An inverse nodal problem for differential pencils with complex spectral parameter dependent boundary conditions

Abstract

In this study, we are concerned with an inverse nodal problem for second order differential pencil on a finite interval with complex spectral parameter dependent boundary conditions by using nodal points. We give some reconstruction formulas for potential functions  and  as a limit.

Keywords

• Inverse nodal problem,differential pencil

Kaynakça

1. V. A. Ambartsumyan, Über eine Frage der Eigenwerttheorie, Zeitschrift für Physik, (1929) 53, 690-695.
2. B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory: Self Adjoint Ordinary Differential Operators, American Mathematical Society, Providence, Rhode Island, (1975).
3. V. Pivovarchik, Direct and inverse three-point Sturm-Liouville problems with parameter dependent boundary conditions, Asymptotic Analysis, (2001) 26(3-4), 219–238.
4. C. T. Shieh, S. A. Buterin and M. Ignatiev, On Hochstadt-Lieberman theorem for Sturm-Liouville operators, Far East Journal of Applied Mathematics, (2011) 52(2), 131-146.
5. R. Hryniv and N. Pronska, Inverse spectral problems for energy-dependent Sturm-Liouville equations, Inverse Problems, (2012) 28(8), 085008.
6. G. Freiling and V. A. Yurko, Inverse Sturm-Liouville problems and their applications, NOVA Science Publishers, New York, (2001).
7. J. Pöschel and E. Trubowitz, Inverse spectral theory, volume 130 of Pure and Applied Mathematics, Academic Press, Inc, Boston, MA, (1987).
8. J. R. McLaughlin, Inverse spectral theory using nodal points as data-a uniqueness result, Journal of Differential Equations, (1988) 73(2), 354-362.

9. C. L. Shen, On the nodal sets of the eigenfunctions of the string equation, SIAM Journal on Mathematical Analysis, (1988) 19(6), 1419-1424.
10. C. K. Law and J. Tsay, On the well-posedness of the inverse nodal problem, Inverse Problems, (2001) 17(5), 1493-1512.
11. Y. H. Cheng and C. K. Law, The inverse nodal problem for Hill’s Equation, Inverse Problems, (2006) 22(3), 891-901.
12. C. K. Law and C. F. Yang, Reconstructing of the potential function and its derivatives using nodal data, Inverse Problems, (1999) 14(2), 299-312.
13. O. H. Hald and J. R. McLaughlin, Solutions of inverse nodal problems, Inverse Problems, (1989) 5(3), 307-347.
14. W. C. Wang, Y. H. Cheng and W. C. Lian, Inverse nodal problems for the p-Laplacian with eigenparameter dependent boundary conditions, Mathematical and Computer modelling, (2011) 54(11-12), 2718-2724.
15. C. T. Shieh and V. A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, Journal of Mathematical Analysis and Applications, (2008) 347(1), 266-272.
16. C. F. Yang, Inverse nodal problems for the Sturm-Liouville operator with eigenparameter dependent boundary conditions, Operators and Matrices, (2012) 6(1), 63-77.
17. H. Koyunbakan, E. S. Panakhov, A uniqueness theorem for inverse nodal problem, Inverse Problems in Science and Engineering, (2007) 15(6), 517-524.
18. S. A. Buterin and C. T. Shieh, Inverse nodal problem for differential pencils, Applied Mathematics Letters, (2009) 22(8), 1240-1247.
19. E. Yilmaz and H. Koyunbakan, Reconstruction of potential function and its derivatives for Sturm-Liouville problem with eigenvalues in boundary condition, Inverse Problems in Science and Engineering, (2010) 18(7), 935-944.
20. Y. V. Kuryshova and C. T. Shieh, An inverse nodal problem for integro-differential operators, Journal of Inverse and III-posed Problems, (2010) 18(4), 357–369.
21. C. F. Yang, Trace Formulae for differential pencils with spectral parameter dependent boundary conditions, Mathematical Methods in the Applications, (2013) 37(9), 1325-1332.
22. M. Jaulent and C. Jean, The inverse s-wave scattering problem for a class of potentials depending on energy, Communications in Mathematical Physics, (1972) 28(3), 177-220.
23. M. G. Gasymov and G. Sh. Guseinov, Determination of a diffusion operator from the spectral data, Doklady Akademii Nauk Azerbaijan SSR, (1981) 37(2), 19-23.
24. A. M. Wazwaz, Partial differential equations: Methods and applications, Balkema, The Netherlands, (2002).
25. L. K. Sharma, P. V. Luhanga and S. Chimidza, Potentials for the Klein-Gordon and Dirac equations, Chiang Mai Journal of Science, (2011) 38(4), 514-526.
26. K. Chadan, D. Colton, L. Päivärinta and W. Rundell, An introduction to inverse scattering and inverse spectral problems, Dokl. Akad. Nauk SSSR, (1985), 285(6), 1292-1296; Society for Industrial and Applied Mathematics, Philadelphia, PA, (1997).
27. G. Sh. Guseinov, On spectral analysis of a quadratic pencil of Sturm-Liouville operators, Soviet Mathematics Doklady, (1985) 32(3), 859-862.
28. I. M. Guseinov and I. M. Nabiev, A class of inverse problems for a quadratic pencil of Sturm-Liouville operators, Differential Equations, (2000) 36(3), 471-473.
29. I. M. Guseinov and I. M. Nabiev, The inverse spectral problem for pencils of differential operators, Sbornik Mathematics, (2007) 198(11), 1579-1598.
30. F. G. Maksudov and G. Sh. Guseinov, On solution of the inverse scattering problem for a quadratic pencil of one dimensional Schrödinger operators on thewhole axis, Soviet Mathematics Doklady, (1987) 34, 34-38.
31. I. M. Nabiev, Multiplicities and relation position of eigenvalues of a quadratic pencil of Sturm-Liouville operators, Mathematical Notes, (2000) 67(3), 309-319.
32. C. F. Yang and A. Zettl, Half inverse problems for quadratic pencils of Sturm-Liouville operators, Taiwanese Journal of Mathematics, (2012) 16(5), 1829-1846.
33. E. Bairamov, Ö. Çakar and A. O. Çelebi, Quadratic pencil of Schrödinger operators with spectral singularities: discrete spectrum and principal functions, Journal of Mathematical Analysis and Applications, (1997) 216(1), 303-320.
34. H. Koyunbakan, Reconstruction of potential function for diffusion operator, Numerical Functional Analysis and Optimization, (2009) 30(1-2), 111-120.
35. H. Koyunbakan and E. Yilmaz, Reconstruction of the potential function and its derivatives for the diffusion operator, Zeitschrift für Naturforchung A, (2008) 63(3-4), 127-130.
36. C. F. Yang, Reconstruction of the diffusion operator from nodal data, Zeitschrift für Naturforchung A, (2010) 65(1-2), 100-106.
37. R. Kh. Amirov and A. A. Nabiev, Inverse Problems for the Quadratic Pencil of the Sturm-Liouville Equations with Impulse, Abstract and Applied Analysis, Volume 2013, Article ID 361989, 10 pp.
38. Y. P. Wang, The inverse problem for differential pencils with eigenparameter dependent boundary conditions from interior spectral data, Applied Mathematics Letters, (2012) 25(7), 1061-1067.