This is a scholarly book with a serious pedagogical objective. The authors are, as it were, proselytes for the method(s) of Green’s functions, noting in their Preface to the book that
the importance of this topic is … somewhat underestimated in contemporary textbooks: not all texts on differential equations cover the topic … [and w]ith standard works on numerical analysis, things are even worse …
They go on to say that their rationale for presenting Green’s functions so as to “provide the user with as many compact and computer-friendly representations of Green’s functions as the format of [a] textbook permits,” is that the according method is remarkably fecund across a number of disciplines, including ordinary differential equations, partial differential equations, numerical analysis, and (ineffable though this description always is) applied mathematics.
It is of course known to every one that Green’s functions arise in hugely important ways in physics, e.g, quantum mechanics, what with PDEs and integral equations present in spades. Indeed, Melnikov and Melnikov state, explicitly, that a primary example is
the solution of a boundary value problem for a partial differential equation[:] By using Green’s functions, the equation can be reduced to an integral equation[, after which a]pplying numerical techniques to solve the reduced problem will be significantly more economical than applying them directly to the original differential equation.
(In gross terms, the idea is that given an inhomogeneous DE, Du = f¸ one tries to construct a Green’s function, say, g, so that the solution u can be written as a suitable convolution of g against f. In other words, the Green’s function becomes the attendant integrating kernel.)
This having been said, the authors are sure to emphasize that despite their objective to give a lot of airplay to the aforementioned practical aspects of the business of Green’s functions, and with their emphasis “not on theoretical aspects,” they do not shortchange rigor. This is obviously a key feature of their treatment not only in and of itself, but also in view of their pedagogical intent: the book is intended to be a text for “elective advanced undergraduate and graduate courses, within the field of applied mathematics and related disciplines.” Thus, the reader should certainly be prepared with a decent knowledge of serious undergraduate level analysis (though Lebesgue probably doesn’t need to make an appearance just yet), and have, say, a course in ODE and a course in “methods of applied mathematics” (as we call it at my university) under his belt, too. The latter should include a first exposure to PDEs, which the sets the stage for what the Melnikovs do in the book under review: while they state that they’re covering “only a limited number of applied partial differential equations[, t]hese include the two-dimensional La[place, the static Klein-Gordon, the biharmonic, the diffusion (heat), and the Black-Scholes equation.” Quite a sampling of very important PDEs.
The book is laid out in eight chapters, chock-full of examples (and rightly so), and every chapter is supplemented with a long set of exercises. There is no other way to learn how to navigate with Green’s functions than to get one’s hands filthily dirty with actual computations, and this book certainly provided that opportunity. It is a well-written book that should have no trouble in meeting its objectives: be it for (very serious) advanced undergraduates or graduate students, or even more experienced mathematicians or fellow-travellers with a need to learn something about Green’s functions, this book serves to present its subject very well indeed.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.