Nonlinear ordinary differential equations and their applications. (English) Zbl 0722.34001

This book discusses the constructive solution, or more often an approximation thereof, of nonlinear ordinary differential equations. As used in the title, “nonlinear” means a single equation of typically second order, analytic in both the independent variables. And the word “applications” is interpreted very much in the classical sense. There are very many examples, generally with a very brief comment as to the source of the equation. A notable exception is the final chapter, giving several detailed applications to partial differential equations. The book proceeds by examples, as might be expected, and not by proving general theorems. The techniques employed are basically well-known: transformations, series solutions, asymptotic behavior, phase plane analysis, etc. The book does not discuss modern discretization methods. This book discusses several topics which are not as readily available elsewhere, in anything like an elementary form. The chapter on shooting methods, for example, emphasizes problems on unbounded intervals. One chapter is devoted to several examples of chaotic behavior. And motivated by the interest in series solutions, the book contains a detailed, up-to- date, readable discussion of the interpretation of such differential equations in the complex plane, up through a lengthy chapter on the Painlevé transcendents. Indeed, a course in complex variables is definitely a prerequisite for reading this book.


34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34A34 Nonlinear ordinary differential equations and systems
34A45 Theoretical approximation of solutions to ordinary differential equations