Spectral Analysis of Discontinuous Boundary-Value Problems with Retarded Argument

Автор(и)

  • Erdoğan Şen

DOI:

https://doi.org/10.15407/mag14.01.078

Ключові слова:

диференцiальне рiвняння iз запiзненням аргументу, власний параметр, умови передачi, асимптотика власних значень, межi власних значень.

Анотація

У данiй статтi ми маємо справу iз спектральними властивостями розривних задач типу Штурма-Лiувiлля iз запiзненням аргументу. Ми розширюємо i узагальнюємо деякi пiдходи i результати класичних регулярних i розривних задач Штурма-Лiувiлля. Спочатку ми вивчаємо спектральнi властивостi задачi Штурма-Лiувiлля на пiвосi й отримуємо нижнi оцiнки для власних значень задачi. Потiм ми вивчаємо спектральнi властивостi задачi Штурма-Лiувiлля з розривною ваговою функцiєю, яка мiстить спектральний параметр в крайових умовах. Ми також отримуємо асимптотичнi формули для власних значень i власних функцiй задачi та межi вiдстанi мiж власними значеннями.

2010:34L15, 34L20, 35R10

Посилання

Z. Akdoğan, M. Demirci, and O.Sh. Mukhtarov, Green function of discontinuous boundary-value problem with transmission conditions, Math. Methods Appl. Sci. 30 (2007), 1719–1738. https://doi.org/10.1002/mma.867

J. Ao, J. Sun, M. Zhang, The finite spectrum of Sturm–Liouville problems with transmission conditions, Appl. Math. Comput. 218 (2011), 1166–1173. https://doi.org/10.1016/j.amc.2011.05.033

A. Bayramov, S. Öztürk Uslu, and S. Kizilbudak C . alis.kan, Computation of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument, Appl. Math. Comp. 191 (2007), 592–600. https://doi.org/10.1016/j.amc.2007.02.118

A. Bayramov and E. Şen, On a Sturm–Liouville type problem with retarded argument, Math. Methods Appl. Sci. 36 (2013), 39–48. https://doi.org/10.1002/mma.2567

R. Bellman and K.L. Cook, Differential-Difference Equations, Academic Press, New York–London, 1963.

P.A. Binding, P.J. Browne, and K. Seddighi, Sturm–Liouville problems with eigenparameter dependent boundary conditions, Proc. Edinb. Math. Soc. (2) 37 (1994), 57–72. https://doi.org/10.1017/S0013091500018691

A. Campbell and S.A. Nazarov, Asymptotics of eigenvalues of a plate with small clamped zone, Positivity 5 (2001), 275–295. https://doi.org/10.1023/A:1011469822255

C.T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), 293–308. https://doi.org/10.1017/S030821050002521X

M. Jabloński and K. Twardowska, On boundary value problems for differential equations with a retarded argument, Univ. Iagel. Acta Math. 26 (1987), 29–36.

M. Kadakal and O. Sh. Mukhtarov, Sturm–Liouville problems with discontinuities at two points, Comput. Math. Appl. 54 (2007), 1367–1379. https://doi.org/10.1016/j.camwa.2006.05.032

B.M. Levitan, Expansion in Characteristic Functions of Differential Equations of the Second Order, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian).

Kh.R. Mamedov, On boundary value problem with parameter in boundary conditions, Spectral Theory of Operator and Its Applications, XI, Izdat. “Èlm”, Baku, 1997, 117–121 (Russian).

Kh.R. Mamedov and N. Palamut, On a direct problem of scattering theory for a class of Sturm–Liouville operator with discontinuous coefficient, Proc. Jangjeon Math. Soc. 12 (2009), 243–251.

V.A. Marchenko, Sturm–Liouville operators and applications (Revised ed.), AMS Chelsea Publishing, Providence, RI, 2011. https://doi.org/10.1090/chel/373

A.D. Miškis, Linear Differential Equations with Retarded Argument, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow–Leningrad, 1951 (Russian).

S.B. Norkin, On boundary problem of Sturm–Liouville type for second-order differential equation with retarded argument, Izv. Vysš. Učebn. Zaved. Matematika, 6 (1958), 203–214 (Russian).

S.B. Norkin, Differential equations of the second order with retarded argument. Some problems of the theory of vibrations of systems with retardation, Translations of Mathematical Monographs, 31, Amer. Math. Soc., Providence, RI, 1972.

Z.I. Rehlickiı̆, Test for boundedness of the solutions of linear differential equations with several lags of the argument, Dokl. Akad. Nauk SSSR 125 (1959), 46–47 (Russian).

E. Şen and A. Bayramov, Calculation of eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument which contains a spectral parameter in the boundary condition, Math. Comput. Modelling 54 (2011), 3090– 3097. https://doi.org/10.1016/j.mcm.2011.07.039

E. Şen and A. Bayramov, Asymptotic formulations of the eigenvalues and eigenfunctions for a boundary value problem, Math. Methods Appl. Sci. 36 (2013), 1512–1519. https://doi.org/10.1002/mma.2699

E. Şen, J.J. Seo, and S. Araci, Asymptotic behaviour of eigenvalues and eigenfunctions of a Sturm–Liouville problem with retarded argument, J. Appl. Math. 2013, Art. ID 306917, 8 pp.

E.C. Titchmarsh, Eigenfunctions Expansion Associated with Second-Order Differential Equations. Part I, Clarendon Press, Oxford, 1962.

E. Tunç and O.Sh. Mukhtarov, Fundamental solutions and eigenvalues of one boundary-value problem with transmission conditions, Appl. Math. Comput. 157 (2004), 347–355. https://doi.org/10.1016/j.amc.2003.08.039

J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z. 133 (1973), 301–312. https://doi.org/10.1007/BF01177870

Downloads

Опубліковано

2018-06-20

Як цитувати

(1)
Şen, E. Spectral Analysis of Discontinuous Boundary-Value Problems With Retarded Argument. J. Math. Phys. Anal. Geom. 2018, 14, 78-99.

Номер

Розділ

Статті

Завантаження

Дані завантаження ще не доступні.