Abstract
In this paper, we study inverse nodal problems for Sturm–Liouville operators with many frozen arguments. We obtain the asymptotics of eigenvalues and nodal points (zeros) of the corresponding eigenfunction. Moreover, we establish a uniqueness theorem concerning the potential under condition
References
[1] S. Akbarpoor, H. Koyunbakan and A. Dabbaghian, Solving inverse nodal problem with spectral parameter in boundary conditions, Inverse Probl. Sci. Eng. 27 (2019), no. 12, 1790–1801. 10.1080/17415977.2019.1597871Search in Google Scholar
[2] S. Akbarpoor Kiasary and E. Yilmaz, Solving an inverse nodal problem with Herglotz–Nevanlinna functions in boundary conditions using the second-kind Chebyshev wavelets method, Math. Methods Appl. Sci. 46 (2023), no. 4, 4437–4448. 10.1002/mma.8768Search in Google Scholar
[3] A. Akyuz Dascioglu and N. Isler, Bernstein collocation method for solving nonlinear differential equations, Math. Comput. Appl. 18 (2013), no. 3, 293–300. 10.3390/mca18030293Search in Google Scholar
[4] N. P. Bondarenko, Finite-difference approximation of the inverse Sturm–Liouville problem with frozen argument, Appl. Math. Comput. 413 (2022), Article ID 126653. 10.1016/j.amc.2021.126653Search in Google Scholar
[5] N. P. Bondarenko, S. A. Buterin and S. V. Vasiliev, An inverse spectral problem for Sturm–Liouville operators with frozen argument, J. Math. Anal. Appl. 472 (2019), no. 1, 1028–1041. 10.1016/j.jmaa.2018.11.062Search in Google Scholar
[6] P. J. Browne and B. D. Sleeman, Inverse nodal problems for Sturm–Liouville equations with eigenparameter dependent boundary conditions, Inverse Problems 12 (1996), no. 4, 377–381. 10.1088/0266-5611/12/4/002Search in Google Scholar
[7] S. Buterin and Y.-T. Hu, Inverse spectral problems for Hill-type operators with frozen argument, Anal. Math. Phys. 11 (2021), no. 2, Paper No. 75. 10.1007/s13324-021-00500-9Search in Google Scholar
[8] S. Buterin and M. Kuznetsova, On the inverse problem for Sturm–Liouville-type operators with frozen argument: rational case, Comput. Appl. Math. 39 (2020), no. 1, Paper No. 5. 10.1007/s40314-019-0972-8Search in Google Scholar
[9] S. A. Buterin and S. V. Vasiliev, On recovering a Sturm–Liouville-type operator with the frozen argument rationally proportioned to the interval length, J. Inverse Ill-Posed Probl. 27 (2019), no. 3, 429–438. 10.1515/jiip-2018-0047Search in Google Scholar
[10] 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 (2023), no. 1, 397–404. 10.55730/1300-0098.3367Search in Google Scholar
[11] X. Chen, Y. H. Cheng and C. K. Law, Reconstructing potentials from zeros of one eigenfunction, Trans. Amer. Math. Soc. 363 (2011), no. 9, 4831–4851. 10.1090/S0002-9947-2011-05258-XSearch in Google Scholar
[12] Y.-H. Cheng, C. K. Law and J. Tsay, Remarks on a new inverse nodal problem, J. Math. Anal. Appl. 248 (2000), no. 1, 145–155. 10.1006/jmaa.2000.6878Search in Google Scholar
[13] H. Ž. Dikinov, A. A. Kerefov and A. M. Nahušev, A certain boundary value problem for a loaded heat equation, Differ. Uravn. 12 (1976), no. 1, 177–179. Search in Google Scholar
[14] O. Dobosevych and R. Hryniv, Reconstruction of differential operators with frozen argument, Axioms 11 (2022), Paper No. 24. 10.3390/axioms11010024Search in Google Scholar
[15] Y. Guo and G. Wei, Inverse problems: Dense nodal subset on an interior subinterval, J. Differential Equations 255 (2013), no. 7, 2002–2017. 10.1016/j.jde.2013.06.006Search in Google Scholar
[16] O. H. Hald and J. R. McLaughlin, Solutions of inverse nodal problems, Inverse Problems 5 (1989), no. 3, 307–347. 10.1088/0266-5611/5/3/008Search in Google Scholar
[17] Y.-T. Hu, N. P. Bondarenko and C.-F. Yang, Traces and inverse nodal problem for Sturm–Liouville operators with frozen argument, Appl. Math. Lett. 102 (2020), Article ID 106096. 10.1016/j.aml.2019.106096Search in Google Scholar
[18] A. D. Iskenderov, The first boundary value problem for a charged system of quasi-linear parabolic equations, Differ. Uravn. 7 (1971), 1911–1913. Search in Google Scholar
[19] B. Keskin and A. S. Ozkan, Inverse nodal problems for Dirac-type integro-differential operators, J. Differential Equations 263 (2017), no. 12, 8838–8847. 10.1016/j.jde.2017.08.068Search in Google Scholar
[20] B. Keskin and Y. P. Wang, On the reconstruction of an integro-differential Dirac operator with parameter-dependent nonlocal integral boundary conditions from the nodal data, Turkish J. Math. 48 (2024), no. 2, 210–220. 10.55730/1300-0098.3502Search in Google Scholar
[21] A. M. Krall, The development of general differential and general differential-boundary systems, Rocky Mountain J. Math. 5 (1975), no. 4, 493–542. 10.1216/RMJ-1975-5-4-493Search in Google Scholar
