By Jean-Claude Nedelec
This booklet is dedicated to the examine of the acoustic wave equation and of the Maxwell method, the 2 most typical wave equations encountered in physics or in engineering. the most target is to offer an in depth research in their mathematical and actual houses. Wave equations are time established. notwithstanding, use of the Fourier trans shape reduces their learn to that of harmonic platforms: the harmonic Helmholtz equation, with regards to the acoustic equation, or the har monic Maxwell procedure. This publication concentrates at the learn of those harmonic difficulties, that are a primary step towards the examine of extra normal time-dependent difficulties. In each one case, we supply a mathematical environment that enables us to turn out life and specialty theorems. we now have systematically selected using variational formulations on the topic of issues of actual strength. We learn the crucial representations of the ideas. those representa tions yield a number of fundamental equations. We research their crucial homes. We introduce variational formulations for those imperative equations, that are the foundation of such a lot numerical approximations. varied elements of this publication have been taught for no less than ten years through the writer on the post-graduate point at Ecole Poly procedure and the collage of Paris 6, to scholars in utilized arithmetic. the particular presentation has been proven on them. I desire to thank them for his or her energetic and positive participation, which has been tremendous necessary, and that i make an apology for forcing them to benefit a few geometry of surfaces.
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Extra info for Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems
132) m 2y;m l-I' 32 2. [)Hr - 1 [)X2 = Cm ( 1 Xl . 4. oHl' OX2 - - -1-- X = - CIm ( Xl [ 21(1 33 . 139) . 1+1/2/ m-l =-z 1 _ 1/2 v (l+m)(I+m-1)H1- I . We then obtain the expressions of oHl' /OXI and oHl' /OX2 by elimination. 8 For m i4 > 0, the following recursion relations hold: [J(l- m)(l- m [2 _ 1. 140) J(I+1)2-t, [;q + 1)(1 - m [J(l- m)(l- m [2 _ 1. 112) by (Xl + iX2) to get 34 2. 144) ( d )1+m+2 d~ (l-e)l+l. 145) (l- m)(l - m - 1) Hm+l( )r2 (l+~)(l-~) 1- 1 X (l + + m + 1)(1 + Tn + 2) Hm+l( )] (l+~)(l+~) 1+1 X .
X 2p - l l-1 (2p - l - m)! 122). 123) is then obtained by adding the first relation times x to the second one. 122) for the value m-1. 132) m 2y;m l-I' 32 2. [)Hr - 1 [)X2 = Cm ( 1 Xl . 4. oHl' OX2 - - -1-- X = - CIm ( Xl [ 21(1 33 . 139) . 1+1/2/ m-l =-z 1 _ 1/2 v (l+m)(I+m-1)H1- I . We then obtain the expressions of oHl' /OXI and oHl' /OX2 by elimination. 8 For m i4 > 0, the following recursion relations hold: [J(l- m)(l- m [2 _ 1. 140) J(I+1)2-t, [;q + 1)(1 - m [J(l- m)(l- m [2 _ 1. 112) by (Xl + iX2) to get 34 2.
P. Using the partition of unity, we only have to prove the corresponding results for the functions Ui. We will establish several preliminary lemmas. 1 Let U(Xl,X2,Z) be a function defined on the complex half space z < o. Let u(~, z) be its partial Fourier transform in the variables (Xl,X2). Then u is in the space Hm(1R 3 -) if and only if: (:z) mU E L2 (1R3 -), (1 . , (1 + 1~12)1/2 (:z) m-l U E L2 (1R3 -), + 1~12)m / 2u E L2(1R3-). The proof is straightforward, using the properties of the Fourier transform, and ordering the partial derivatives.