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Volterra Integral Equations

Hermann Brunner
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Monographs on Applied and Computational Mathematics
[Reviewed by
Allen Stenger
, on

Volterra integral equations come in three kinds, The first kind is \[\int_0^t H(t,s) u(s) \, ds = g(t) \quad (0 \le t \le T),\] where \(H\) is a known function (called the kernel) and \(g\) is a known function, and we seek the function \(u\) on \([0,T]\). Volterra thought of this problems as “inverting the definite integral”. The second and third kinds of Volterra equations are similar but more complicated. The other main type of integral equation is the Fredholm, which is the same except that the interval of integration is fixed instead of depending on \(t\).

The present book is an introduction to the theory of Volterra equations; it makes only brief mention of Fredholm equations. It covers existence and uniqueness of solutions, and what inferences can be made about the behavior of \(u\) given the behavior of \(H\). It does not cover any solution methods, and I believe no equation is actually solved anywhere in the book.

The theory splits into many branches depending on the form of the equation and any particular properties that are known about the kernel. The present book aims for breadth rather than depth and tries to say something about all the common variations. The book is strong on history, and the first chapter explains the theory as it was viewed by Volterra himself (Abel, du Bois-Reymond, and Bôcher were also pioneers in this field and get some coverage).

Each chapter ends with a long list of exercises, at various levels of difficulty, and research problems (happily the latter as labelled as such). No solutions are given.

The last chapter covers applications of integral equations; this is interesting but very brief. There are about 30 applications, and each one gives a brief explanation of the problem and shows the form of the integral equation that arises, but does not show the modeling or the solution.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

1. Linear Volterra integral equations
2. Regularity of solutions
3. Nonlinear Volterra integral equations
4. Volterra integral equations with highly oscillatory kernels
5. Singularly perturbed and integral-algebraic Volterra equations
6. Qualitative theory of Volterra integral equations
7. Cordial Volterra integral equations
8. Volterra integral operators on Banach spaces
9. Applications of Volterra integral equations
Appendix. A review of Banach space tools