on April 5, 2000
Barenblatt's book, Scaling, self similarity and intermediate asymptotics, addresses the understanding of physical processes and the interpretation of calculations revealing these processes, two mental problems intertwined closely with the deeper more general issues raised by the recognition of patterns. The book contains excellent problems that are considered in detail and then followed by brilliant generalizations that inspire and provoke reflection.
This book contains many deep examples of analytic solutions to various problems, including propagation of heat from a source in linear and nonlinear cases, and energy propagation from a localized explosion, in which dimensions of the constants that characterize the medium and the dimensions of energy determine uniquely the exponents of the self-similar solutions. By introducing losses, however, the problems change, so that now the conservation of energy does not hold, but the self-similarity remains.
Problems of the non-linear propagation of waves on the surface of a heavy fluid, described by the Kortweg-de Vries equation, are excellent. This example is remarkable in that theorems exist proving the stability of solitons even after these solitons collide. The solutions giving the asymptotic behavior of generalized initial distributions are then transformed beautifully into a sequence of solitons.
In general problems included in this book are focused, cleverly presented and are exemplary. Many are non-linear, and their special solutions represent the asymptotics of a wider class of other more general solutions corresponding to many different initial conditions.
The great value of this book is that the problems introduce general concepts in a unique and memorable way and serve to tie the book together. As a rule the special solutions of the selected problems represent the asymptotics of a larger class of general solutions, the value of the special solutions as asymptotics depending, of course, on their stability.
on September 28, 2002
Last semester I took professor Barrenblatt's graduate course Math 275 at UC Berkeley: "Advanced topics in Applied Mathematics." The topics covered therein were more or less what is covered in this book. I am not a math major, but a civil engineering one, and the course a lot of times got way over my head. Nevertheless, it was a truly amazing experience. I learned a lot. But enough about the course...
This is a truly great book! The introduction (Chapter 0) is a little overwhelming because it attempts to present an overview of topics covered in the following chapters of the book, but the brevity and lack of rigor (it is a summary) may result in confusion. This was the one and only weak point in the book. So... what did I do? I skipped the intro chapter. You can go back to it after you have read the book (or a good part of it) and things will make a lot more sense. From chapter 1 forward, the book is excellent. The ideas are very interesting (this is an applied math book, and the author documents real world examples of where the ideas are applicable) and the concepts presented with sufficient rigor and lucidity that one expects from a mathematics book. Barenblatt is a truly brilliant mathematician and an excellent educator as well, and provides deep insight about dimensional analysis, scaling, similarity, and intermediate asymptotics in this book.