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Numerical solutions to inverse nodal problems for the Sturm–Liouville operator and their applications

  • Yu Ping Wang EMAIL logo , Tzong-Mo Tsai , Shahrbanoo Akbarpoor and Chung-Tsun Shieh
Published/Copyright: January 13, 2025

Abstract

In this paper, we apply numerical methods to study inverse nodal problems for the Sturm–Liouville operator. At first, we find an approximate function of the potential from the nodal points of the ( n + 1 ) -th eigenfunction and three constants via the second kind Chebyshev wavelet method (SCW). We have a sharp condition for uniform convergence of Chebyshev series and an error estimate between the approximate solution and exact solution of the potential. Then, a numerical example is provided to show that the approximate solutions become more accurate and the errors decrease as the values of n increase. Also, we show the comparison of SCW with Bernstein method. Finally, we present an application of numerical solutions to reconstruct the potential from parts of nodal set and its mean value. Compared with some well-known results, the nodal data used here is least.

MSC 2020: 34A55; 34B99; 47E05

References

[1] H. Adibi and P. Assari, Chebyshev wavelet method for numerical solution of Fredholm integral equations of the first kind, Math. Probl. Eng. 2010 (2010), Article ID 138408. 10.1155/2010/138408Search in Google Scholar

[2] 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

[3] N. Bondarenko and S. Buterin, Numerical solution and stability of the inverse spectral problem for a convolution integro-differential operator, Commun. Nonlinear Sci. Numer. Simul. 89 (2020), Article ID 105298. 10.1016/j.cnsns.2020.105298Search in Google Scholar

[4] 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

[5] S. A. Buterin and C. T. Shieh, Inverse nodal problem for differential pencils, Appl. Math. Lett. 22 (2009), no. 8, 1240–1247. 10.1016/j.aml.2009.01.037Search in Google Scholar

[6] S. A. Buterin and C.-T. Shieh, Incomplete inverse spectral and nodal problems for differential pencils, Results Math. 62 (2012), no. 1–2, 167–179. 10.1007/s00025-011-0137-6Search in Google Scholar

[7] 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

[8] 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

[9] S. Currie and B. A. Watson, Inverse nodal problems for Sturm–Liouville equations on graphs, Inverse Problems 23 (2007), no. 5, 2029–2040. 10.1088/0266-5611/23/5/013Search in Google Scholar

[10] G. Freiling and V. Yurko, Inverse Sturm–Liouville Problems and Their Applications, Nova Science, Huntington, 2001. Search in Google Scholar

[11] F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential. II. The case of discrete spectrum, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2765–2787. 10.1090/S0002-9947-99-02544-1Search in Google Scholar

[12] T. Gulsen, E. Yilmaz and S. Akbarpoor, Numerical investigation of the inverse nodal problem by Chebyshev interpolation method, Thermal Science 22 (2018), 123–136. 10.2298/TSCI170612278GSearch in Google Scholar

[13] 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

[14] 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

[15] 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

[16] N. A. Lahmar, O. Belhamiti and S. M. Bahri, A new Legendre decomposition method for solving PDEs, Malaya J. Mat. 1 (2014), 72–81. 10.26637/mjm201/009Search in Google Scholar

[17] Q. Y. Li, N. C. Wang and D. Y. Yi, Numerical Analysis, 4th. ed., Tsinghua University Press, Beijing, 2001. Search in Google Scholar

[18] K. Maleknejad, E. Saeedipoor and R. Dehbozorgi, Legendre wavelets direct method for the numerical solution of Fredholm integral equation of the first kind, Proceedings of the World Congress on Engineering 2016. Vol I, WCE, London (2016), 35–38. Search in Google Scholar

[19] 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

[20] S. Mosazadeh and H. Koyunbakan, On the stability of the solution of the inverse problem for Dirac operator, Appl. Math. Lett. 102 (2020), Article ID 106118. 10.1016/j.aml.2019.106118Search in Google Scholar

[21] M. T. Rashed, Numerical solutions of the integral equations of the first kind, Appl. Math. Comput. 145 (2003), no. 2–3, 413–420. 10.1016/S0096-3003(02)00497-6Search in Google Scholar

[22] X. Shang and D. Han, Numerical solution of Fredholm integral equations of the first kind by using linear Legendre multi-wavelets, Appl. Math. Comput. 191 (2007), no. 2, 440–444. 10.1016/j.amc.2007.02.108Search in Google Scholar

[23] 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

[24] 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

[25] 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

[26] 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, Paper No. 4204. 10.3390/math10224204Search in Google Scholar

[27] Y. P. Wang and C.-T. Shieh, Inverse problems for Sturm–Liouville operators on a compact equilateral graph by partial nodal data, Math. Methods Appl. Sci. 44 (2021), no. 1, 693–704. 10.1002/mma.6775Search in Google Scholar

[28] Y. P. Wang, C.-T. Shieh and X. Wei, Partial inverse nodal problems for differential pencils on a star-shaped graph, Math. Methods Appl. Sci. 43 (2020), no. 15, 8841–8855. 10.1002/mma.6574Search in Google Scholar

[29] 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

[30] 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

[31] X. Wei, H. Miao, C. Ge and C. Zhao, An inverse problem for Sturm–Liouville operators with nodal data on arbitrarily-half intervals, Inverse Probl. Sci. Eng. 29 (2021), no. 3, 305–317. 10.1080/17415977.2020.1779711Search in Google Scholar

[32] 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

[33] X.-F. Yang, A new inverse nodal problem, J. Differential Equation 169 (2001), 633–653. 10.1006/jdeq.2000.3911Search in Google Scholar

[34] 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

[35] 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

Received: 2024-11-13
Revised: 2024-12-10
Accepted: 2024-12-12
Published Online: 2025-01-13
Published in Print: 2025-04-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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