Perturbation Methods in Applied Mathematics

by J. Kevorkian, J.D. Cole

This book is a revised and updated version, including a substantial portion of new material, of J. D. Cole's text Perturbation Methods in Applied Mathe­ matics, Ginn-Blaisdell, 1968. We present the material at a level which assumes some familiarity with the basics of ordinary and partial differential equations. Some of the more advanced ideas are reviewed as needed; therefore this book can serve as a text in either an advanced undergraduate course or a graduate level course on the subject. The applied mathematician, attempting to understand or solve a physical problem, very often uses a perturbation procedure. In doing this, he usually draws on a backlog of experience gained from the solution of similar examples rather than on some general theory of perturbations. The aim of this book is to survey these perturbation methods, especially in connection with differ­ ential equations, in order to illustrate certain general features common to many examples. The basic ideas, however, are also applicable to integral equations, integrodifferential equations, and even to_difference equations. In essence, a perturbation procedure consists of constructing the solution for a problem involving a small parameter B, either in the differential equation or the boundary conditions or both, when the solution for the limiting case B = 0 is known. The main mathematical tool used is asymptotic expansion with respect to a suitable asymptotic sequence of functions of B.

Hardcover

First published March 19, 1985

Reviews

[Stuart Marlatt · Rating 3 out of 5]

Perturbation methods are a powerful tool which is often lost in this day of numerical methods and readily available computational horsepower. This is unfortunate, because the derivation and implementation of a well-constructed perturbation solution can tell one much about the physics and dynamics of the phenomenon in question - which is easily lost amidst the terabytes of data resulting from a numerical solution.

All that said, I'm somewhat ambivalent about Kevorkian and Cole. All the material is here, but the presentation is dense and often seems to be unnecessarily difficult. For instance, the second paragraphy on the first page begins:

"Let x be fixed. We say phi = O(psi) if there exists k(x) such that |phi|
If you are new to the field, I might suggest Van Dyke's "Perturbation Methods in Fluid Mechanics" as a more accessible introduction.