Matched asymptotic expansions. Ideas and techniques
The contents of this book justify its second title: ``Ideas and techniques''. Indeed one could hardly find a single proved theorem in this book. The reviewer would call this a book of mathematical ``know- how''. It is composed of three parts: an introductory one, one devoted to ordinary differential equations and one to partial differential equations. The introductory part of the book is concerned with the basic concepts of asymptotic expansions. Based on the case of a simple linear second-order differential equation the following notions are introduced in an informal way: regular and singular perturbations, secular and layer problems, gauge functions, validity domain of an expansion, matched asymptotic expansions. The part devoted to ordinary differential equations deals essentially with model problems from science and engineering; the equations are second-order, linear and nonlinear (the model of Friedrichs, the flow at low Reynolds number, relaxation oscillations). The chapter focuses on the ideas of S. Kaplun concerning such variable changes as \(x_{\eta}=(x-x_ d)/\eta\) where \(\eta\) (\(\epsilon)\) is a gauge function and taking the limit for \(\epsilon\) \(\to 0\) while keeping x fixed. One can find here also the conjecture of Kaplun concerning the domain of validity of an expansion. In this part only layer problems are studied with one exception where secular problems are noted. The part concerning partial differential equation is devoted to layer problems on elliptic, parabolic and hyperbolic problems in two variables (e.g. KdV equation, shocklike phenomena). As a whole this book reminds of the ``old good'' computational mathematics of more than 100 years ago when people were eager to find than to prove. It shows how difficult and subtle mathematics could be even remaining not very abstract and not very formal. It is a stimulating book.
scientific article; zbMATH DE number 41890
scientific article