Introduction -- Fourier Spectral Methods for Periodic Problems -- Orthogonol Polynomials and Related Approximation Results -- Second-Order Two-Point Boundary Value Problems -- Integral Equations -- High-Order Differential Equations -- Problems in Unbounded Domains -- Multi-Dimensional Domains -- Mathematical Preliminaries -- Basic iterative methods -- Basic time discretization schemes -- Instructions for routines in Matlab.  .

Along with finite differences and finite elements, spectral methods are one of the three main methodologies for solving partial differential equations on computers. This book provides a detailed presentation of basic spectral algorithms, as well as a systematical presentation of  basic convergence theory and error analysis for spectral methods. Readers of this book will be exposed to a unified framework for designing and analyzing spectral algorithms for a variety of problems, including in particular high-order differential equations and problems in unbounded domains. The book contains a large number of figures which are designed to illustrate various concepts stressed in the book. A set of basic matlab codes has been made available online to help the readers to develop their own spectral codes for their specific applications.