The classical theory of integral equations. A concise treatment.

*(English)*Zbl 1270.45001
Basel: Birkhäuser (ISBN 978-0-8176-8348-1/hbk; 978-0-8176-8349-8/ebook). xiii, 344 p. (2012).

This book covers the classical theory of linear, scalar Fredholm and Volterra equations. A brief resumé follows.

Chapter 1 deals with Fredholm equations with separable kernels and gives the classical Fredholm theorems. Chapter 2 considers the Fredholm theorems for general nonseparable complex-valued kernels \(K(x,t)\). Chapter 3 presents the theory for Fredholm equations with Hermitian kernels. The classical eigenvalue and eigenfunction theory is given, including the \(L_2\)-theory. These first three chapters comprise half the book. Chapter 4 gives some elementary Volterra theory. Chapter 5 presents techniques for converting differential equations to integral equations and vice versa. Chapter 6 looks at nonlinear equations and includes some illustrative examples. Chapter 7 lists some equations with singular kernels and mentions Laplace and Fourier transforms. A brief look at linear systems of integral equations makes up the final Chapter 8. To conclude the book, the author offers brief French, German and Italian mathematical vocabularies and sample translations.

All chapters include numerous simple examples, some explicitly solvable, some to which numerical methods are applied. No applications to, e.g., physics or engineering are given.

In its content, the book is reminiscent of the classical work by F. G. Tricomi [Integral equations. Pure and Applied Mathematics, Vol. 5. New York: Interscience Publishers (1957; Zbl 0078.09404)].

All integrals in this volume are assumed to be Riemann integrals, not Lebesgue integrals. Perhaps unavoidably, this entails some errors as the statement that the set of functions having finite square Riemann integrals \(\int|f(t)|^2\,dt\) is complete in the \(L_2\)-norm. The book is somewhat too elementary for the intended audience of advanced undergraduate or early graduate students of mathematics but may be useful for engineering students wishing to learn the fundamentals of integral equation theory.

Chapter 1 deals with Fredholm equations with separable kernels and gives the classical Fredholm theorems. Chapter 2 considers the Fredholm theorems for general nonseparable complex-valued kernels \(K(x,t)\). Chapter 3 presents the theory for Fredholm equations with Hermitian kernels. The classical eigenvalue and eigenfunction theory is given, including the \(L_2\)-theory. These first three chapters comprise half the book. Chapter 4 gives some elementary Volterra theory. Chapter 5 presents techniques for converting differential equations to integral equations and vice versa. Chapter 6 looks at nonlinear equations and includes some illustrative examples. Chapter 7 lists some equations with singular kernels and mentions Laplace and Fourier transforms. A brief look at linear systems of integral equations makes up the final Chapter 8. To conclude the book, the author offers brief French, German and Italian mathematical vocabularies and sample translations.

All chapters include numerous simple examples, some explicitly solvable, some to which numerical methods are applied. No applications to, e.g., physics or engineering are given.

In its content, the book is reminiscent of the classical work by F. G. Tricomi [Integral equations. Pure and Applied Mathematics, Vol. 5. New York: Interscience Publishers (1957; Zbl 0078.09404)].

All integrals in this volume are assumed to be Riemann integrals, not Lebesgue integrals. Perhaps unavoidably, this entails some errors as the statement that the set of functions having finite square Riemann integrals \(\int|f(t)|^2\,dt\) is complete in the \(L_2\)-norm. The book is somewhat too elementary for the intended audience of advanced undergraduate or early graduate students of mathematics but may be useful for engineering students wishing to learn the fundamentals of integral equation theory.

Reviewer: Stig-Olof Londen (Aalto)

##### MSC:

45B05 | Fredholm integral equations |

45D05 | Volterra integral equations |

45-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to integral equations |

45G10 | Other nonlinear integral equations |