The term spectral methods in the title refers to the use of Fourier transforms, specifically Fourier series, for the solution of linear partial differential equations. The Fourier series are applied to the space coordinates--which are restricted to one dimension--reducing the problem to the solution of a system of ordinary differential equations. The book consists of two parts. Part A deals with the use of Fourier series for the solution of partial differential equations with periodic coefficients. Part B discusses the use of orthogonal polynomials, specifically Chebyshev polynomials, for approximating solutions in a finite interval.

The treatment is purely theoretical without reference to physical applications or numerical examples. The reader is assumed to be familiar with the basic facts of Hilbert and Sobolev spaces. Part A starts with a brief introduction to Fourier series, with emphasis on convergence and error terms. The results are then applied to the advection operator L u ≡ a ∂ u &slash; ∂ x + ∂ ( a u ) &slash; ∂ x, with a periodic coefficient a ( x ), for the solution of ∂ u &slash; ∂ t + L u = 0. After briefly discussing the Galerkin method, that is, Fourier’s original method based on integration, Mercier introduces an interpolation operator P _C u which replaces the function u ( x ) with the finite Fourier series obtained from the values of u ( x ) at 2 N + 1 equidistant points x _i in the interval - &pgr; ≤ x &pgr; with ( P _C u ) ( x _i ) = u ( x _i ). This approach leads to a collocation method for solving the differential equation, which is more efficient because the fast Fourier transformation (FFT) can be used and is also more accurate. Either method leads to a system of ordinary differential equations for the Fourier coefficients as functions of time, which can be solved with standard methods. Next, the author adds a diffusion operator A u ≡ - ∂ &slash; ∂ x ( b ( x ) ∂ u &slash; ∂ x ) + e ( x ) u ( x ) and discusses the solution of ∂ u &slash; ∂ t + ( &egr; A + L ) u = 0. The method can also be applied to the one-dimensional elliptic equation A u = f ( x ).

Part B starts with a brief discussion of systems of orthogonal polynomials and the related Gauss, Gauss-Radau, and Gauss-Lobatto integration formulas. The rest of this part is devoted to approximations with Chebyshev polynomials; since Chebyshev polynomials and trigonometric functions are closely related, the results from Part A, including the use of FFT, also apply to these approximations. The author includes a list of references but no index.

This book is based on a course on spectral methods at the Université Pierre et Marie Curie, Paris, and was originally published in French in 1981 as a report of the French Atomic Energy Commission. The contents are typical French university lectures: elegant, precise, and requiring undivided attention from the student. No exercises are included. The material is well suited for a graduate seminar and will appeal to everyone who can appreciate beautiful mathematics elegantly presented. Anyone interested primarily in numerical results, however, will be disappointed by the lack of examples and the limited scope, although many of the ideas and tools discussed can be used for more general classes of problems.