IJPAM: Volume 113, No. 4 (2017)

Title

TRANSFORMATION OPERATORS FOR
THE FOURTH-ORDER DIFFERENTIAL EQUATIONS
WITH NON-ANALYTIC COEFFICIENTS

Authors

Baltabek E. Kanguzhin$^1$, Gulzhan E. Abduakhitova$^2$
$^1$Institute of Mathematics and Mathematical Modeling
Almaty, KAZAKHSTAN
$^{1,2}$Al-Farabi Kazakh National University
Almaty, KAZAKHSTAN

Abstract

This work is dedicated to constructive method of building cores of transformation operator. In this article a solution of a linear differential equation of the fourth order is investigated. Also, a method of building a solution in the form of integral representation is given.

History

Received: December 15, 2016
Revised: January 30, 2017
Published: March 30, 2017

AMS Classification, Key Words

AMS Subject Classification: 34A12, 34A55, 34B05, 34B24
Key Words and Phrases: transformation operators, holomorphic function, transformation operator's core, integral representation

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Bibliography


1
V.A. Marchenko, Nekotorye voprosy teorii differencifal'nogooperatora vtorogo poryadka, DAN SSSR, 72, No. 3(1952), 457-460 (in Russian).

2
V.A. Marchenko, Nekotorye voprosy teorii odnomernykh lineinykhdifferencial'nyh operatorov vtorogo poryidka, Trudy MMO,1, (1952), 327-420 (in Russian).

3
V.A. Marchenko, Vosstanovlenie potencial'noi energii po fazamrasseyannykh voln, DAN SSSR, 104, No. 5 (1955),695-698 (in Russian).

4
V.A. Marchenko, Spektral'naya teoriya operatorovShurma-Liuvillya, Naukova dumka, Kiev (1972) (in Russian).

5
V.A. Marchenko, Operatory Shturma-Liuvillya i ikhprilojeniya, Naukova dumka, Kiev (1977) (in Russian).

6
J. Delsarte, J. Lions, Transmutations d'operateurs differentielesdans le domaine complexe, Comment. Math. Helv., 32,No. 2 (1957), 113-128.

7
I.G. Hachatryan, Operatory preobrazovaniya dlya kraevykh zadach,Izv. AN Arm.SSR. Seriya matem., 15, No. 1 (1980),30-44 (in Russian).

8
I.G. Hachatryan, Ob operatorakh preobrazovaniya dlyadifferencial'nykh uravnenii vysshikh poryadkov, Izv. AN Arm.SSR. Seriya matem., 13, No. 3 (1978), 215-237 (inRussian).

9
V.I. Matsayev, O sushestvovanii operatora preobrazovaniya dlyadifferencial'nykh uravnenii vysshikh poryadkov, DAN SSSR,130, No. 3 (1960), 499-502 (in Russian).

10
L.A. Sakhnovich, Obratnaya zadacha dlya differencial'nykhoperatorov poryadka $n>2$ s analiticheskimi koefficientami, . Sb., 46(88), No. 1 (1958), 61-76 (in Russian).

11
L.A. Sakhnovich, Neobkhodimyye usloviya nalichiya operatorapreobrazovaniya dlya uravneniya chetvertogo poryadka, Uspekhimatem. nauk, 16, No. 5 (1961), 199-204 (in Russian).

12
M.M. Malamud, Neobkhodimye usloviya sushestvovaniya operatorapreobrazovaniya dlya uravnenii vysshikh poryadkov, Func.Analiz i ego priloj., 16, No. 3 (1982), 74-75 (inRussian).

13
M.K. Fage, Reshenie odnoi zadachi Koshi putem uvelichivaniyachisla nezavisimykh peremennykh, Trudy tret'ego vsesoiuzn.matem. s''ezda, 1 (1956), 71-72 (in Russian).

14
M.K. Fage, Integral'nye predstavleniya operatorno-analiticheskikhfunkcii odnoi nezavisimoi peremennoi, Trudy MMO, 8(1959), 3-48 (in Russian).

15
A.F. Leont'ev, Ocenka rosta resheniya odnogo differencial'nogouravneniya pri bol'shykh po modulyu znacheniyakh parametra i eeprimeneniya k nekotorym voprosam teorii funkcii, Sib. Matem.J., 1, No. 3 (1960), 456-487 (in Russian).

16
A.F. Leont'ev, Obobsheniya ryadov eksponent, Nauka, Moskva(1981) (in Russian).

17
A.P. Khromov, Ob odnom predstavlenii yader rezol'ventvol'terrovskikh operatorov i ego primeneniyakh, Matem. sb.,89(131), No. 2 (1972), 207-226 (in Russian).

18
B.E. Kanguzhin, M.A. Sadybekov, Differencial'nye operatory naotrezke. Raspredelenie sobstvennyh funkcii, Gylym, Shymkent(1996), (in Russian).

19
M.A. Naimark, Lineinye differencial'nye operatory, Nauka,Moskva (1969), (in Russian).

20
B.M. Levitan, Obratnye zadachi Shturma-Liuvillya, Nauka,Moskva (1984), (in Russian).

21
N.I. Shetinin, Teoriya integral'nyh uravnenii, Izd. MGU,Moskva (1968), (in Russian).

22
A.B. Vasil'eva, N.A. Tikhonov, Integral'nye uravneniya, Izd.MGU, Moskva (1989), (in Russian).

23
I.G. Petrovskii, Lekcii po teorii integral'nykh uravnenii,Izd. MGU, Moskva (1984), (in Russian).

24
B.E. Kanguzhin, G.E. Abduakhitova, Transformation operators forthe fourth-order differential equations with non-analyticcoefficients, Kazakh Mathematical Journal, 17, No.1 (2017), 35-36.

How to Cite?

DOI: 10.12732/ijpam.v113i4.8 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 113
Issue: 4
Pages: 71 - 86


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