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Formulation variationnelle espace-temps pour le calcul par potentiel retardé de la diffraction d’une onde acoustique. I. (French) Zbl 0618.35069
The paper discusses the properties of the so-called ”first kind” integral equation associated to the simple layer retarded potential: \[ sp(t,x):=(1/4\pi)\int_{\Gamma}(p(t-(x-y),y)/(x-y))d\sigma_ y=g(t,x) \] where \(\Gamma\) is a closed surface in \({\mathbb{R}}^ 3\), \((t,x)\in {\mathbb{R}}_+\times \Gamma.\)
By a Fourier-Laplace transform technique, the authors prove an existence and uniqueness theorem for this equation in a convenient space-time functional framework, and show a variational formulation to the equation with a coercive bilinear form. This enables them to propose new numerical schemes for the equation, based on the Galerkin method. They show that, with well chosen basis functions, the schemes are explicit and of the marching-in-time type, conserving the convolution character of the operator. It seems that those are the first schemes on boundary transient integral equations which are mathematically analysed.
In a companion paper [ibid. 8, 598-608 (1986)] the integral equation associated to the double layer retarded potential for the resolution of a Neumann scattering problem is also dealt with in the same techniques. The results of these papers are since improved into a finite-time framework by the second author (to appear).

MSC:
35L20 Initial-boundary value problems for second-order hyperbolic equations
76Q05 Hydro- and aero-acoustics
35A15 Variational methods applied to PDEs
45A05 Linear integral equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
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