Numerical Approaches to the Inverse Nodal Problem for Diffusion Operators on Star Graphs with Nonhomogeneous Edges

Öz

Abstract: This paper presents numerical approaches to the inverse nodal problem for a diffusion operator defined on a star graph with nonhomogeneous edges. Based on the asymptotic behavior of sufficiently large eigenvalues, nodal data are employed to reconstruct the potential functions and the parameters appearing in the boundary conditions. The numerical analysis is carried out separately for odd and even star graph configurations, with distinct treatments on the subintervals and . This interval-based strategy enables an effective investigation of the influence of graph parity and edge nonhomogeneity on the inverse reconstruction process. Several numerical examples are provided to illustrate the accuracy and efficiency of the proposed methods.

Anahtar Kelimeler

• inverse nodal problem
• Chebyshev interpola tion method
• approximate solutions

Homojen Olmayan Kenarlara Sahip Yıldız Grafikler Üzerindeki Difüzyon Operatörleri İçin Ters Nodal Probleme Sayısal Yaklaşımlar

Öz

Bu çalışma, homojen olmayan kenarlara sahip bir yıldız grafik üzerinde tanımlanan bir difüzyon operatörü için ters nodal probleme yönelik sayısal yaklaşımlar sunmaktadır. Yeterince büyük özdeğerlerin asimptotik davranışına dayanarak, nodal veriler potansiyel fonksiyonların ve sınır koşullarında yer alan parametrelerin yeniden yapılandırılmasında kullanılmaktadır. Sayısal analiz, tek ve çift yıldız grafik konfigürasyonları için ayrı ayrı gerçekleştirilmiş olup, ve alt aralıklarında farklı yaklaşımlar uygulanmıştır. Bu aralığa dayalı strateji, grafik paritesinin ve kenar homojen olmamasının ters yeniden yapılandırma süreci üzerindeki etkisinin etkin bir biçimde incelenmesini sağlamaktadır. Önerilen yöntemlerin doğruluğunu ve etkinliğini göstermek amacıyla çeşitli sayısal örnekler sunulmuştur.

Anahtar Kelimeler

• Ters nodal problem
• Chebyshev enterpolasyon yöntemi
• Yaklaşık çözümler

Kaynakça

1. Albeverio S, Gesztesy F, Hoegh-Kron R, Holden H. Solvable models in quantum mechanics, Springer, New York-Berlin; 1988. p. 414-452.
2. Amirov R.Kh, Guseinov I. M. Boundary value problems for a class of Sturm-Liouville operator with nonintegrable potential, Differ. Equ. 2002; 38(8): 1195–1197.
3. Savchuk M. On the eigenvalues and eingenfunctions of the Sturm-Liouville operator with singular potential, Math. Notes.2001;69:2.
4. Ibadzadeh Ch.G, Mammadova L. I.,.Nabiev I. M, Inverse problem of spectral analysis for diffusion operatorwith nonseparated boundary conditions and spectral parameter in boundary condition, Azerbaijan J. Math. 2019. 9 (1), 171–189.
5. Durak S. Inverse nodal problem for diffusion operator onastar graph with nonhomogeneous edges J. Inverse Ill-Posed Probl. 2024; 32(3): 485–496
6. Nabiev I. M. Reconstruction of the differential operator with spectral parameter in the boundary condition, Mediterr. J. Math. 2022 19 (3): 124.
7. Mammadova L. I, Nabiev I. M., Rzayeva Ch.H. Uniqueness of the solution of the inverse problem for differential operator with semiseparated boundary conditions, Baku Math. 2022. 1 ( 1): 47–52.
8. Gasymov M. G. and Guseinov G. Sh.. Determination of a diffusion operator from spectral data, In Dokl.Akad. Nauk Azerb SSR. 1981. 37 (2): 19–23.

