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A Multiscale Model Reduction Method for Partial Differential Equations

Maolin Ci, Thomas Y. Hou, Zuoqiang Shi (2014)

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

We propose a multiscale model reduction method for partial differential equations. The main purpose of this method is to derive an effective equation for multiscale problems without scale separation. An essential ingredient of our method is to decompose the harmonic coordinates into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Such a decomposition plays a key role in our construction of the effective equation. We show...

A way of estimating the convergence rate of the Fourier method for PDE of hyperbolic type

Evgenii Pustylnik (2001)

Czechoslovak Mathematical Journal

The Fourier expansion in eigenfunctions of a positive operator is studied with the help of abstract functions of this operator. The rate of convergence is estimated in terms of its eigenvalues, especially for uniform and absolute convergence. Some particular results are obtained for elliptic operators and hyperbolic equations.

An energy analysis of degenerate hyperbolic partial differential equations.

William J. Layton (1984)

Aplikace matematiky

An energy analysis is carried out for the usual semidiscrete Galerkin method for the semilinear equation in the region Ω (E) ( t u t ) t = i , j = 1 ( a i j ( x ) u x i ) x j - a 0 ( x ) u + f ( u ) , subject to the initial and boundary conditions, u = 0 on Ω and u ( x , 0 ) = u 0 . (E) is degenerate at t = 0 and thus, even in the case f 0 , time derivatives of u will blow up as t 0 . Also, in the case where f is locally Lipschitz, solutions of (E) can blow up for t > 0 in finite time. Stability and convergence of the scheme in W 2 , 1 is shown in the linear case without assuming u t t (which can blow up as t 0 is...

Caractérisation des problèmes mixtes hyperboliques bien posés

Jacques Chazarain, Alain Piriou (1972)

Annales de l'institut Fourier

On considère le problème mixte dans un quadrant pour un opérateur différentiel hyperbolique P en supposant que P et les opérateurs au bord sont homogènes à coefficients constants. On caractérise les conditions au bord pour avoir existence et unicité de la solution du problème mixte, en se plaçant successivement dans le cadre des fonctions C , puis, lorsque P est strictement hyperbolique, dans le cadre des espaces de Sobolev. Ces caractérisations s’expriment au moyen d’une condition dite de Lopatinski,...

Distributional derivatives of functions of two variables of finite variation and their application to an impulsive hyperbolic equation

Dariusz Idczak (1998)

Czechoslovak Mathematical Journal

We give characterizations of the distributional derivatives D 1 , 1 , D 1 , 0 , D 0 , 1 of functions of two variables of locally finite variation. Then we use these results to prove the existence theorem for the hyperbolic equation with a nonhomogeneous term containing the distributional derivative determined by an additive function of an interval of finite variation. An application of the above theorem to a hyperbolic equation with an impulse effect is also given.

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