Introduction to integral equations with applications. (English) Zbl 0623.45001
The author states in the preface: “The material in this text is intended to present the subject of integral equations, and their basic methods of solution, on a level close to that of a first course in ordinary differential equations”. The contents: Chapter I: Integral equations, their origin and classification; Chapter II: Modeling of problems as integral equations; Chapter III: Volterra integral equations (including some numerical methods of solution); Chapter IV: The Green’s function (Fredholm integral equations are naturally obtained); Chapter V: Fredholm integral equations (degenerate kernels, symmetric kernels, numerical procedures); Chapter VI: Existence of solutions (basic fixed point theorems).
Appendix A is concerned with Fourier and Hankel transforms. Appendix B is dedicated to the solution of boundary value problems by Green’s function method (hence, the connection with integral equations). Appendix C is concerned with Green’s function for partial differential equations. Appendix D contains the proof of existence in \(L_ 2\) for the solution of the classical Fredholm equation, for small values of the parameter (based on Banach’s contraction mapping).
We emphasize the significance of Chapter II, which contains interesting modeling examples involving integral equations, in such fields as population dynamics, control theory, mechanics, as well as the presentation of rather general procedures of reducing initial value or boundary value problems to integral equations. The list of references contains more than 60 titles, mostly books on integral equations, integral transforms, Green’s function method etc. A final section contains answers to the exercises proposed in the textbook. Some misspelling appear in the names of Soviet authors who are quoted (for instance, Krasnar instead of Krasnov a.o.).
This book constitutes a valuable addition to the existing literature on integral equation (as an undergraduate teaching subject). There are very few titles in this category, and the one that comes to our attention is M. L. Krasnov, A. I. Kiselev and G. I. Makarenko [Integral equations (1968; Zbl 0159.410)]. It would be certainly interesting to introduce a course on integral equations in the undergraduate program. The problem is how many schools can afford the luxury ? In the positive case, the text could definitely render valuable services.
Appendix A is concerned with Fourier and Hankel transforms. Appendix B is dedicated to the solution of boundary value problems by Green’s function method (hence, the connection with integral equations). Appendix C is concerned with Green’s function for partial differential equations. Appendix D contains the proof of existence in \(L_ 2\) for the solution of the classical Fredholm equation, for small values of the parameter (based on Banach’s contraction mapping).
We emphasize the significance of Chapter II, which contains interesting modeling examples involving integral equations, in such fields as population dynamics, control theory, mechanics, as well as the presentation of rather general procedures of reducing initial value or boundary value problems to integral equations. The list of references contains more than 60 titles, mostly books on integral equations, integral transforms, Green’s function method etc. A final section contains answers to the exercises proposed in the textbook. Some misspelling appear in the names of Soviet authors who are quoted (for instance, Krasnar instead of Krasnov a.o.).
This book constitutes a valuable addition to the existing literature on integral equation (as an undergraduate teaching subject). There are very few titles in this category, and the one that comes to our attention is M. L. Krasnov, A. I. Kiselev and G. I. Makarenko [Integral equations (1968; Zbl 0159.410)]. It would be certainly interesting to introduce a course on integral equations in the undergraduate program. The problem is how many schools can afford the luxury ? In the positive case, the text could definitely render valuable services.
Reviewer: C.Corduneanu
MSC:
| 45-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to integral equations |
| 45D05 | Volterra integral equations |
| 45B05 | Fredholm integral equations |
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
| 34B27 | Green’s functions for ordinary differential equations |
| 45F10 | Dual, triple, etc., integral and series equations |
| 65R20 | Numerical methods for integral equations |
| 92D25 | Population dynamics (general) |
| 93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |
