Description

Nonlinear differential equations arise as mathematical models of various phenomena. Here, various methods of solving and approximating linear and nonlinear differential equations are examined. Since analytical solutions to nonlinear differential equations are rare and difficult to determine, approximation methods have been developed. Initial and boundary value problems will be discussed. Several linear and nonlinear techniques to approximate or solve the linear or nonlinear problems are demonstrated. Regular and singular perturbation theory and Magnus expansions are our particular focus. Each section offers several examples to show how each technique is implemented along with the use of visuals to demonstrate the accuracy, or lack thereof, of each technique. These techniques are integral in applied mathematics and it is shown that correct employment allows us to see the behaviour of a differential equation when the exact solution may not be attainable. Finding and interpreting the solutions of differential equations is a central and essential part of applied mathematics. This book aims to enable the reader to develop the required skills needed for a thorough understanding of the subject. This book bridges the gap between elementary courses and research literature. The basic concepts necessary to study differential equations critical points and equilibrium, periodic solutions, invariant sets and invariant manifolds are discussed. Prepared with an eye toward the needs of applied mathematicians, engineers, and physicists, the treatment is equally valuable as a reference for professionals. The focuses of the book is developing analytical skills needed for research in this area. Designed for senior undergraduates and graduate students, as well as professionals the text emphasizes practical application and analytical techniques rather than abstract formulations.