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A nonlinear plate control without linearization

Kenan Yildirim, Ismail Kucuk (2017)

Open Mathematics

In this paper, an optimal vibration control problem for a nonlinear plate is considered. In order to obtain the optimal control function, wellposedness and controllability of the nonlinear system is investigated. The performance index functional of the system, to be minimized by minimum level of control, is chosen as the sum of the quadratic 10 functional of the displacement. The velocity of the plate and quadratic functional of the control function is added to the performance index functional as...

A nonlocal coagulation-fragmentation model

Mirosław Lachowicz, Dariusz Wrzosek (2000)

Applicationes Mathematicae

A new nonlocal discrete model of cluster coagulation and fragmentation is proposed. In the model the spatial structure of the processes is taken into account: the clusters may coalesce at a distance between their centers and may diffuse in the physical space Ω. The model is expressed in terms of an infinite system of integro-differential bilinear equations. We prove that some results known in the spatially homogeneous case can be extended to the nonlocal model. In contrast to the corresponding local...

A nonlocal elliptic equation in a bounded domain

Piotr Fijałkowski, Bogdan Przeradzki, Robert Stańczy (2004)

Banach Center Publications

The existence of a positive solution to the Dirichlet boundary value problem for the second order elliptic equation in divergence form - ∑ i , j = 1 n D i ( a i j D j u ) = f ( u , ∫ Ω g ( u p ) ) , in a bounded domain Ω in ℝⁿ with some growth assumptions on the nonlinear terms f and g is proved. The method based on the Krasnosel’skiĭ Fixed Point Theorem enables us to find many solutions as well.

A note of uniqueness on the Cauchy problem for Schrödinger or heat equations with degenerate elliptic principal parts

Hideki Takuwa (2004)

Bollettino dell'Unione Matematica Italiana

We study the local uniqueness in the Cauchy problem for Schrödinger or heat equations whose principal parts are nonnegative. We show the compact uniqueness under a weak form of pseudo convexity. This makes up for the known results under the conormal pseudo convexity given by Tataru, Hörmander, Robbiano- Zuily and L. T'Joen. Our method is based on a kind of integral transform and a weak form of Carleman estimate for degenerate elliptic operators.

A note on ( 2 𝖪 + 1 ) -point conservative monotone schemes

Huazhong Tang, Gerald Warnecke (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

First–order accurate monotone conservative schemes have good convergence and stability properties, and thus play a very important role in designing modern high resolution shock-capturing schemes. Do the monotone difference approximations always give a good numerical solution in sense of monotonicity preservation or suppression of oscillations? This note will investigate this problem from a numerical point of view and show that a ( 2 K + 1 ) -point monotone scheme may give an oscillatory solution even though...

A note on (2K+1)-point conservative monotone schemes

Huazhong Tang, Gerald Warnecke (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

First–order accurate monotone conservative schemes have good convergence and stability properties, and thus play a very important role in designing modern high resolution shock-capturing schemes. Do the monotone difference approximations always give a good numerical solution in sense of monotonicity preservation or suppression of oscillations? This note will investigate this problem from a numerical point of view and show that a (2K+1)-point monotone scheme may give an oscillatory solution even...

A note on a critical problem with natural growth in the gradient

Boumediene Abdellaoui, Ireneo Peral (2006)

Journal of the European Mathematical Society

The paper analyzes the influence on the meaning of natural growth in the gradient of a perturbation by a Hardy potential in some elliptic equations. Indeed, in the case of the Laplacian the natural problem becomes − Δ u − Λ N u | x | 2 = ∇ u + N − 2 2 u | x | 2 x 2 | x | ( N − 2 ) / 2 + λ f ( x ) in Ω , u = 0 on ∂ Ω , Λ N = ( ( N − 2 ) / 2 ) 2 . This problem is a particular case of problem (2). Notice that ( N − 2 ) / 2 is optimal as coefficient and exponent on the right hand side.

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