A functional method applied to operator equations.
A Multiscale Model Reduction Method for Partial Differential Equations
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 non-local problem with integral conditions for hyperbolic equations.
A note on a theorem of Jörgens.
A note on hyperbolic partial differential equations. II.
A simple construction of a parametrix for a regular hyperbolic operator
A survey of partial differential equations with piecewise continuous arguments.
A way of estimating the convergence rate of the Fourier method for PDE of hyperbolic type
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.
An energy analysis is carried out for the usual semidiscrete Galerkin method for the semilinear equation in the region (E) , subject to the initial and boundary conditions, on and . (E) is degenerate at and thus, even in the case , time derivatives of will blow up as . Also, in the case where is locally Lipschitz, solutions of (E) can blow up for in finite time. Stability and convergence of the scheme in is shown in the linear case without assuming (which can blow up as is...
An explicit determination of the Petrov type N space-times on which the conformally invariant scalar wave equation satisfies Huygens' principle
An invariance method for constructing fundamental solutions for
Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients
Boundary controllability of hyperbolic equations.
Boundary value problems for partial differential equations with piecewise constant delay.
Caractérisation des problèmes mixtes hyperboliques bien posés
On considère le problème mixte dans un quadrant pour un opérateur différentiel hyperbolique en supposant que 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 , puis, lorsque est strictement hyperbolique, dans le cadre des espaces de Sobolev. Ces caractérisations s’expriment au moyen d’une condition dite de Lopatinski,...
Consequences of the validity of Huygens' principle for the conformally invariant scalar wave equation, Weyl's neutrino equation and Maxwell's equations on Petrov type II space-times
Dispersive waves and caustics.
Distributional derivatives of functions of two variables of finite variation and their application to an impulsive hyperbolic equation
We give characterizations of the distributional derivatives , , 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.
Estimates for a solution to one differential inequality.
Existence des opérateurs d'ondes pour les systèmes hyperboliques avec un potentiel périodique en temps