The convergence of a Galerkin approximation scheme for an extensible beam

Geveci, Tunc; Christie, I.

doi:10.1051/m2an/1989230405971

Cited by 8 publications

(4 citation statements)

References 10 publications

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“…Let us apply relations (27) together with (31) and (33) in (29). In addition, we take into consideration definitions (15) and (25). Computations yield…”

Section: Formula For N 1 and Nmentioning

confidence: 99%

“…The projection and difference methods, various iteration processes, Greenâs function method and other methods and numerical experiments were performed. One can get acquainted with these and other numerical approaches to the solution of the corresponding problems in the works of S. Bilbao [6], M. Chipot [7], S. M. Choo et al [8], T. Geveci, I. Christie [15], T. Gudi [16], N. Kachakhidze et al [22], T. F. Ma [27], J. Peradze [33][34][35][36], J. Rogava, M. Tsiklauri [40], J. Sladek, V. Sladek [41]. The interest in Kirchhoff class equations is of a permanent nature.…”

Section: Introductionmentioning

confidence: 99%

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A Kirchhoff type equation in a nonlinear model of shell vibration

Peradze

2016

Z Angew Math Mech

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The paper deals with the study of the initial boundary value problem for the Donnell-Mushtari-Vlasov nonlinear system of differential equations, which describes large deflections of a rectangular shallow shell. The sought functions u i = u i (x 1 , x 2 , t), i = 1, 2, and w = w(x 1 , x 2 , t) are respectively longitudinal and transverse displacements of points of the shell midsurface. It is assumed that Dirichlet and Neumann type homogeneous conditions are fulfilled for functions u 1 and u 2 . After expressing u 1 and u 2 through w, i.e. after constructing the relations u i = ψ i (w), i = 1, 2, with a help of Fourier series method, we come to a nonlinear integro-differential hyperbolic equation for the transverse displacement function w, which by its structure is analogous to Kirchhoff string, Woinowsky-Krieger beam and Berger plate equations. These equations are united under the common name -Kirchhoff type equations. The advantage of the described technique consists in splitting the initial boundary value problem at the end into two subproblems. From the integro-differential equation we first find w and using the explicit relations u i = ψ i (w), i = 1, 2, we construct u 1 and u 2 . The issue of the use of this approach in the general case not restricted by the rectangular domain and Dirichlet and Neumann conditions on the boundary is discussed. It is shown that in order to obtain formulas u i = ψ i (w), i = 1, 2, we have to solve a linear plane problem of elasticity with respect to the functions u 1 and u 2 .

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“…Let us apply relations (27) together with (31) and (33) in (29). In addition, we take into consideration definitions (15) and (25). Computations yield…”

Section: Formula For N 1 and Nmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Kirchhoff type equation in a nonlinear model of shell vibration

Peradze

2016

Z Angew Math Mech

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“…Geveci and Christie [69] examined the stability and convergence of a semidiscrete PEG method and two fully discrete Crank-Nicols on-type methods for the approximate solution of the undamped extensible beam equation [171] Utt…”

Section: One Dimensional Problemsmentioning

confidence: 99%

Numerical Methods for Schrödinger-Type Problems

Fairweather¹,

Khebchareon²

2002

Applied Optimization

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Several physical phenomena are modeled by initial-boundary value problems which can be formulated as SchrOdinger -type systems of partial differential equations. In this paper, two classes of problems of this kind, Schrodinger equations, which arise in various areas of physics, and certain vibration problems from civil and mechanical engineering, are considered. A survey of numerical methods for solving linear and nonlinear problems in one and several space variables is presented, with special attention being devoted to the parabolic wave equation, the cubic Schrodinger equation, and to fourth order parabolic equations arising in vibrating beam and plate problems. Recently developed finite element methods for solving SchrOdinger-type systems are also outlined.Keywords: Schrodinger systems, linear and nonlinear Schrodinger equations, parabolic wave equation, vibrating beams and plates, fourth order parabolic equations, finite difference methods, finite element Galerkin methods, orthogonal spline collocation methods, spectral methods IntroductionWe consider the numerical solution of problems that can be written as initialboundary value problems (IBVPs) for real systems of partial differential equations of the form

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“…A Galerkin approximation for a nonlinear extensible beam was studied by Geveci and Christie [5]. Choo and Chung [6] proposed a finite difference approximation to the solution of a strongly damped extensible beam equation and, in [7], error estimates were obtained for a finite element approximation.…”

Section: Introductionmentioning

confidence: 99%