Inverse Nodal Problem for Sturm- Liouville Boundary Value Problem

Yıl 2025, Cilt: 15 Sayı: 1, 308 - 315, 01.03.2025

Merve Arslantaş

https://doi.org/10.21597/jist.1525499

Öz

Inverse nodal problems has been studied for Sturm-Liouville equations with point δ coaction. First, the eigenvalues of the problem are obtained. Then, the solution of the inverse problem is given by obtaining potential function and the parameters in the boundary conditions with the help of a dense set of nodal points. Lastly, the uniqueness theorem is proven and a constructive procedure for solutions is provided.

Anahtar Kelimeler

Singular Sturm-Liouville Operatör , inverse problem , nodel problem

Kaynakça

• Amirov, RKh., Arslantaş, M. & Durak, S. (2024). Inverse nodal problem for singular Sturm-Liouville operator on a star graph. Journal of Inverse and Ill-Posed Problems, 32 (1), 1-8.
• Amirov, RKh. & Durak, S. (2024). Inverse nodal problems for singular diffusion equation. Mathematical Methods in the Applied Sciences, 47 (11), 9067-9083.
• Browne, PJ. & Sleeman, BD. (1996). Inverse nodal problems for Sturm−Liouville equations with eigenparameter dependent boundary conditions. InverseProblems, 12 (4), 377-381.
• Buterin, SA. & Shieh, CT. (2009). Inverse nodal problem for differential pencils. Applied Mathematics Letters, 22 (8), 1240-1247.
• Buterin, SA. & Shieh, CT. (2012). Incomplete inverse spectral and nodal problems for differential pencils. Results in Mathematics, 62, 167-179.
• Chen, X., Cheng, YH. & Law, CK. (2011). Reconstructing potentials from zeros of one eigenfunction. Transactions of the American Mathematical Society, 363 (9), 4831-4851.
• Çakmak, Y. & Keskin, B. (2023). Inverse nodal problem for the quadratic pencil of the Sturm−Liouville equations with parameter-dependent nonlocal boundary condition. Turkish Journal of Mathematics, 47 (1), 397-404.
• Durak, S. (2022). Inverse nodal problem for Sturm-Liouville operator on a star graph with nonequal edges. Turkish Journal of Mathematics, 46 (6), 2178-2192.
• Freiling, G. & Yurko, VA. (2001). Inverse Sturm-Liouville Problems and Their Applications. New York: NOVA Science Publications.
• Guo, Y. & Wei, G. (2013) Inverse Problems dense nodal subset on an interior subinterval. Journal of Differential Equations, 255, 2002-2017.
• Hald, OH. & McLaughlin, JR. (1989). Solutions of inverse nodal problems. Inverse Problems, 5 (3), 307-347.
• Hald, OH. & McLaughlin, JR. (1998). Inverse problems: recovery of BV coefficients from nodes. Inverse Problems, 14 (2), 245-273.
• Hu, YT., Yang, CF. & Xu, XC. (2017). Inverse nodal problems for the Sturm−Liouville operator with nonlocal integral conditions. Journal of Inverse and Ill-Posed Problems, 25 (6), 799-806.
• Law, CK. & Yang, CF. (1998). Reconstucting the potential function and its derivatives using nodal data. Inverse Problems, 14, 299-312.
• Law, CK., Shen, CL. & Yang, CF. (1999) The inverse nodal problem on the smoothness of the potential function. Inverse Problems, 15 (1), 253-263.
• Levitan, BM. (1984). Inverse Sturm-Liouville Problems. Moskow: Nauka.
• Manafov, MDzh. & Kablan, A. (2015). Inverse spectral and inverse nodal problems for energy dependent Sturm-Liouville equations with δ- interaction. Electronic Journal of Differential Equations, 1-10.
• Manafov, MDzh. (2019). Inverse spectral and inverse nodal problems for Sturm-Liouville equations with point δ and δ′-interactions. Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 45 (2), 286-294.
• Marchenko, VA. (1977) Sturm-Liouville Operators and Their Applications. Kiev: Naukova Dumka.
• McLaughlin, JR. (1988). Inverse spectral theory using nodal points as data-A uniqueness result. Journal of Differential Equations,73 (2), 354-362.
• Poschel, J. & Trubowitz E. (1987). Inverse Spectral Theory, Academic Press, New York.
• Qin, X., Gao, Y. & Yang, C. (2019). Inverse nodal problems for the Sturm−Liouville operator with some nonlocal integral conditions. Journal of Applied Mathematics and Physics, 7 (1), 111-122.
• Sadovnichy, VA., Sultanaev, YT. & Akhtyamov, A.M. (2009). Inverse Sturm-Liouville Problems with Nonseparated Boundary Conditions. MSU, Moscow.
• Shieh, CT. & Yurko, VA. (2008). Inverse nodal and inverse spectral problems for discontinuous boundary value problems. Journal of Mathematical Analysis and Applications, 347, 266-272.
• Wang, YP. & Yurko, VA. (2016). On the inverse nodal problems for discontinuous Sturm−Liouville operators. Journal of Differential Equations, 260 (5), 4086-4109.
• Wang, YP., Shieh, CT. & Wei, X. (2020). Partial inverse nodal problems for differential pencils on a star-shaped graph. Mathematical Methods in the Applied, 43 (15), 8841–8855.
• Xu, XJ. & Yang, CF. (2019). Inverse nodal problem for nonlocal differential operators. Tamkang Journal of Mathematics, 50 (3), 337-347.
• Yang, CF. (2010). Inverse nodal problem for a class of nonlocal Sturm−Liouville operator. Mathematical Modelling and Analysis, 15 (3), 383-392.
• Yang, CF. (2010). Reconstruction of the diffusion operator with nodal data. Zeitschrift für Naturforschung A, 65, 100-106.
• Yang, CF. (2013). Inverse nodal problems of discontinuous Sturm-Liouville operator. Journal of Differential Equations, 254, 1992-2014.
• Yang, CF. (2014). An inverse problem for a differential pencil using nodal points as data. Israel Journal of Mathematics, 204, 431-446.
• Yang, XF. (1997). A solution of the nodal problem. Inverse Problems, 13 (1), 203-213.
• Yurko, VA. (2002). Method of Spectral Mappings in the Inverse Problem Theory. Journal of Inverse and Ill Posed Problems.
• Zhdanovich, VF. (1960). Formulas for the zeros of Dirichlet polynomials and quasipolinomials. Doklady Akademii Nauk SSSR, 135 (8), 1046-1049.