Lipschitz Stability of Inverse Nodal Problem for Energy-Dependent Sturm-Liouville Equation

Abstract

In this study, we solve the reconstruction and some stability problems for diffusion operator using nodal set ofeigenfunctions. Moreover, we show that the space of all potential functions q is homeomorphic to the partition set of allasymptotically equivalent nodal sequences induced by an equivalence relation. To show this stability which is known Lipschitzstability, we have to construct two metric spaces and a mapΦdi fbetween these spaces. We find thatΦdi fis a homeomorphism whenthe corresponding metrics are magnified by the derivatives of q. Basically, this method is similar to [1] and [] which is given forSturm-Liouville and Hill operators, respectively and depends on the explicit asymptotic expansions of nodal points and nodal lengths

Keywords

• Inverse Nodal Problem, Diffusion Equation, Lipschitz Stability

Lipschitz Stability of inverse nodal problem for energy-dependent Sturm-Liouville equation

Öz

References

1. C. K. Law and J. Tsay, On the well-posedness of the inverse nodal problem, Inverse Problems, (2001) 17, 1493-1512.
2. Y. H. Cheng and C. K. Law, The inverse nodal problem for Hill’s Equation, Inverse Problems, (2006) 22, 891-901.
3. G. Freiling and V. A. Yurko, Inverse Sturm-Liouville problems and their applications, NOVA Science Publishers, New York, (2001).
4. V. A. Ambartsumyan, ¨Uber eine frage der eigenwerttheorie, Zeitschrift f¨ur Physik, (1929) 53, 690-695.
5. B. M. Levitan and I. S. Sargsjan, Introduction to spectral theory: self adjoint ordinary differential operators, American
6. Mathematical Society, Providence, Rhode Island, (1975).
7. J. R. McLaughlin, Analytic methods for recovering coefficients in differential equations from spectral data, SIAM Review, (1986) 28, 53-72.
8. J. P¨oschel and E. Trubowitz, Inverse spectral theory, volume 130 of Pure and Applied Mathematics, Academic Press, Inc, Boston, MA, (1987).

9. V. Pivovarchik, Direct and inverse three-point Sturm-Liouville problems with parameter-dependent boundary conditions,
10. Asymptotic Analysis, (2001) 26, 219–238
11. C. T. Shieh, S. A. Buterin and M. Ignatiev, On Hochstadt-Liebermann theorem for Sturm-Liouville operators, Far East Journal of
12. Applied Mathematics, (2011) 52(2), 131-146.
13. J. R. McLaughlin, Inverse spectral theory using nodal points as data-a uniqueness result, Journal of Differential Equations, (1988) 73, 342-362.
14. C. L. Shen, On the nodal sets of the eigenfunctions of the string equations, SIAM Journal on Mathematical Analysis, (1988) 19, 1419-1424.
15. O. H. Hald and J. R. McLaughlin, Solutions of the inverse nodal problems, Inverse Problems, (1989) 5, 307-347.
16. 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(27), 2718-2724.
17. C. F. Yang, Inverse nodal problems for the Sturm-Liouville operator with eigenparameter dependent boundary conditions, Operators and Matrices, (2012) 6(1), 63-77.
18. 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.
19. E. Yılmaz 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. H. Koyunbakan, E. S. Panakhov, A uniqueness theorem for inverse nodal problem, Inverse Problems in Science and Engineering, (2007) 12(6), 517-524.
21. M. G. Gasymov and G. Sh. Guseinov, Determination of a diffusion operator from the spectral data, Doklady Akademii Nauk Azerbaijan SSSR, (1981) 37(2), 19-23.
22. C. F. Yang, New trace formulae for a quadratic pencil of the Schr¨odinger operator, Journal of Mathematical Physics, (2010) 51, 033506.
23. G. Sh. Guseinov, On spectral analysis of a quadratic pencil of Sturm-Liouville operators, Soviet Mathematics Doklady, (1985) 32(3), 859-862.
24. 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.
25. I. M. Guseinov and I. M. Nabiev, The inverse spectral problem for pencils of differential operators, Sbornik Mathematics, (2007) 198(11), 1579-1598.
26. F. G. Maksudov and G. Sh. Guseinov, On solution of the inverse scattering problem for a quadratic pencil of one dimensional Schr¨odinger operators on whole axis, Soviet Mathematics Doklady, (1987) 34, 34-38.
27. I. M. Nabiev, Multiplicities and relation position of eigenvalues of a quadratic pencil of Sturm-Liouville operators, Mathematical Notes, (2000) 67(3), 369-381.
28. 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.
29. R. Hryniv and N. Pronska, Inverse spectral problems for energy-dependent Sturm-Liouville equations, Inverse Problems, (2012) 28(8), 085008.
30. E. Bairamov, ¨O. C¸ akar and A. O. C¸ elebi, Quadratic pencil of Schr¨odinger operators with spectral singularities: discrete spectrum and principal functions, Journal of Mathematical Analysis and Applications, (1997) 216(1), 303-320.
31. H. Koyunbakan, Inverse nodal problem for p−Laplacian energy-dependent Sturm-Liouville equation, Boundary Value Problems, (2013), 2013:272.
32. H. Koyunbakan, Reconstruction of potential function for diffusion operator, Numerical Functional Analysis and Optimization, (2008) 29(7-8), 1-10.
33. H. Koyunbakan, A new inverse problem for diffusion operator, Applied Mathematics Letters, (2006) 19(10), 995-999.
34. S. A. Buterin and C. T. Shieh, Inverse nodal problem for differential pencils, Applied Mathematics Letters, (2008) 22, 1240-1247.
35. C. F. Yang, Reconstruction of the diffusion operator from nodal data, Verlag der Zeitschrift f¨ur Naturforch, (2010) 65(a), 100-106.
36. H. Koyunbakan and E. Yilmaz, Reconstruction of the potential function and its derivatives for the diffusion operator, Verlag der Zeitschrift f¨ur Naturforch, (2008) 63(a), 127-130.
37. Rauf Kh. Amirov and A. Adiloglu Nabiev, Inverse Problems for the Quadratic Pencil of the Sturm-Liouville Equations with Impulse, Abstract and Applied Analysis, Volume 2013, Article ID 361989.
38. Y. P. Wang, The inverse problem for differential pencils with eigenparameter dependent boundary conditions from interior spectral data, Applied Mathematics Letters, (2012) 25 1061-1067.
39. M. Sat and E. S. Panakhov, Spectral problem for diffusion operator, Applicable Analysis, (2014) 93(6) 1178-1186.
40. M. Jaulent and C. Jean, The inverses-wave scattering problem for a class of potentials depending on energy, Communications in Mathematical Physics, (1972) 28(3), 177-220.
41. A. Wazwaz, Partial differential equations methods and applications, Taylor and Francis, (2002).
42. 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.
43. K. Chadan, D. Colton, L. Paivarinta and W. Rundell, An introduction to inverse scattering and inverse spectral problems, Society for Industrial and Applied Mathematics, (1997).
44. V. A. Marchenko and K. V. Maslov, Stability of the problem of recovering the Sturm-Liouville operator from the spectral function, Mathematics of the USSR Sbornik, (1970) 81(123), 475-502.
45. J. R. McLaughlin, Stability theorems for two inverse spectral problems, Inverse Problems, (1988) 4, 529-540.
46. E. Yilmaz and H Koyunbakan, On the high order Lipschitz stability of inverse nodal problem for string equation, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, (2014) 21, 79-88.
47. C. K. Law and C. F. Yang, Reconstruction of the potential function and its derivatives using nodal data, Inverse Problems, (1999) 14, 299-312.