9. E, Yilmaz, Koyunbakan H, Ambarzumyan S. Type theorems for bessel operator on a f inite interval, Differ. Equ. Dyn. Syst. 2019. 27: 553–559.
10. Bartels S.C, Currie .S, Watson B. A. Sturm-liouville problems with transfer condition Herglotz dependent on the eigenparameter: eigenvalue asymptotics, Complex Anal, Oper. Theory. 2021. 15: 71.
11. Kuznetsova M. A. On recovering the Sturm-Liouville differential operators on time scales, Math. Notes. 2021. 109: 74–88.
12. Bondarenko P, Chitorkin E. E., Inverse Sturm-Liouville problem with spectral parameter in the boundary conditions, Mathematics. 2023. 11(5): 1138.
13. Du G, Gao C, Wang J. Spectral analysis of discontinuous Sturm-Liouville operators with Herglotzs transmission, Electron. Res. Arch. 2023. 31(4): 2108–2119.
14. Guliyev N. J. Inverse square singularities and eigenparameter-dependent boundary condi tions are two sides of the same coin, 2023, DOI 10.1093/qmath/haad004.
15. Guliyev N. J. Schr¨odinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter, J. Math. Phys. 2019. 60(6): 063501.
16. Guliyev N. J. Essentially isospectral transformations and their applications, Ann. Mat. Pura Appl. 2020. 199 (4): 1621–1648.
17. Guliyev N. J. On two-spectra inverse problems, Proc. Amer. Math. Soc. 2020. 148 (10): 4491–4502.
18. McLaughlin J. R. Inverse spectral theory using nodal points as data- a uniqueness result, J. Differ. Equ. 1988.73 (1): 354–362.
19. Hald O. H, McLaughlin J. R. Solutions of inverse nodal problems, Inverse Probl. 1989. 5:307–347.
20. Yang C. F. A solution of the nodal problem, Inverse Probl. 1997. 13:203–213.
21. Buterin S. A, Chung T. Inverse nodal Problem for differential pencils, Appl. Math. Lett. 2009. 22: 1240–1247.
22. Browne P. J, Sleeman B. D. Inverse nodal problem for Sturm–Liouville equation with eigenparameter depend boundary conditions, Inverse Probl. 1996. 12: 377–381.
23. Buterin S. A, Shieh C. T. Incomplete inverse spectral and nodal problems for differential pencil, Results Math. 2012, . 62: 67–179.
24. Cheng Y. H, Law C. K, Tsay J. Remarks on a new inverse nodal problem, J. Math. Anal. Appl. 2000. 248: 145–155.
25. Currie L.S, Watson B. A. Inverse nodal problems for Sturm–Liouville equations on graphs, Inverse Probl. 2007. 23: 2029–2040.
26. Guo Y. X, Wei G. S. Inverse problems: Dense nodal subset on an interior subinterval, J. Differ. Equ. 2013. 255: 2002–2017.
27. Law C. K, Tsay J. On the well-posedness of the inverse nodal problem, Inverse Probl. 2001.17: 1493–1512.
28. Ozkan A. S. Half inverse Sturm-Liouville problem with boundary and discontinuity condi tions dependent on the spectral parameter, Inverse Probl. Sci. Eng. 2014. 22: 848–859.
29. Shen C. L, Shieh C. T. An inverse nodal problem for vectorial Sturm–Liouville equation, Inverse Probl. 2000. 16: 349–356..
30. Shieh C. T, Yurko V. A. Inverse nodal inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl. 2008. 347: 266–272.
31. Yang C. F. A new inverse nodal problem, J. Differ. Equ. 2001. 169: 633–653.
32. Yang C. F, Yang X. P. Inverse nodal problems for the Sturm–Liouville equation with polynomially dependent on the eigenparameter, Inverse Probl. Sci. Eng. 2011. 19: 951–961.
33. Amirov R., Arslantaş M. and Durak S. Inverse nodal problem for singular Sturm--Liouville operator on a star graph. Journal of Inverse and Ill-posed Problems, vol. 32, no. 1, (2024), pp. 1-8