal problem for Sturm-Liouville operators on star-type
graphs [J], Journal of Inverse and Ill-Posed Problems, 16 (2008), 715-722.
[32] V. A. Yurko, An inverse problem for integro-differential operators [J], Matem.Zametki, 50 (1991), 134-146(Russian); English transl. An inverse problem for integro-differential operators [J], in Mathematical Notes, 50
(1991), 1188-1197.
[2] N. Bondarenko, S. A. Buterin, An inverse spectral problem for Sturm- Liouville operators with frozen argument [J], J. Math. Anal. Appl., 472 (2019), 1028-1041.
[3] S. A. Buterin, On an inverse spectral problem for a convolution integro- differential operator [J], Results in Mathematics, 50 (2007), 173-181.
[4] S. A. Buterin, C.-T. Shieh, Inverse nodal problem for differential pencils [J], Appl. Math. Lett., 22 (2009), 1240-1247.
[5] S. A. Buterin, C.-T. Shieh, Incomplete inverse spectral and nodal problems for differential pencils [J], Res. Math. 62 (2012), no.1-2, 167-179.
[6] S. A. Buterin, S. V. Vasiliev, On recovering a Sturm-Liouville-type operator with the frozen argument rationally proportioned to the interval length, J. Inv. Ill-posed Probl., (2019) https://doi.org/10.1515/jiip-2018-0047.
[7] B. Chanane, Computing the eigenvalues of a class of nonlocal Sturm-Liouville Problems [J], Mathematical and Computer Modelling, 50 (2009), 225-232.
[8] Y. H. Cheng, C. K. Law, J. Tsay, Remarks on a new inverse nodal problem [J], Journal of Mathematical Analysis and Applications, 248 (2000), 145-155.
[9] R. Ciegis, A. Stikonas, O. Stikoniene and O. Suboc, Stationary problems with nonlocal boundary conditions [J], Mathematical Modelling and Analysis, 6
(2001), 178-191.
[10] R. Ciegis, A. Stikonas, O. Stikoniene and O. Suboc, A monotonic finite differ-
ence scheme for a parabolic problem with nonlocal conditions [J], Differential
Equations, 38 (2002), 1027-1037.
[11] S. Currie, B. A. Watson, Inverse nodal problems for Sturm-Liouville equations
on graphs [J], Inverse Problems, 23 (2007), 2029-2040.
[12] W. Feller, The parabolic differential equations and the associated semigroups
of transformations [J], Ann.Math., 55 (1952), 468-519.
[13] Y Guo, G Wei, Inverse problem for differential pencils with incompletely
spectral information [J], Taiwanese J. Math., 19 (2015), 927-942.
[14] Y Guo, G Wei, Determination of differential pencils from dense nodal subset
in an interior subinterval [J], Israel J. Math., 206 (2015), no.1, 213-231.
[15] O. H. Hald, J. R. McLaughlin, Inverse problems using nodal position data- uniqueness results, algorithms and bounds [J], Proc. Centre Math. Anal. Aus-
tral. Nat. Univ., 17 (1988), 32-59.
[16] N. I. Ionkin, The solution of a certain boundary value problem of the theory of
heat conduction with a nonclassical boundary condition [J], Diff. Equations,
13 (1997), 294-304. (in Russian)
[17] H. Y. Liu, J. Zou, Zeros of the Bessel and spherical Bessel functions and their
applications for uniqueness in inverse acoustic obstacle scattering [J], IMA J.
Appl. Math., 72 (2007), no. 6, 817-831.
[18] H. Y. Liu, J. Zou, Uniqueness in an inverse acoustic obstacle scattering prob-
lem for both sound-hard and sound-soft polyhedral scatterers [J], Inverse
Problems, 22 (2006), no. 2, 515-524.
[19] H. Y. Liu, J. Zou, On unique determination of partially coated polyhedral
scatterers with far field measurements [J], Inverse Problems, 23 (2007), no. 1, 297-308.
[20] H. Y. Liu, On recovering polyhedral scatterers with acoustic far-field mea- surements [J], IMA J. Appl. Math., 74 (2009), no. 2, 264-272.
[21] H. Koyunbakan, A new inverse problem for the diffusion operator [J], Applied Mathematics Letters, 19 (2006), 995-999.
[22] K. V. Kravchenko, On differential operators with nonlocal boundary condi- tions [J], Differ. Uravn. 36 (2000), 464-469; English transl. in Differ. Equa- tions, 36 (2000), 517-523.
[23] J. R. McLaughlin, Inverse spectral theory using nadal points as data-a unique- ness result [J], Journal of Differential Equations, 73 (1988), 354-362.
[24] L. P. Nizhnik, Inverse eigenvalue problems for nonlocal Sturm-Liouville op- erators [J], Methods Funct. Anal. Topology, 15 (2009), 41-47.
[25] L. P. Nizhnik, Inverse nonlocal Sturm-Liouville problem [J], Inverse Problems, 26 (2010), 635-684.
[26] A. S. Ozkan, B. Keskin, Inverse nodal problems for Sturm-Liouville equation with eigenparameter-dependent boundary and jump conditions [J], Inverse Problems in Science and Engineering, 23 (2015), 1306-1312.
[27] A. D. Wentzell, On boundary conditions for multi-dimensional diffusion pro- cesses [J], Teor. Veroyatnost. Primenen., 4 (1959) 172-185.(in Russian)
[28] C. F. Yang, Inverse nodal problem for a class of nonlocal Sturm-Liouville operator [J], Mathematical Modelling and Analysis, 15 (2010), 383-392.
[29] C. F. Yang, Inverse nodal problems for the Sturm-Liouville operator with a
constant delay [J], Journal of Differential Equations, 257 (2014), 1288-1306.
[30] C. F. Yang, On the quasinodal map for the diffusion operator [J], Journal of
Functional Analysis, 266 (2014), 4236-4265.
[31] V. A. Yurko, Inverse nodal problem for Sturm-Liouville operators on star-type
graphs [J], Journal of Inverse and Ill-Posed Problems, 16 (2008), 715-722.
[32] V. A. Yurko, An inverse problem for integro-differential operators [J], Matem.Zametki, 50 (1991), 134-146(Russian); English transl. An inverse problem for integro-differential operators [J], in Mathematical Notes, 50
(1991), 1188-1197.