##
**The numerical solution of integral equations of the second kind.**
*(English)*
Zbl 0899.65077

Cambridge Monographs on Applied and Computational Mathematics 4. Cambridge: Cambridge University Press. xvi, 552 p. (1997).

This outstanding monograph presents a comprehensive and “state-of-the-art” introduction to the numerical solution of Fredholm integral equations of the second kind and of boundary integral equations arising from reformulations of Laplace’s equation in \(\mathbb{R}^2\) and \(\mathbb{R}^3\). According to its preface, it is directed to all numerical analysts working on the numerical solution of integral equations; toward applied mathematicians both interested in integral equations and in solving elliptic boundary value problems via boundary integral techniques; and toward “that very large group of engineers needing to solve problems involving integral equations”.

Its content reveals the very significant progress the field has experienced since the author’s earlier book [A survey of numerical methods for the solution of Fredholm integral equations of the second kind (1976; Zbl 0353.65069)]; it also reflects the major contributions of the author himself during the last twenty years to the numerical analysis of Fredholm integral equations in \(\mathbb{R}^3\) and of boundary integral equations.

A brief summary of the content of the book follows.

Chapter 1 (about 20 pages) introduces the various types of integral equations treated in the book (Volterra and Fredholm equations of the first and second kind; Wiener-Hopf equations; Cauchy singular integral equations; boundary integral equations) and the underlying theory of the corresponding integral operators in a functional analytic setting. Some key results from Functional Analysis are summarized in a brief appendix at the end of the book.

Chapter 2 (24 pages) deals with degenerate kernel methods for solving Fredholm integral equations of the second kind. The general theory is complemented by a discussion of Taylor series and interpolatory degenerate kernel approximations, orthogonal expansions, and the conditioning and the approximate calculation of the resulting linear systems.

Chapter 3 (50 pages) presents the theory of projection methods (collocation and Galerkin methods; iterated projection methods) for second-kind Fredholm integral equations. Much attention is given to the properties of the condition numbers and other properties of the systems of linear algebraic equations arising from these methods.

Chapter 4 (55 pages) is dedicated to a thorough analysis of the Nyström method for Fredholm integral equations of the second kind. Here, a preliminary discussion of the error analysis is followed by the general theory of collectively compact operator approximations. The extension of Nyström methods to equations with noncontinuous kernels leads to product integration methods; their use on graded meshes, and their relationship with collocation methods are described in detail. The chapter ends with two sections on discrete collocation and discrete Galerkin methods.

In Chapter 5 (85 pages) the author extends the material of the previous chapters to multivariable integral equations. The main focus is on integral equations defined on surfaces in \(\mathbb{R}^3\), and this prepares the ground for the later discussion (in Chapter 9) of boundary integral equations on such surfaces. The necessary tools for the analysis of numerical methods (interpolation and numerical integration over triangles; spherical polynomials and numerical integration on the sphere) for these equations are introduced in a lucid way.

Chapter 6 (65 pages) is dedicated to iteration methods for solving the (often large) systems of linear equations arising in the discretization of Fredholm integral equations of the second kind. These methods include two-grid methods for the Nyström method and the collocation method, as well as multigrid iteration for collocation methods. The chapter concludes with a detailed discussion of the conjugate gradient method.

The last three chapters give, on some 210 pages, a comprehensive account of the theory and the numerical treatment of boundary integral equations (BIEs) arising from reformulations of Laplace’s equation in \(\mathbb{R}^2\) and \(\mathbb{R}^3\). This account begins in Chapter 7 with a discussion of the theory of BIEs of the first and second kind on smooth planar boundaries (it includes a brief introduction to pseudodifferential equations and operators). The core sections are given to a good discussion of the abstract framework for the finite element method and the boundary element method for BIEs. In Chapter 8 this is extended to BIEs on piecewise planar boundaries: after describing elements of the relevant theory the focus is on the Galerkin, collocation, and Nyström methods.

The final Chapter 9 on boundary integral equations in \(\mathbb{R}^3\) includes a thorough analysis of the boundary element collocation and Galerkin method on smooth surfaces. This is one of the areas where the author has made substantial contributions in recent years.

Each chapter includes numerical illustrations and ends with a short discussion of the relevant literature, guiding the reader to a comprehensive bibliography of close to 600 items.

