Method of Averaging for Differential Equations on an Infinite Interval: Theory and Applications

ISBN-13: 9781584888741 / Angielski / Miękka / 2007 / 360 str.

Vladimir Burd; V. Sh Burd

Opis

In recent years, mathematicians have detailed simpler proofs of known theorems, have identified new applications of the method of averaging, and have obtained many new results of these applications. Encompassing these novel aspects, Method of Averaging of the Infinite Interval: Theory and Applications rigorously explains the modern theory of the method of averaging and provides a solid understanding of the results obtained when applying this theory.
The book starts with the less complicated theory of averaging linear differential equations (LDEs), focusing on almost periodic functions. It describes stability theory and Shtokalo's method, and examines various applications, including parametric resonance and the construction of asymptotics. After establishing this foundation, the author goes on to explore nonlinear equations. He studies standard form systems in which the right-hand side of a system is proportional to a small parameter and proves theorems similar to Banfi's theorem. The final chapters are devoted to systems with a rapidly rotating phase.
Covering an important asymptotic method of differential equations, this book provides a thorough understanding of the method of averaging theory and its resulting applications.

Spis treści

PREFACE

AVERAGING OF LINEAR DIFFERENTIAL EQUATIONS
Periodic and Almost Periodic Functions. Brief Introduction
Bounded Solutions
Lemmas on Regularity and Stability
Parametric Resonance in Linear Systems
Higher Approximations. Shtokalo Method
Linear Differential Equations with Fast and Slow Time
Asymptotic Integration
Singularly Perturbed Equations

AVERAGING OF NONLINEAR SYSTEMS
Systems in Standard Form. First Approximation
Systems in Standard Form. First Examples
Pendulum Systems with an Oscillating Pivot
Higher Approximations of the Method of Averaging
Averaging and Stability
Systems with a Rapidly Rotating Phase
Systems with a Fast Phase. Resonant Periodic Oscillations
Systems with Slowly Varying Parameters

APPENDICES
Almost Periodic Functions
Stability of the Solutions of Differential Equations
Some Elementary Facts from the Functional Analysis

REFERENCES
INDEX

O autorze

Vladimir Burd