[22] M. Kuznetsova, Necessary and sufficient conditions for the spectra of the Sturm–Liouville operators with frozen argument, Appl. Math. Lett. 131 (2022), Article ID 108035. 10.1016/j.aml.2022.108035Search in Google Scholar
[23] M. Kuznetsova, Uniform stability of recovering Sturm–Liouville-type operators with frozen argument, Results Math. 78 (2023), no. 5, Paper No. 169. 10.1007/s00025-023-01945-zSearch in Google Scholar
[24] G. G. Lorentz, Bernstein Polynomials, Chelsea, New York, 1986. Search in Google Scholar
[25] J. R. McLaughlin, Inverse spectral theory using nodal points as data—a uniqueness result, J. Differential Equations 73 (1988), no. 2, 354–362. 10.1016/0022-0396(88)90111-8Search in Google Scholar
[26] A. M. Nakhushev, An approximate method for solving boundary value problems for differential equations and its application to the dynamics of ground moisture and ground water, Differ. Uravn. 18 (1982), no. 1, 72–81. Search in Google Scholar
[27] A. M. Nakhushev, Loaded Equations and Their Applications, Nauka, Moscow, 2012. Search in Google Scholar
[28] A. M. Nakhushev and V. N. Borisov, Boundary value problems for loaded parabolic equations and their applications to the prediction of ground water level, Differ. Uravn. 13 (1977), no. 1, 105–110. Search in Google Scholar
[29] C. L. Shen, On the nodal sets of the eigenfunctions of the string equation, SIAM J. Math. Anal. 19 (1988), no. 6, 1419–1424. 10.1137/0519104Search in Google Scholar
[30] C.-T. Shieh and T.-M. Tsai, Inverse spectral problems for Sturm–Liouville operators with many frozen arguments, Appl. Math. Comput. 492 (2025), Article ID 129235. 10.1016/j.amc.2024.129235Search in Google Scholar
[31] C.-T. Shieh and V. A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl. 347 (2008), no. 1, 266–272. 10.1016/j.jmaa.2008.05.097Search in Google Scholar
[32] F. Song, Y. Wang and S. Akbarpoor, Inverse nodal problems for Dirac operators and their numerical approximations, Electron. J. Differential Equations 2023 (2023), Paper No. 81. 10.58997/ejde.2023.81Search in Google Scholar
[33] Y. Tang, H. Ni, F. Song and Y. P. Wang, Numerical solutions of inverse nodal problems for a boundary value problem, Mathematics 10 (2022), no. 22, Article ID 4204. 10.3390/math10224204Search in Google Scholar
[34] T.-M. Tsai, H.-F. Liu, S. Buterin, L.-H. Chen and C.-T. Shieh, Sturm–Liouville-type operators with frozen argument and Chebyshev polynomials, Math. Methods Appl. Sci. 45 (2022), no. 16, 9635–9652. 10.1002/mma.8327Search in Google Scholar
[35] Y. Wang and L. Zhu, SCW method for solving the fractional integro-differential equations with a weakly singular kernel, Appl. Math. Comput. 275 (2016), 72–80. 10.1016/j.amc.2015.11.057Search in Google Scholar
[36] Y. P. Wang, S. Akbarpoor Kiasary and E. Yılmaz, Solving inverse nodal problem with frozen argument by using second Chebyshev wavelet method, Appl. Math. 69 (2024), no. 3, 339–354. 10.21136/AM.2024.0038-21Search in Google Scholar
[37] Y. P. Wang, E. Yilmaz and S. Akbarpoor, The numerical solution of inverse nodal problem for integro-differential operator by Legendre wavelet method, Int. J. Comput. Math. 100 (2023), no. 1, 219–232. 10.1080/00207160.2022.2108708Search in Google Scholar
[38] Y. P. Wang and V. A. Yurko, On the inverse nodal problems for discontinuous Sturm–Liouville operators, J. Differential Equations 260 (2016), no. 5, 4086–4109. 10.1016/j.jde.2015.11.004Search in Google Scholar
[39] Y. P. Wang, M. Zhang, W. Zhao and X. Wei, Reconstruction for Sturm–Liouville operators with frozen argument for irrational cases, Appl. Math. Lett. 111 (2021), Article ID 106590. 10.1016/j.aml.2020.106590Search in Google Scholar
[40] C.-F. Yang, Solutions to open problems of Yang concerning inverse nodal problems, Israel J. Math. 204 (2014), no. 1, 283–298. 10.1007/s11856-014-1093-0Search in Google Scholar
[41] X.-F. Yang, A solution of the inverse nodal problem, Inverse Problems 13 (1997), no. 1, 203–213. 10.1088/0266-5611/13/1/016Search in Google Scholar
[42] V. Yurko, Inverse nodal problems for Sturm–Liouville operators on star-type graphs, J. Inverse Ill-Posed Probl. 16 (2008), no. 7, 715–722. 10.1515/JIIP.2008.044Search in Google Scholar
[43] F. Zhou and X. Xu, Numerical solution of the convection diffusion equations by the second kind Chebyshev wavelets, Appl. Math. Comput. 247 (2014), 353–367. 10.1016/j.amc.2014.08.091Search in Google Scholar
[44] L. Zhu and Q. Fan, Numerical solution of nonlinear fractional-order-Volterra integro-differential equations by SCW, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), no. 5, 1203–1213. 10.1016/j.cnsns.2012.09.024Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston