Solution of Differential Equations by Lindstedt Poincare Techniques

Abstract

Many of the problems facing physicists, engineers, and applied mathematicians involve such difficulties as nonlinear boundary conditions at complex known or unknown boundaries that preclude solving them exactly. Consequently, solutions are approximated using numerical techniques, analytic techniques, and combinations of both. Foremost among the analytic techniques are the systematic methods of perturbations (asymptotic expansions) in terms of a small or a large parameter or coordinate. Although the techniques are described by means of examples that start with simple ordinary equations that can be solved exactly and progress toward complex partial-differential equations, the material is concise and advanced and therefore is intended for researchers and advanced graduate students only. The purpose of this techniques, however, is to present the material in an elementary way that makes it easily accessible to advanced undergraduates and first-year graduate students in a wide variety of scientific and engineering fields. As a result of teaching perturbation methods for eight years to first-year and advanced graduate students at Virginia Poly-techniques and amplified their description considerably. The techniques are described by means of simple examples that consist mainly of algebraic and ordinary-differential equations. The material in Chapters 1 and 2 is about solution of differential equations by different perturbation Methods. Chapter 1 discusses “The Straightforward Expansions” method.