Asymptotic Methods for Investigating Quasiwave Equations of Hyperbolic Type

Mitropol’skii, Yu. A.; Khoma, G. P.; Gromyak, M.

doi:10.1007/978-94-011-5752-0

Cited by 18 publications

(27 citation statements)

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“…Rigorous estimations of asymptotic convergence for the perturbation technique are given by Mitropolsky et al [26]. The accuracy of the approach of arti"cial small parameter was veri"ed by comparison with numerical data.…”

Section: Discussionmentioning

confidence: 99%

“…In"nite systems such as (13) may be obtained in various ways (e.g., by means of Galerkin method [6,9,24,25], by multiple time scales technique [15}23] or by the averaging procedure [26]. Existence of non-trivial solutions in the case of internal resonance is shown in papers [1}5].…”

Section: Vibrations Of Continuous Structuresmentioning

confidence: 99%

“…Another powerful asymptotic approach is the averaging method, and its strong theoretical bases relating to PDEs were given by Mitropolsky et al [26].…”

Section: Introductionmentioning

confidence: 99%

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“…If the nonlinear functions f and r 1 satisfy the Lipschitz condition, then the system of integral Volterra equations has a unique continuous solution. According to [2], this means the existence of a smooth solution of problem (1) …”

Section: W(x T) = ~-O(t -2nl X Xmentioning

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On the existence of a solution of a quasilinear mixed problem for a quasiwave equation

Kolomiets

Komashinskaya

1997

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We establish conditions for the existence and uniqueness of a solution of a quasilinear mixed problem for a quasiwave equation.Consider the following mixed problem for a quasilinear differential equation with nonlinear friction:(1)where e e (0, e0 ], 0 < e 0 < 1, is a small parameter, and b, h, l, and T are constants.If we know u for x = 0, then we can easily represent a solution u(x, t) by using the d'Alembert method. For x = 0, this representation is the identity u(0, t) = 0(t).The problem is now to obtain 0 such that condition (3) is satisfied for x = 0. By using the d'Alembert method and the continuation method [1], we can find a solution of the problemu(l,t) = O, 0 < t < T, by using the following system of integral equations:I Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. 2 Poltava Pedagogic University, Poltava. x + b(t-'~)x -b (t -"c) w t -

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“…">Existence of a SolutionIn many works (see, e.g., [1,2]), a boundary-value periodic problemis considered and special spaces of functions are indicated in which this problem can be solved. In [2], it is shown that problem (1)-(3) can have a unique solution only in three spaces A l, A 2, and A 3 of functions which correspond to theperiods TI =(2p-1)rc/q, T:=4rcp/(2s-1), and T3 =2(2p-1)Tz/(2s-l), p ~ 7~, q,s~ ~.…”

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confidence: 99%

Existence of the Vejvoda-Shtedry spaces

Khoma¹,

Khoma²,

Botyuk³

1997

Ukr Math J

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We investigate the linear periodic problem u tt - Uxx = F(x, t ), u (x + 2~, t) = u (x, t + T) = u (x, t), (x, t)a IR 2, and establish conditions for the existence of its classical solution in spaces that are subspaces of the Vejvoda-Shtedry spaces.We investigate the solvability of the linear problem 2re-periodic in a variable x and T-periodic in a variable t for a linear hyperbolic equation of the second order. Existence of a SolutionIn many works (see, e.g., [1,2]), a boundary-value periodic problemis considered and special spaces of functions are indicated in which this problem can be solved. In [2], it is shown that problem (1)-(3) can have a unique solution only in three spaces A l, A 2, and A 3 of functions which correspond to theperiods TI =(2p-1)rc/q, T:=4rcp/(2s-1), and T3 =2(2p-1)Tz/(2s-l), p ~ 7~, q,s~ ~. Let us prove that, in subspaces A ~ of the spaces A k, there exists a classical solution of the periodic problemu (x + 2rC, t) = u(x,t), u(x,t+T) = u(x,t),and establish analytic conditions of the appearance of the subspaces A ~ To study the existence of a solution of problem (4)- (6), we consider the operator (5) ix i (RF)(x,t,b) =-~ d~ {F(~,t+~-rl)+F(~,t-~+rl)}dr 1 oTemopol' Pedagogical Institute, Ternopol'.

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Asymptotic Methods for Investigating Quasiwave Equations of Hyperbolic Type

Cited by 18 publications

References 0 publications

Asymptotic Approach for Non-Linear Periodical Vibrations of Continuous Structures

Asymptotic Approach for Non-Linear Periodical Vibrations of Continuous Structures

On the existence of a solution of a quasilinear mixed problem for a quasiwave equation

Existence of the Vejvoda-Shtedry spaces

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