Differential Equations: An Introduction to Basic Concepts, Results and Applications

✍ Scribed by Ioan I. Vrabie

• Publisher: World Scientific
• Year: 2004
• Tongue: English
• Leaves: 420
• Category: Library

✦ Synopsis

This book presents the main concepts and results of differential equations, and offers the reader another point of view concerning a possible way to approach the problems of existence, uniqueness, approximation, and continuation of the solutions to a Cauchy problem. In addition, it contains simple introductions to some topics which are not usually included in classical textbooks: the exponential formula, conservation laws, generalized solutions, Caratheodory solutions, differential inclusions, variational inequalities, viability, invariance, gradient systems.

✦ Table of Contents

Cover
Title
Preface
List of Symbols
Contents
1. Generalities
1.1 Brief History
1.1.1 The Birth of the Discipline
1.1.2 Major Themes
1.2 Introduction
1.3 Elementary Equations
1.3.1 Equations with Separable Variables
1.3.2 Linear Equations
1.3.3 Homogeneous Equations
1.3.4 Bernoulli Equations
1.3.5 Riccati Equations
1.3.6 Exact Differential Equations
1.3.7 Equations Reducible to Exact Differential Equations
1.3.8 Lagrange Equations
1.3.9 Clairaut Equations
1.3.10 Higher-Order Differential Equations
1.4 Some Mathematical Models
1.4.1 Radioactive Disintegration
1.4.2 The Carbon Dating Method
1.4.3 Equations of Motion
1.4.4 The Harmonic Oscillator
1.4.5 The Mathematical Pendulum
1.4.6 Two Demographic Models
1.4.7 A Spatial Model in Ecology
1.4.8 The Prey-Predator Model
1.4.9 The Spreading of a Disease
1.4.10 Lotka Model
1.4.11 An Autocatalytic Generation Model
1.4.12 An RLC Circuit Model
1.5 Integral Inequalities
1.6 Exercises and Problems
2. The Cauchy Problem
2.1 General Presentation
2.2 The Local Existence Problem
2.3 The Uniqueness Problem
2.3.1 The Locally Lipschitz Case
2.3.2 The Dissipative Case
2.4 Saturated Solutions
2.4.1 Characterization of Continuable Solutions
2.4.2 The Existence of Saturated Solutions
2.4.3 Types of Saturated Solutions
2.4.4 The Existence of Global Solutions
2.5 Continuous Dependence on Data and Parameters
2.5.1 The Dissipative Case
2.5.2 The Locally Lipschitz Case
2.5.3 Continuous Dependence on Parameters
2.6 Problems of Differentiability
2.6.1 Differentiability with Respect to the Data
2.6.2 Differentiability with Respect to the Parameters
2.7 The Cauchy Problem for the nth-Order Differential Equation
2.8 Exercises and Problems
3. Approximation Methods
3.1 Power Series Method
3.1.1 An Example
3.1.2 The Existence of Analytic Solutions
3.2 The Successive Approximations Method
3.3 The Method of Polygonal Lines
3.4 Euler Implicit Method. Exponential Formula
3.4.1 The Semigroup Generated by A
3.4.2 Two Auxiliary Lemmas
3.4.3 The Exponential Formula
3.5 Exercises and Problems
4. Systems of Linear Differential Equations
4.1 Homogeneous Systems. The Space of Solutions
4.2 Non-homogeneous Systems. Variation of Constants Formula
4.3 The Exponential of a Matrix
4.4 A Method to Find etA
4.5 The nth-Order Linear Differential Equation
4.6 The nth-order Linear Differential Equation with Constants Coefficients
4.7 Exercises and Problems
5. Elements of Stability
5.1 Types of Stability
5.2 Stability of Linear Systems
5.3 The Case of Perturbed Systems
5.4 The Lyapunov Function Method
5.5 The Case of Dissipative Systems
5.6 The Case of Controlled Systems
5.7 Unpredictability and Chaos
5.8 Exercises and Problems
6. Prime Integrals
6.1 Prime Integrals for Autonomous Systems
6.2 Prime Integrals for Non-Autonomous Systems
6.3 First Order Partial Differential Equations
6.4 The Cauchy Problem for Quasi-Linear Equations
6.5 Conservation Laws
6.5.1 Some Examples
6.5.2 A Local Existence and Uniqueness Result
6.5.3 Weak Solutions
6.6 Exercises and Problems
7. Extensions and Generalizations
7.1 Distributions of One Variable
7.2 The Convolution Product
7.3 Generalized Solutions
7.4 Caratheodory Solutions
7.5 Differential Inclusions
7.6 Variational Inequalities
7.7 Problems of Viability
7.8 Proof of Nagumo‘s Viability Theorem
7.9 Sufficient Conditions for Invariance
7.10 Necessary Conditions for Invariance
7.11 Gradient Systems. Frobenius Theorem
7.12 Exercises and Problems
8. Auxiliary Results
8.1 Elements of Vector Analysis
8.2 Compactness in C ([a, b];Rn)
8.3 The Projection of a Point on a Convex Set
Solutions
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Bibliography
Index