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NUMERICAL INVESTIGATION OF THE INVERSE NODAL PROBLEM BY CHEBISYHEV INTERPOLATION METHOD

ABSTRACT
In this study, we deal with the inverse nodal problem for Sturm-Liouville equation with eigenparameter-dependent and jump conditions. Firstly, we obtain reconstruction formulas for potential function, q, under a condition and boundary data, α, as a limit by using nodal points to apply the Chebyshev interpolation method. Then, we prove the stability of this problem. Finally, we calculate approximate solutions of the inverse nodal problem by considering the Chebyshev interpolation method. We then present some numerical examples using Matlab software program to compare the results obtained by the classical approach and by Chebyshev polynomials for the solutions of the problem.
KEYWORDS
PAPER SUBMITTED: 2017-06-12
PAPER REVISED: 2017-11-21
PAPER ACCEPTED: 2017-11-22
PUBLISHED ONLINE: 2018-01-07
DOI REFERENCE: https://doi.org/10.2298/TSCI170612278G
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2018, VOLUME 22, ISSUE Supplement 1, PAGES [S123 - S136]
REFERENCES
  1. Levitan, B. M., Sargsjan, I. S., Introduction to spectral theory: Self adjoint ordinary differential operators. American Mathematical Society, Providence, Rhode Island, 1975
  2. Lowe, B. D., et al. The recovery of potentials from finite spectral data. SIAM Journal on Mathematical Analysis, 23 (1992), 2, pp. 482-504
  3. Hryniv, R., Pronska, N., Inverse spectral problems for energy-dependent Sturm-Liouville equations, Inverse Problems, 28 (2012), 8, 085008
  4. McLaughlin, J. R., Inverse spectral theory using nodal points as data-a uniqueness result, Journal of Differential Equations, 73 (1988), 2, pp. 342-362
  5. Shen, C. L., On the nodal sets of the eigenfunctions of the string equations, SIAM Journal on Mathematical Analysis,19 (1988), 6, pp. 1419-1424
  6. Browne, P. J., Sleeman, B. D., Inverse nodal problem for Sturm-Liouville equation with eigenparameter depend boundary conditions, Inverse Problems, 12 (1996), 4, pp. 377-381
  7. Shieh, C. T.,Yurko, V. A., Inverse nodal and inverse spectral problems for discontinuous boundary value problems, Journal of Mathematical Analysis and Applications, 347 (2008), pp. 266-272
  8. Koyunbakan, H., Yilmaz, E., Reconstruction of the potential function and its derivatives for the diffusion operator, Verlag der Zeitschrift für Naturforsch, 63 (2008), a, pp. 127-130
  9. Guo, Y. X., et al., Inverse nodal problems for Sturm-Liouville equations with boundary conditions depending polynomially on the spectral parameter, Chinese Annals of Mathematics Series A, 33 (2012), 6, pp. 705-718
  10. Yilmaz, E., et al., Inverse nodal problem for the differential operator with a singularity at zero, CMES-Computer Modeling in Engineering, 92 (2013), 3, pp. 301-313
  11. Ozkan, A. S., Keskin, B., Spectral problems for Sturm-Liouville operator with boundary and jump conditions linearly dependent on the eigenparameter, Inverse Problems in Science and Engineering, 20 (2012), 6, pp. 799-808
  12. Ozkan, A. S., Keskin, B., Inverse nodal problems for Sturm-Liouville equation with eigenparameter- dependent boundary and jump conditions, Inverse Problems in Science and Engineering, 23 (2014), 8, pp. 1306-1312
  13. Gladwell, G. M. L., The application of Schur's algorithm to an inverse eigenvalue problem, Inverse Problems, 7 (1991), 4, pp. 557-565
  14. Fabiano, R. H., et al., A finite-difference algorithm for an inverse Sturm-Liouville problem, IMA Journal of Numerical Analysis, 15 (1995), 1, pp. 75-88
  15. Gao, Q., et al., Modified Numerov's method for inverse Sturm-Liouville problems, Journal of Computational and Applied Mathematics, 253 (2013), pp. 181-199
  16. Efremova, L., Freiling, G., Numerical solution of inverse spectral problems for Sturm-Liouville operators with discontinuous potentials, Central European Journal of Mathematics, 11 (2013), 11, pp. 2044-2051
  17. Drignei, M. C., A Newton-type method for solving an inverse Sturm-Liouville problem, Inverse Problems in Science and Engineering, 23 (2015), 5, pp. 851-883
  18. Ignatiev, M., Yurko, V., Numerical methods for solving inverse Sturm-Liouville problems, Results in Mathematics, 52 (2008), 1-2, pp. 63-74
  19. Hald, O. H., The inverse Sturm-Liouville problem and the Rayleigh-Ritz method, Mathematics of Computation, 32 (1978), 143, pp. 687-705
  20. Sacks, P. E., An iterative method for the inverse Dirichlet problem, Inverse Problems, 4 (1988), 4, pp. 1055-1069
  21. Rundell, W., Sacks P. E., Reconstruction techniques for classical inverse Sturm-Liouville problems, Mathematics of Computation, 58 (1992), 92, pp. 161-183
  22. Ro¨hrl, N., A least-squares functional for solving inverse Sturm-Liouville problems, Inverse Problems, 21 (2005), 6, pp. 2009-2017
  23. Andrew, A. L., Numerov's method for inverse Sturm-Liouville problem, Inverse Problems, 21 (2005), 1, pp. 223-238
  24. Rashed, M. T., Numerical solution of a special type of integro-differential equations, Applied Mathematics and computation, 143 (2003), 1, pp. 73-88
  25. Marchenko, V. A., Maslov, K. V., Stability of the problem of recovering the Sturm-Liouville operator from the spectral function, Mathematics of the USSR Sbornik, 81 (1970), 4, pp. 475-502
  26. McLaughlin, J. R., Stability theorems for two inverse spectral problems, Inverse Problems, 4 (1988), 2, pp. 529-540
  27. Law, C. K., Tsay, J., On the well-posedness of the inverse nodal problem, Inverse Problems, 17 (2001), 5, pp. 1493-1512
  28. Cheng, Y. H., Law, C. K., The inverse nodal problem for Hill's Equation, Inverse Problems, 22 (2006), 3, pp. 891-901
  29. Yilmaz, E., Koyunbakan, H., On the high order Lipschitz stability of inverse nodal problem for string equation, Dynamics of Continuous, Discrete and Impulsive Systems, 21 (2014), 1, pp. 79-88
  30. Yilmaz, E., et al., On the Lipschitz stability of inverse nodal problem for p-Laplacian Schrödinger equation with energy dependent potential, Boundary Value Problems, 2015:32 (2015), 8 pp.
  31. Yang, C. F., Stability in the inverse nodal solution for the interior transmission problem, Journal of differential equations, 260 (2016), 3, pp. 2490-2506
  32. Baumeister, J., Stable solution of inverse problems, Advanced Lectures in Mathematics, Vieweg, Braunschweig, 1986

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence