Ordinary Differential Equations With Applications (third Edition) (Series On Applied Mathematics)

✍ Scribed by Sze-bi Hsu, Kuo-chang Chen

• Publisher: WSPC
• Year: 2022
• Tongue: English
• Leaves: 378
• Edition: 3
• Category: Library

✦ Synopsis

Written in a straightforward and easily accessible style, this volume is suitable as a textbook for advanced undergraduate or first-year graduate students in mathematics, physical sciences, and engineering. The aim is to provide students with a strong background in the theories of Ordinary Differential Equations, Dynamical Systems and Boundary Value Problems, including regular and singular perturbations. It is also a valuable resource for researchers. This volume presents an abundance of examples in physical and biological sciences, and engineering to illustrate the applications of the theorems in the text. Readers are introduced to some important theorems in Nonlinear Analysis, for example, Brouwer fixed point theorem and fundamental theorem of algebras. A chapter on Monotone Dynamical Systems takes care of the new developments in Ordinary Differential Equations and Dynamical Systems. In this third edition, an introduction to Hamiltonian Systems is included to enhance and complete its coverage on Ordinary Differential Equations with applications in Mathematical Biology and Classical Mechanics.

✦ Table of Contents

Contents
Preface to the First Edition
Preface to the Second Edition
Preface to the Third Edition
1. INTRODUCTION
1.1 Where do ODEs arise
2. FUNDAMENTAL THEORY
2.1 Introduction and Preliminaries
2.2 Local Existence and Uniqueness of Solutions of I.V.P.
2.3 Continuation of Solutions
2.4 Continuous Dependence Properties
2.5 Differentiability of I.C. and Parameters
2.6 Differential Inequalities
2.7 Exercises
3. LINEAR SYSTEMS
3.1 Introduction
3.2 Fundamental Matrices
3.3 Linear Systems with Constant Coefficients
3.4 Two-Dimensional Linear Autonomous Systems
3.5 Linear Systems with Periodic Coefficients
3.6 Adjoint Systems
3.7 Exercises
4. STABILITY OF NONLINEAR SYSTEMS
4.1 Definitions
4.2 Linearization
4.3 Saddle Point Property
4.4 Orbital Stability
4.5 Traveling Wave Solutions
4.6 Exercises
5. METHOD OF LYAPUNOV FUNCTIONS
5.1 An Introduction to Dynamical Systems
5.2 Lyapunov Functions
5.3 Simple Oscillatory Phenomena
5.4 Gradient Vector Fields
5.5 Exercises
6. TWO-DIMENSIONAL SYSTEMS
6.1 Poincaré-Bendixson Theorem
6.2 Levinson-Smith Theorem
6.3 Hopf Bifurcation
6.4 Exercises
7. SECOND ORDER LINEAR EQUATIONS
7.1 Sturm's Comparison Theorem and Sturm-Liouville Boundary Value Problem
7.2 Distributions
7.3 Green's Function
7.4 Fredholm Alternative
7.5 Exercises
8. THE INDEX THEORY AND BROUWER DEGREE
8.1 Index Theory in the Plane
8.2 Introduction to the Brouwer Degree in Rn
8.3 Lienard Equation with Periodic Forcing
8.4 Exercises
9. PERTURBATION METHODS
9.1 Regular Perturbation Methods
9.2 Singular Perturbation: Boundary Value Problem
9.3 Singular Perturbation: Initial Value Problem
9.4 Exercises
10. INTRODUCTION TO MONOTONE DYNAMICAL SYSTEMS
10.1 Monotone Dynamical System with Applications to Cooperative Systems and Competitive Systems
10.2 Uniform Persistence
10.3 Application: Competition of Two Species in a Chemostat with Inhibition
10.4 Two Species Competition Models
10.5 Exercises
11. INTRODUCTION TO HAMILTONIAN SYSTEMS
11.1 Definitions and Classic Examples
11.2 Linear Hamiltonian Systems
11.3 First Integrals and Poisson Bracket
11.4 Symplectic Transformations
11.5 Generating Functions and Hamilton-Jacobi's Method
11.6 Exercises
APPENDIX A
A.1
A.2
A.3
APPENDIX B
Bibliography
Index