This well-produced book represents a major milestone in the list of books on the numerical solution of integral equations; it will have to form part of the library not only of the audience to which it is directed in the first place but also deserves to be on the shelf of any researcher and graduate student interested in the numerical solution of elliptic boundary value problems.

Its content reveals the very significant progress the field has experienced since the author’s earlier book [A survey of numerical methods for the solution of Fredholm integral equations of the second kind (1976; Zbl 0353.65069)]; it also reflects the major contributions of the author himself during the last twenty years to the numerical analysis of Fredholm integral equations in \(\mathbb{R}^3\) and of boundary integral equations.

A brief summary of the content of the book follows.

Chapter 1 (about 20 pages) introduces the various types of integral equations treated in the book (Volterra and Fredholm equations of the first and second kind; Wiener-Hopf equations; Cauchy singular integral equations; boundary integral equations) and the underlying theory of the corresponding integral operators in a functional analytic setting. Some key results from Functional Analysis are summarized in a brief appendix at the end of the book.

Chapter 2 (24 pages) deals with degenerate kernel methods for solving Fredholm integral equations of the second kind. The general theory is complemented by a discussion of Taylor series and interpolatory degenerate kernel approximations, orthogonal expansions, and the conditioning and the approximate calculation of the resulting linear systems.

Chapter 3 (50 pages) presents the theory of projection methods (collocation and Galerkin methods; iterated projection methods) for second-kind Fredholm integral equations. Much attention is given to the properties of the condition numbers and other properties of the systems of linear algebraic equations arising from these methods.

Chapter 4 (55 pages) is dedicated to a thorough analysis of the Nyström method for Fredholm integral equations of the second kind. Here, a preliminary discussion of the error analysis is followed by the general theory of collectively compact operator approximations. The extension of Nyström methods to equations with noncontinuous kernels leads to product integration methods; their use on graded meshes, and their relationship with collocation methods are described in detail. The chapter ends with two sections on discrete collocation and discrete Galerkin methods.

In Chapter 5 (85 pages) the author extends the material of the previous chapters to multivariable integral equations. The main focus is on integral equations defined on surfaces in \(\mathbb{R}^3\), and this prepares the ground for the later discussion (in Chapter 9) of boundary integral equations on such surfaces. The necessary tools for the analysis of numerical methods (interpolation and numerical integration over triangles; spherical polynomials and numerical integration on the sphere) for these equations are introduced in a lucid way.

Chapter 6 (65 pages) is dedicated to iteration methods for solving the (often large) systems of linear equations arising in the discretization of Fredholm integral equations of the second kind. These methods include two-grid methods for the Nyström method and the collocation method, as well as multigrid iteration for collocation methods. The chapter concludes with a detailed discussion of the conjugate gradient method.

The last three chapters give, on some 210 pages, a comprehensive account of the theory and the numerical treatment of boundary integral equations (BIEs) arising from reformulations of Laplace’s equation in \(\mathbb{R}^2\) and \(\mathbb{R}^3\). This account begins in Chapter 7 with a discussion of the theory of BIEs of the first and second kind on smooth planar boundaries (it includes a brief introduction to pseudodifferential equations and operators). The core sections are given to a good discussion of the abstract framework for the finite element method and the boundary element method for BIEs. In Chapter 8 this is extended to BIEs on piecewise planar boundaries: after describing elements of the relevant theory the focus is on the Galerkin, collocation, and Nyström methods.

The final Chapter 9 on boundary integral equations in \(\mathbb{R}^3\) includes a thorough analysis of the boundary element collocation and Galerkin method on smooth surfaces. This is one of the areas where the author has made substantial contributions in recent years.

Each chapter includes numerical illustrations and ends with a short discussion of the relevant literature, guiding the reader to a comprehensive bibliography of close to 600 items.

This well-produced book represents a major milestone in the list of books on the numerical solution of integral equations; it will have to form part of the library not only of the audience to which it is directed in the first place but also deserves to be on the shelf of any researcher and graduate student interested in the numerical solution of elliptic boundary value problems.

Reviewer: H.Brunner (St.John’s)

### MSC:

65R20 | Numerical methods for integral equations |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

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

45Exx | Singular integral equations |

65N38 | Boundary element methods for boundary value problems involving PDEs |

35C15 | Integral representations of solutions to PDEs |

45B05 | Fredholm integral equations |