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Two-mode bifurcation in solution of a perturbed nonlinear fourth order differential equation

Ahmed Abbas Mizeal, Mudhir A. Abdul Hussain (2012)

Archivum Mathematicum

In this paper, we are interested in the study of bifurcation solutions of nonlinear wave equation of elastic beams located on elastic foundations with small perturbation by using local method of Lyapunov-Schmidt.We showed that the bifurcation equation corresponding to the elastic beams equation is given by the nonlinear system of two equations. Also, we found the parameters equation of the Discriminant set of the specified problem as well as the bifurcation diagram.

Two-sided bounds of eigenvalues of second- and fourth-order elliptic operators

Andrey Andreev, Milena Racheva (2014)

Applications of Mathematics

This article presents an idea in the finite element methods (FEMs) for obtaining two-sided bounds of exact eigenvalues. This approach is based on the combination of nonconforming methods giving lower bounds of the eigenvalues and a postprocessing technique using conforming finite elements. Our results hold for the second and fourth-order problems defined on two-dimensional domains. First, we list analytic and experimental results concerning triangular and rectangular nonconforming elements which...

Un théorème d'existence en théorie non linéaire des coques minces

Philippe G. Ciarlet, Daniel Coutand (1999)

Journées équations aux dérivées partielles

Les équations bidimensionnelles d'une coque non linéairement élastique «en flexion» ont été récemment justifiées par V. Lods et B. Miara par la méthode des développements asymptotiques formels appliquée aux équations de l'élasticité non linéaire tridimensionnelle. Ces équations se mettent sous la forme d'un problème de point critique d'une fonctionnelle dont l'intégrande est une expression quadratique en termes de la différence exacte entre les tenseurs de courbure des surfaces déformée et non déformée,...

Uncertain input data problems and the worst scenario method

Ivan Hlaváček (2007)

Applications of Mathematics

An introduction to the worst scenario method is given. We start with an example and a general abstract scheme. An analysis of the method both on the continuous and approximate levels is discussed. We show a possible incorporation of the method into the fuzzy set theory. Finally, we present a survey of applications published during the last decade.

Uniform controllability for the beam equation with vanishing structural damping

Ioan Florin Bugariu (2014)

Czechoslovak Mathematical Journal

This paper is devoted to studying the effects of a vanishing structural damping on the controllability properties of the one dimensional linear beam equation. The vanishing term depends on a small parameter ε ∈ ( 0 , 1 ) . We study the boundary controllability properties of this perturbed equation and the behavior of its boundary controls v ε as ε goes to zero. It is shown that for any time T sufficiently large but independent of ε and for each initial data in a suitable space there exists a uniformly bounded...

Uniform stabilization of some damped second order evolution equations with vanishing short memory

Louis Tebou (2014)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a damped abstract second order evolution equation with an additional vanishing damping of Kelvin–Voigt type. Unlike the earlier work by Zuazua and Ervedoza, we do not assume the operator defining the main damping to be bounded. First, using a constructive frequency domain method coupled with a decomposition of frequencies and the introduction of a new variable, we show that if the limit system is exponentially stable, then this evolutionary system is uniformly − with respect to the calibration...

Unilateral dynamic contact of von Kármán plates with singular memory

Igor Bock, Jiří Jarušek (2007)

Applications of Mathematics

The solvability of the contact problem is proved provided the plate is simply supported. The singular memory material is assumed. This makes it possible to get a priori estimates important for the strong convergence of gradients of velocities of solutions to the penalized problem.

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