Differential equations arise in various branches of science and engineering, such as fluid and solid mechanics, biology, material sciences, economics, ecology, and computer science. Some well-known examples include the Navier-Stokes equations in fluid dynamics, bi-harmonic equations for stress, and the Maxwell equations in electromagnetics. It is often too difficult to obtain analytic solutions (closed-form solutions) to many differential equations; in these cases, numerical methods are used. The most widely used solutions for differential equations are numerical approximation methods. Three well-known numerical methods used to solve these problems are finite difference, finite element, and spectral methods. Plenty of books exist for finite difference and finite element methods, but there are fewer books on spectral methods.
This is a self-contained presentation on the construction, implementation, and analysis of spectral methods for various differential and integral equations, with wide applications in science and engineering. The analysis is based on non-uniformly weighted Sobolev spaces, and yields simplified analysis and more precise estimates, particularly for problems with corner singularities. Efficient spectral algorithms are developed for Volterra integral equations and for higher-order differential equations. Error analysis is also presented.
In the preface, the authors discuss existing books and monographs on spectral methods. Chapter 1 presents a detailed introduction to spectral methods and the formulation of spectral methods in a generalized way using residuals. Several important tools like discrete transform and spectral differentiation are described. Chapter 2 studies Fourier spectral methods for periodic problems, including both computational and theoretical aspects. The authors next present orthogonal polynomials--including Jacobi, Legendre, and Chebyshev--and related approximation results, which are essential for understanding polynomial-based spectral methods.
Chapter 4 presents spectral methods for second-order, two-point boundary value problems and the corresponding error estimates. Spectral methods for Volterra integral equations are presented in chapter 5, specifically spectral algorithms using the Legendre-collocation method and the Jacobi-Galerkin method for Volterra integral equations with regular kernels, and the Jacobi-collocation method for Volterra integral equations with weakly singular kernels. Chapter 6 deals with the spectral methods for higher-order differential equations in steady and unsteady type problems, including the Cahn-Hilliard equation and the Korteweg-de-Vries (KdV) equation.
In chapter 7, the authors delve into unbounded domains, discussing such topics as Laguerre and Hermite polynomials and functions. The next chapter turns to an efficient spectral algorithm for solving second-order elliptic problems in separable geometries, and a basic framework for error analysis of multidimensional spectral methods. Chapter 9 applies the spectral methods to various differential equations that arise in practical applications in multidimensional domains. These problems include steady-state problems involving Helmholtz and Stokes equations, as well as time-dependent problems like the Allen-Cahn equation.
Every chapter ends with a set of problems for practice. MATLAB code for applying spectral methods to various types of problems is available online.
This excellent and very well-written book could be used as a graduate textbook in mathematics and other engineering disciplines. It would also be a good reference book for active practitioners and researchers of spectral methods.
