By Jean-Claude Nedelec

ISBN-10: 1441928898

ISBN-13: 9781441928894

ISBN-10: 1475743939

ISBN-13: 9781475743937

This booklet is dedicated to the learn of the acoustic wave equation and of the Maxwell approach, the 2 most typical wave equations encountered in physics or in engineering. the most aim is to offer an in depth research in their mathematical and actual houses. Wave equations are time established. although, use of the Fourier trans shape reduces their research to that of harmonic platforms: the harmonic Helmholtz equation, when it comes to the acoustic equation, or the har monic Maxwell method. This ebook concentrates at the learn of those harmonic difficulties, that are a primary step towards the examine of extra common time-dependent difficulties. In every one case, we supply a mathematical environment that enables us to end up lifestyles and area of expertise theorems. we have now systematically selected using variational formulations relating to issues of actual power. We research the quintessential representations of the options. those representa tions yield numerous fundamental equations. We study their crucial homes. We introduce variational formulations for those essential equations, that are the root of so much numerical approximations. assorted elements of this e-book have been taught for no less than ten years by way of the writer on the post-graduate point at Ecole Poly process and the collage of Paris 6, to scholars in utilized arithmetic. the particular presentation has been demonstrated on them. I desire to thank them for his or her lively and positive participation, which has been super worthwhile, and that i say sorry for forcing them to benefit a few geometry of surfaces.

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M (I (fx-) + m)! 80) 1 -m (1 _ x 2)1 . Their parity is I + m. For m fixed, they are mutually orthogonal, i. , 1: 1 lPL:'(x)lPG(x)dx = 0, if h 1= 12, and also satisfy the following orthogonality relations +1 lPml (x )lP m2 (x) 1 1 dx = 0 if ml 1= m2 and ml 1= -m2. 4 . The Case of the Sphere in lR 3 25 These spherical harmonics constitute an orthogonal basis of the space L2(S), also orthogonal in the space Hl(S). 64). 83) holds. 86) = (l- m)(l + m + l)}[m. It is easily checked that the operator L3 is hermitian and that the two operators L+ and L_ are mutual adjoints and thus Is (L+L_ }[m) Y;"da Is [L_ }[m[2 da, Is (L_L+}[m) Y;" da Is [L+}[m[2 da.

2. 3 Associated Legendre functions We have just seen that the Legendre polynomials are exactly the spherical harmonics invariant by rotation around the axis (0, X3) . It is convenient to use the variables

We then localize using the partition of unity. 61). 68) {3 az We only need to build the lifting in the lower half space, for a known au/ az and a vanishing u. 2). 69) admits VI for normal derivative and satisfies IluIIH'"(R3- ) :::; J(m + l)((m - 1/2)2 + 1/2)/31IvIIIHm-3/2(R2). 73) • We have built a lifting operator for the first two traces. We can now proceed by induction. 74) 54 2. The Helmholtz Equation We only have to consider the case of the half space for which we introduce the operator (here l = 2) u(~, z) = ~zl:;j(Oe(l + 1~12)1/2z, which is easily seen to be continuous in the adequate norms.

### Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems by Jean-Claude Nedelec

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