NUMERICAL SOLUTION OF SINGULAR INVERSE NODAL PROBLEM BY USING CHEBYSHEV POLYNOMIALS

Abstract

In this study, we consider Sturm-Liouville problem in two cases: the first case having no singularity and the second case having a singularity at zero. Then, we calculate the eigenvalues and the nodal points and present the uniqueness theorem for the solution of the inverse problem by using a dense subset of the nodal points in two given cases. Also, we use Chebyshev polynomials of the first kind for calculating the approximate solution of the inverse nodal problem in these cases. Finally, we present the numerical results by providing some examples.

Keywords

• Inverse nodal problem; singularity,numerical method,Chebyshev

References

1. [1] Ambartsumyan VA. Über eine frage der eigenwerttheorie. Zeitschrift fr Physik. 1929;53:690-695.
2. [2] Andrew AL. Numerov's method for inverse Sturm-Liouville problem. Inverse Problems. 2005;21:223-238.
3. [3] Aygar Y. Investigation of spectral analysis of matrix quantum difference equations with spectral singularities. Hacettepe Journal of Mathematics and Statistics. DOI: 10. 15672/HJMS.20164513107.
4. [4] Browne PJ, Sleeman BD. Inverse nodal problem for Sturm-Liouville equation with eigenparameter dependent boundary conditions. Inverse Problems. 1996;12:377-381.
5. [5] Drignei MC. A Newton-type method for solving an inverse Sturm-Liouville problem. Inverse Problems in Science and Engineering. http://dx.doi.org/10.1080/17415977.2014.947478.
6. [6] Efremova L, Freiling G. Numerical solution of inverse spectral problems for Sturm-Liouville operators with discontinuous potentials. Cent. Eur. J. Math. 2013;11:2044-2051.
7. [7] Fabiano RH, Knobel R, Lowe BD. A nite-difference algorithm for an inverse Sturm-Liouville problem. IMA J. Numer. Anal. 1995;15:75-88. doi:10.1093/imanum/15.1.75.
8. [8] Freiling G, Yurko V. Inverse Sturm-Liouville problems and their applications. New York:NOVA science publishers; 2001.

9. [9] Gao Q, Cheng X, Huang Z. Modied Numerov's method for inverse Sturm-Liouville problems. Journal of Computational and Applied Mathematics. 2013;253:181-199.
10. [10] Gladwell GML. The application of Schur's algorithm to an inverse eigenvalue problem. Inverse Problems. 1991;7:557-565.
11. [11] Gulsen T, Panakhov ES. Dirac systems that contain discontinuity conditions. AIP Conference Proceeding. 2016;1759:1-6.
12. [12] Gulsen T, Yilmaz E, Panakhov ES. On a Lipschitz stability problem for p-Laplacian Bessel equation. Communications, Series A1; Mathematics and Statistics. 2017;66:253-262.
13. [13] Hald OH. The inverse Sturm-Liouville problem and the Rayleigh-Ritz method. Math. Comp. 1978;32:687-705.
14. [14] Hald OH, McLaughlin JR. Solutions of inverse nodal problems. Inverse Problems. 1989;5:307-347.
15. [15] Ignatiev M, Yurko V. Numerical methods for solving inverse Sturm-Liouville problems. Result. Math. 2008;52:63-74.
16. [16] Kerimov NB, Goktas S, Maris EA. Uniform convergence of the spectral expansions in terms of root functions for a spectral problem. Electronic Journal of Differential Equations. 2016;80:1-14.
17. [17] Koyunbakan H. A new inverse problem for the diffusion operator. Applied Mathematics Letters. 2006;19:995-999.
18. [18] Koyunbakan H, Panakhov ES. Solution of a discontinuous inverse nodal problem on a nite interval. Mathematical and Computer Modelling. 2006;44:204-209.
19. [19] Koyunbakan H, Yilmaz E. Reconstruction of the potential function and its derivatives for the diffusion operator. Verlag der Zeitschriftfur Naturforch. 2008;63(a):127-130.
20. [20] Law CK, Yang CF. Reconstruction of the potential function and its derivatives using nodal data. Inverse Problems. 1999;14:299-312.
21. [21] Levitan BM, Sargsjan IS. Introduction to spectral theory: Self adjoint ordinary differential operators. American Mathematical Society. Providence, Rhode Island; 1975.
22. [22] Lowe BD, Pilant M, Rundell W. The recovery of potentials from nite spectral data. SIAM J. Math. Anal. 1992;23:482-504. doi:10.1137/0523023.
23. [23] Marchenko VA, Maslov KV. Stability of the problem of recovering the Sturm- Liouville operator from the spectral function. Mathematics of the USSR Sbornik. 1970;81:475-502.
24. [24] McLaughlin JR. Inverse spectral theory using nodal points as data-a uniqueness result. Journal of Differential Equations. 1988;73:342-362.
25. [25] McLaughlin JR. Stability theorems for two inverse spectral problems. Inverse Problems. 1988;4:529-540.
26. [26] Neamaty A, Akbarpoor Sh. Numerical solution of inverse nodal problem with an eigenvalue in the boundary condition. Inverse Problems in Science and Engineering. 2017;25:978-994.
27. [27] Neamaty A, Akbarpoor Sh, Dabbaghian A. Uniqueness theorem for the inverse aftereffect problem and representation the nodal points form. Journal of Mathematical Extension. 2015;9:37-49.
28. [28] Panakhov ES, Gulsen T. On discontinuous Dirac systems with eigenvalue dependent boundary conditions. AIP Conference Proceeding. 2015;1648:1-4.
29. [29] Pivovarchik V. Direct and inverse three-point Sturm-Liouville problems with parameter dependent boundary conditions. Asymptotic Analysis. 2001;26:219-238.
30. [30] Poschel J, Trubowitz E. Inverse spectral theory. volume 130 of Pure and Applied Mathematics. Academic Press, Inc, Boston, MA; 1987.
31. [31] Rashed MT. Numerical solution of a special type of integro-dierential equations. Applied Mathematics and computation. 2003;143:73-88.
32. [32] Rohrl N. A least-squares functional for solving inverse Sturm-Liouville problems. Inverse Problems. 2005;21:2009{2017. doi:10.1088/0266-5611/21/6/013.
33. [33] Rundell W, Sacks PE. Reconstruction techniques for classical inverse Sturm-Liouville problems. Math. Comp. 1992;58:161-183.
34. [34] Sacks PE. An iterative method for the inverse Dirichlet problem. Inverse Problems. 1988;4:1055{1069. doi:10.1088/0266-5611/4/4/009.
35. [35] Shen CL. On the nodal sets of the eigenfunctions of the string equations. SIAM Journal on Mathematical Analysis. 1988;19:1419-1424.
36. [36] Shieh CT, Yurko VA. Inverse nodal and inverse spectral problems for discontinuous boundary value problems. Journal of Mathematical Analysis and Applications. 2008;347:266-272.
37. [37] Yang CF. Reconstruction of the diffusion operator from nodal data. Verlag der Zeitschrift für Naturforch. 2010;65a:100-106.
38. [38] Yang XF. A solution of the inverse nodal problem. Inverse Problems. 1997;13:203-213.
39. [39] Yang XF. A new inverse nodal problem. J. Differential Equations. 2001;169:633-653.
40. [40] Yilmaz E, Koyunbakan H, Ic U. Inverse nodal problem for the differential operator with a singularity at zero. Computer Modeling in engineering and Sciences. 2013;92:303-313.