Computational methods for integral equations.

*(English)*Zbl 0592.65093
Cambridge etc.: Cambridge University Press. XII, 376 p. £35.00; $ 69.50 (1985).

There are not many systematic books on the numerical treatment of integral equations. Regarding the last ten years before this book appeared, there existed only the volume by C. T. H. Baker [The numerical treatment of integral equations (1977; Zbl 0373.65060)] and the recent book by P. Linz [Analytical and numerical methods for Volterra equations (1985; Zbl 0566.65094)]. Certainly, a new book on the whole subject is justified. First of all, a summary of the contents: After the introducig Chapter 0, Chapters 1,3 and 6 cover the mathematical background. Chapter 2 deals with numerical quadrature, Chapters 4 and 5 with the application of quadrature to Fredholm and Volterra integral equations of the second kind. Chapter 7 is devoted to expansion methods, in particular Ritz-Galerkin and in Chapter 8 the numerical performance of these methods is described. In Chapter 9 the Galerkin method is analysed again with orthogonal basis. Chapter 10 deals with the practical point of view of different algorithms for second kind Fredholm integral equations. In Chapter 11 the reader finds singular integral equations, in Chapter 12 integral equations of the first kind and in Chapter 13 integro- differential equations.

This short specification shows, that the authors restrict themselves essentially to two numerical methods: Quadrature and Ritz-Galerkin methods. This choice is explained in preface and introduction partly by personal likes and dislikes and partly by the opinion, that these methods fulfill the following criteria optimally: (i) Ability to obtain rapid convergence; (ii) Existence of reliable and cheaply computable error estimates; (iii) Potential for extension to ”nonstandard” equations. Within this self-made frame the authors want to present - using their own words - ”the elements of the theory of linear integral equations ab initio and at the same time develop various numerical methods for their solution. The theory is developed within the framework given by the space of \(L^ 2\) functions and operators”. The level which can be reached is described by the intention that the book should serve as a reference text for the practising numerical mathematician, scientist or engineer and also as a textbook for third year undergraduate and M. Sc. students attending a course on the numerical solution of integral equations. So, the mathematical demands are limited, for example as to some more sophisticated results on error bounds. The treatment of eigenvalue problems, which are usually the more pretentious ones, goes a bit short in the reviewer’s eyes. But for the practising mathematician, the book is very helpful. The numerical performance, comparison of algorithms, error estimates for practical use are in the foreground. Integral equations of the first kind are treated in a relatively extended manner, regularization methods are mentioned. The same holds for singular integral equations, including the standard methods of product integration and subtraction of the singularity. It is also worth mentioning the detailed discussion of the methods applied to integro-differential equations in the last chapter. There are many computed examples in the book and exercises at the end of each chapter. The exercises are not too difficult to solve and many of them are meant to enhance practical experience.

The book is conveniently readable, printing and outfit are good. In the reviewer’s opinion, the selection is limited and the weights are distributed according to the individual taste and computer experience of the authors. Conscious of this general line, the book is very useful for students and for practising mathematicians as well, as a text and exercise book and as a practical guide.

This short specification shows, that the authors restrict themselves essentially to two numerical methods: Quadrature and Ritz-Galerkin methods. This choice is explained in preface and introduction partly by personal likes and dislikes and partly by the opinion, that these methods fulfill the following criteria optimally: (i) Ability to obtain rapid convergence; (ii) Existence of reliable and cheaply computable error estimates; (iii) Potential for extension to ”nonstandard” equations. Within this self-made frame the authors want to present - using their own words - ”the elements of the theory of linear integral equations ab initio and at the same time develop various numerical methods for their solution. The theory is developed within the framework given by the space of \(L^ 2\) functions and operators”. The level which can be reached is described by the intention that the book should serve as a reference text for the practising numerical mathematician, scientist or engineer and also as a textbook for third year undergraduate and M. Sc. students attending a course on the numerical solution of integral equations. So, the mathematical demands are limited, for example as to some more sophisticated results on error bounds. The treatment of eigenvalue problems, which are usually the more pretentious ones, goes a bit short in the reviewer’s eyes. But for the practising mathematician, the book is very helpful. The numerical performance, comparison of algorithms, error estimates for practical use are in the foreground. Integral equations of the first kind are treated in a relatively extended manner, regularization methods are mentioned. The same holds for singular integral equations, including the standard methods of product integration and subtraction of the singularity. It is also worth mentioning the detailed discussion of the methods applied to integro-differential equations in the last chapter. There are many computed examples in the book and exercises at the end of each chapter. The exercises are not too difficult to solve and many of them are meant to enhance practical experience.

The book is conveniently readable, printing and outfit are good. In the reviewer’s opinion, the selection is limited and the weights are distributed according to the individual taste and computer experience of the authors. Conscious of this general line, the book is very useful for students and for practising mathematicians as well, as a text and exercise book and as a practical guide.

Reviewer: G.HĂ¤mmerlin

##### MSC:

65R20 | Numerical methods for integral equations |

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

65D32 | Numerical quadrature and cubature formulas |

45B05 | Fredholm integral equations |

45C05 | Eigenvalue problems for integral equations |

45Exx | Singular integral equations |

45J05 | Integro-ordinary differential equations |