a 'sleeper', receiving only modest attention for 50 years before emerging as a cornerstone of communications engineering in more recent times. Roughly one in six of Walsh's 281 publications are included, photographically reproduced. Reproduction is excellent except for one paper from 1918. The book also reproduces three brief papers about Walsh and his work, by W. E. Sewell, D. V. Widder and Morris Marden. The first two were written for a special issue of the SIAM Journal celebrating Walsh's 70th birthday; the third is an obituary.

Matrix iterative analysis (2nd edn), by Richard S. Varga. Springer Series in Computational Mathematics 27. Pp. 358. £55. 2000. ISBN 3 540 66321 5 (Springer Verlag).

Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials

It is well known that spectral methods (tau, Galerkin, collocation) have a condition number of $O(N^{4})$
 (
 $N$
 is the number of retained modes of polynomial approximations). This paper presents some efficient spectral algorithms, which have a condition number of $O(N^{2})$
 , based on the Jacobi–Galerkin methods of second-order elliptic equations in one and two space variables. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. The complexities of the algorithms are a small multiple of $N^{d+1}$
 operations for a $d$
 -dimensional domain with $(N-1)^d$
 unknowns, while the convergence rates of the algorithms are exponentials with smooth solutions.

Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials

In this paper, we develop a spectral method based on generalized Hermite functions with weight $\chi(x)\equiv1$. We also establish some basic results on generalized Hermite orthogonal approximations, which play an important role in spectral methods. As examples, the generalized Ginzburg-Landau equation in a population problem and an elliptic equation with a harmonic potential are considered. Related spectral schemes are proposed, and their convergence is proved. Numerical results demonstrate the spectral accuracy of this approach.

Generalized Hermite Spectral Method and its Applications to Problems in Unbounded Domains

This paper is concerned with fast spectral-Galerkin Jacobi algorithms for solving one- and two-dimensional elliptic equations with homogeneous and nonhomogeneous Neumann boundary conditions. The paper extends the algorithms proposed by Shen (SIAM J Sci Comput 15:1489–1505, 1994) and Auteri et al. (J Comput Phys 185:427–444, 2003), based on Legendre polynomials, to Jacobi polynomials $P_{n}^{(\alpha,\beta)}(x)$
 with arbitrary α and β. The key to the efficiency of our algorithms is to construct appropriate basis functions with zero slope at the endpoints, which lead to systems with sparse matrices for the discrete variational formulations. The direct solution algorithm developed for the homogeneous Neumann problem in two-dimensions relies upon a tensor product process. Nonhomogeneous Neumann data are accounted for by means of a lifting. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented.

Jacobi spectral Galerkin method for elliptic Neumann problems

Nonisotropic heat transfer in an infinite plate is considered. A weak formulation is derived, which is appropriate for its numerical simulation. The mixed Legendre-Hermite spectral method is proposed for this problem. Its convergence is proved. Numerical results show the efficiency of this new approach.

Mixed legendre-hermite spectral method for heat transfer in an infinite plate

In this paper, we study numerical methods for a class of two-dimensional semilinear elliptic boundary value problems with variable coefficients in a union of rectangular domains. A compact finite difference method with an anisotropic mesh is proposed for the problems. The existence of a maximal and a minimal compact difference solution is proved by the method of upper and lower solutions, and two sufficient conditions for the uniqueness of the solution are also given. The optimal error estimate in the discrete L ∞ norm is obtained under certain conditions. The error estimate shows the fourth-order accuracy of the proposed method when two spatial mesh sizes are proportional. By using an upper solution or a lower solution as the initial iteration, an "almost optimal" Picard type of monotone iterative algorithm is developed for solving the resulting nonlinear discrete system efficiently. Numerical results are presented to confirm our theoretical analysis.

Fourth-order compact finite difference methods and monotone iterative algorithms for semilinear elliptic boundary value problems

In this paper, we change differential equations on the half line to certain singular problems on finite interval, and then use a specific Jacobi approximation for solving the resulting problems numerically. Theoretical analysis and numerical results demonstrate the efficiency of this method.

Ben-Yu Guo

Papers

Mixed legendre-hermite spectral method for heat transfer in an infinite plate

Fourth-order compact finite difference methods and monotone iterative algorithms for semilinear elliptic boundary value problems

Jacobi spectral method for differential equations with Rough asymptotic behaviors at infinity