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Matching of asymptotic expansions for waves propagation in media with thin slots II: The error estimates

Patrick Joly, Sébastien Tordeux (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We are concerned with a 2D time harmonic wave propagation problem in a medium including a thin slot whose thickness ε is small with respect to the wavelength. In a previous article, we derived formally an asymptotic expansion of the solution with respect to ε using the method of matched asymptotic expansions. We also proved the existence and uniqueness of the terms of the asymptotics. In this paper, we complete the mathematical justification of our work by deriving optimal error estimates between...

Mesures limites pour l’équation de Helmholtz dans le cas non captif

Jean-François Bony (2009)

Annales de la faculté des sciences de Toulouse Mathématiques

Cet article est consacré à l’étude des mesures limites associées à la solution de l’équation de Helmholtz avec un terme source se concentrant en un point. Le potentiel est supposé C et l’opérateur non-captif. La solution de l’équation de Schrödinger semi-classique s’écrit alors micro-localement comme somme finie de distributions lagrangiennes. Sous une hypothèse géométrique, qui généralise l’hypothèse du viriel, on en déduit que la mesure limite existe et qu’elle vérifie des propriétés standard....

Modeling of vibration for functionally graded beams

Gülsemay Yiğit, Ali Şahin, Mustafa Bayram (2016)

Open Mathematics

In this study, a vibration problem of Euler-Bernoulli beam manufactured with Functionally Graded Material (FGM), which is modelled by fourth-order partial differential equations with variable coefficients, is examined by using the Adomian Decomposition Method (ADM).The method is one of the useful and powerful methods which can be easily applied to linear and nonlinear initial and boundary value problems. As to functionally graded materials, they are composites mixed by two or more materials at a...

Monge solutions for discontinuous hamiltonians

Ariela Briani, Andrea Davini (2005)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an Hamilton-Jacobi equation of the form H ( x , D u ) = 0 x Ω N , ( 1 ) where H ( x , p ) is assumed Borel measurable and quasi-convex in p . The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation (1) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also discussed.

Monge solutions for discontinuous Hamiltonians

Ariela Briani, Andrea Davini (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an Hamilton-Jacobi equation of the form

 H ( x , D u ) = 0 x Ω N , ( 1 ) 
 where H(x,p) is assumed Borel measurable and quasi-convex in p. The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation ([see full text]) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also...

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