Sinc methods for quadrature and differential equations.

*(English)*Zbl 0753.65081
Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. x, 304 p. (1992).

The sinc function is defined by \(\text{sinc}(z)=(\pi z)^{-1}\sin(\pi z)\), \(z\neq 0\), \(\text{sinc}(0)=1\), and has the property that the set \(\{h^{-{1\over 2}}\text{sinc}(h^{-1}z-k)\}\), \(k\in\mathbb{Z}\) is an orthonormal set for \(L^ 2(\mathbb{R})\) and is complete for the Paley-Wiener class of functions \(B(h)\), the subspace of \(L^ 2(\mathbb{R})\) of functions exponentially bounded of type \(\pi/h\) i.e. \(| f(z)|\leq K\exp(\pi| z|/h)\) for some \(K>0\). For this class of functions expansion in sinc functions is the basis of some elegant numerical methods. These methods can be applied to functions not in \(B(h)\) but obviously there will be some deterioration in the accuracy obtained. The aim of this well written and clearly printed book is to show how to exploit as fully as possible the potential of the method.

Chapter 1 provides a résumé of the necessary material from Fourier analysis and theory of a complex variable. Chapter 2 describes the application of sinc methods to a class of functions analytic on a strip of width \(2\pi d\) parallel to the real axis for which the order of convergence of the sinc expansion is \(\exp(-2\pi d/h)\). Chapter 3 applies the method to an arc in the complex plane by mapping it to the whole real axis in such a way that the exponential order of convergence is preserved, although multiplied by a larger constant. Chapter 4 on the sinc-Galerkin method shows how to approximate the two-point boundary value problem for a second-order differential equation. Chapter 5 applies this to solve steady problems in one and two dimensions. Chapter 6 discusses the heat equation and the viscous Burgers’ equation. These latter three chapters require rather more concentration on the part of the reader as the method generates full matrices whose structure is not easy at first sight to grasp.

The case for the method made in this book is persuasive and if one is not wholly convinced it is partly because there is no comparison with spectral methods of a more familiar kind. Many of the problems have singularities at the end points of intervals which would spoil the convergence of any method if applied blindly, but which can easily be taken account of once their presence has been recognized. Unfortunately in many cases doing so renders the problem trivial so that an ordinary spectral method would produce very rapid convergence or in some cases the exact solution.

Anyone who wishes to look into the sinc method more closely will find a large number of references at the end of each chapter, while the student who wants to learn about the method will find the book a readily accessible account of it.

Chapter 1 provides a résumé of the necessary material from Fourier analysis and theory of a complex variable. Chapter 2 describes the application of sinc methods to a class of functions analytic on a strip of width \(2\pi d\) parallel to the real axis for which the order of convergence of the sinc expansion is \(\exp(-2\pi d/h)\). Chapter 3 applies the method to an arc in the complex plane by mapping it to the whole real axis in such a way that the exponential order of convergence is preserved, although multiplied by a larger constant. Chapter 4 on the sinc-Galerkin method shows how to approximate the two-point boundary value problem for a second-order differential equation. Chapter 5 applies this to solve steady problems in one and two dimensions. Chapter 6 discusses the heat equation and the viscous Burgers’ equation. These latter three chapters require rather more concentration on the part of the reader as the method generates full matrices whose structure is not easy at first sight to grasp.

The case for the method made in this book is persuasive and if one is not wholly convinced it is partly because there is no comparison with spectral methods of a more familiar kind. Many of the problems have singularities at the end points of intervals which would spoil the convergence of any method if applied blindly, but which can easily be taken account of once their presence has been recognized. Unfortunately in many cases doing so renders the problem trivial so that an ordinary spectral method would produce very rapid convergence or in some cases the exact solution.

Anyone who wishes to look into the sinc method more closely will find a large number of references at the end of each chapter, while the student who wants to learn about the method will find the book a readily accessible account of it.

Reviewer: J.D.P.Donnelly (Oxford)

##### MSC:

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

65T40 | Numerical methods for trigonometric approximation and interpolation |

65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |

34B05 | Linear boundary value problems for ordinary differential equations |

35Q53 | KdV equations (Korteweg-de Vries equations) |

35K05 | Heat equation |

65D32 | Numerical quadrature and cubature formulas |