Inverse problem for differential systems having a singularity and turning point of even or odd order

Abstract

In this paper, the canonical property of the solutions and the inverse problem for a system of differential equations having a singularity and turning point of even or odd order are investigated. First, we study the infinite product representation of the solutions of the system in turning case, and derive the corresponding dual equations. Then, by a replacement, we transform the system of differential equations to a second-order differential equation with a singularity and find the canonical product representation of its solution, and provide a procedure for constructing the solution of the inverse problem. We present a new approach to solve the inverse problems having a singularity inside the interval.

Keywords

• Maxwell’s equations
• singularity
• turning point
• dual equations
• factorization theorem

References

1. [1] R.Kh. Amirov, A.S. Ozkan and B. Keskin , Inverse problems for impulsive Sturm- Liouville operator with spectral parameter linearly contained in boundary conditions, Integral Transforms Spec. Funct. 20 (8), 607-618, 2009.
2. [2] K. Aydemir and O.Sh. Mukhtarov, Second-order differential operators with interior singularity, Adv. Difference Equ. 2015 (26), 1-10, 2015.
3. [3] J. Behrndt, P. Schmitz and C. Trunk, Spectral bounds for indefinite singular Sturm- Liouville operators with uniformly locally integrable potentials, J. Differential Equations 267 (1), 468-493, 2019.
4. [4] W. Eberhard, G. Freiling and A. Schneider, Connection formulae for second-order differential equations with a complex parameter and having an arbitrary number of turning points, Math. Nachr. 165, 205-229, 1994.
5. [5] W. Eberhard, G. Freiling and K. Wilcken, Indefinite eigenvalue problems with several singular points and turning points, Math. Nachr. 229 (1), 51-71, 2001.
6. [6] A. Fedoseev, An inverse problem for Sturm-Liouville operators on the half-line having Bessel-type singularity in an interior point, Cent. Eur. J. Math. 11 (12), 2203-2214, 2013.
7. [7] G. Freiling and V. Yurko, On the determination of differential equations with singularities and turning points, Results Math. 41, 275-290, 2002.
8. [8] F. Gesztesy, L.L. Littlejohn and R. Nichols, On self-adjoint boundary conditions for singular Sturm-Liouville operators bounded from below, J. Differential Equations 269 (9), 6448-6491, 2020.

9. [9] S.G. Halvorsen, A function theoretic property of solutions of the equation $ x''+(\lambda w-q)x=0 $, Q. J. Math. 38 (1), 73-76, 1987.
10. [10] A. Jodayree Akbarfam and A.B. Mingarelli, The canonical product of the solution of the Sturm-Liouville equation in one turning point case, Can. Appl. Math. Q. 8 (4), 305-320, 2000.
11. [11] N.D. Kazarinoff, Asymptotic theory of second order differential equations with two simple turning points, Arch. Ration. Mech. Anal. 2, 129-150, 1958.
12. [12] H. Kheiri, A. Jodeyree Akbarfam and A.B. Mingarelli, The uniqueness of the solution of dual equations of an inverse indefinite Sturm-Liouville problem, J. Math. Anal. Appl. 306 (1), 269-281, 2005.
13. [13] H. Koyunbakan and E.S. Panakhov, Half inverse problem for operators with singular potentials, Integral Transforms Spec. Funct. 18 (10), 765-770, 2007.
14. [14] R.E. Langer, On the asymptotic solution of ordinary differential equations with an application to the Bessel functions of large order, Trans. Amer. Math. Soc. 33, 23-64, 1931.
15. [15] B.Ja. Levin, Distribution of Zeros of Entire Functions, Transl. Math. Monogr., Amer. Math. Soc., 1964.
16. [16] O.N. Litvinenko, V.I. Soshnikov, The Theory of Heterogenious Lines and their Applications in Radio Engineering, Moscow: Radio (in Russian), 1946.
17. [17] H.R. Marasi and A. Jodeyree Akbarfam, On the canonical solution of indefinite problem with m turning points of even order, J. Math. Anal. Appl. 332 (2), 1071-1086, 2007.
18. [18] V.P. Meshanov and A.L. Feldstein, Automatic Design of Directional Couplers, Moscow: Sviza (in Russian), 1980.
19. [19] S. Mosazadeh, The stability of the solution of an inverse spectral problem with a singularity, Bull. Iranian Math. Soc. 41 (5), 1061-1070, 2015.
20. [20] S. Mosazadeh, Energy levels of a physical system and eigenvalues of an operator with a singular potential, Rep. Math. Phys. 82 (2), 137-148, 2018.
21. [21] A. Neamaty, The canonical product of the solution of the Sturm-Liouville problems, Iran. J. Sci. Technol. Trans. A Sci. 23 (1), 141-146, 1999.
22. [22] A. Neamaty and S. Mosazadeh, On the canonical solution of Sturm-Liouville problem with singularity and turning point of even order, Canad. Math. Bull. 54 (3), 506-518, 2011.
23. [23] F.W.J. Olver, Second-order linear differential equation with two turning points, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 278, 137-174, 1975.
24. [24] A.S. Ozkan and B. Keskin, Spectral problems for Sturm-Liouville operator with boundary and jump conditions linearly dependent on the eigenparameter, Inverse Probl. Sci. Eng. 20 (6), 799-808, 2012.
25. [25] A.G. Sveshnikov and A.S. Il’inskii, Design problems in electrodynamics, Dokl. Akad. Nauk SSSR. 204 (5), 1077-1080, 1972.
26. [26] X.-J. Xu, N. Bondarenko and C.-F. Yang, Inverse spectral problems for Bessel operators with interior transmission conditions, J. Math. Anal. Appl. 504 (2), 125435, 2021. https://doi.org/10.1016/j.jmaa.2021.125435.
27. [27] X.-J. Xu and C.-F. Yang, Inverse spectral problems for the Sturm-Liouville operator with discontinuity, J. Differential Equations 262 (3), 3093-3106, 2017.
28. [28] M. Zhang, K. Li and H. Song, Regular approximation of singular Sturm-Liouville problems with eigenparameter dependent boundary conditions, Bound. Value Probl. 2020 (1), 2020. https://doi.org/10.1186/s13661-019-01316-0