Advanced Differential Equations

1st Edition - April 13, 2022
Author: Youssef N. Raffoul
Language: English
Paperback ISBN:
9 7 8 - 0 - 3 2 3 - 9 9 2 8 0 - 0
eBook ISBN:
9 7 8 - 0 - 3 2 3 - 9 9 2 8 1 - 7

Advanced Differential Equations provides coverage of high-level topics in ordinary differential equations and dynamical systems. The book delivers difficult material in an access… Read more

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Advanced Differential Equations provides coverage of high-level topics in ordinary differential equations and dynamical systems. The book delivers difficult material in an accessible manner, utilizing easier, friendlier notations and multiple examples. Sections focus on standard topics such as existence and uniqueness for scalar and systems of differential equations, the dynamics of systems, including stability, with examples and an examination of the eigenvalues of an accompanying linear matrix, as well as coverage of existing literature. From the eigenvalues' approach, to coverage of the Lyapunov direct method, this book readily supports the study of stable and unstable manifolds and bifurcations.

Additional sections cover the study of delay differential equations, extending from ordinary differential equations through the extension of Lyapunov functions to Lyapunov functionals. In this final section, the text explores fixed point theory, neutral differential equations, and neutral Volterra integro-differential equations.

Cover image
Title page
Table of Contents
Copyright
Dedication
Preface
Chapter 1: Preliminaries and Banach spaces
Abstract
1.1. Preliminaries
1.2. Escape velocity
1.3. Applications to epidemics
1.4. Metrics and Banach spaces
1.5. Variation of parameters
1.6. Special differential equations
1.7. Exercises
Bibliography
Chapter 2: Existence and uniqueness
Abstract
2.1. Existence and uniqueness of solutions
2.2. Existence on Banach spaces
2.3. Existence theorem for linear equations
2.4. Continuation of solutions
2.5. Dependence on initial conditions
2.6. Exercises
Bibliography
Chapter 3: Systems of ordinary differential equations
Abstract
3.1. Existence and uniqueness
3.2. x′=A(t)x
3.3. x′=A(t)x+g(t)
3.4. Discussion
3.5. Exercises
Bibliography
Chapter 4: Stability of linear systems
Abstract
4.1. Definitions and examples
4.2. x′=A(t)x
4.3. Floquet theory
4.4. Exercises
Chapter 5: Qualitative analysis of linear systems
Abstract
5.1. Preliminary theorems
5.2. Near-constant systems
5.3. Perturbed linear systems
5.4. Autonomous systems in the plane
5.5. Hamiltonian and gradient systems
5.6. Exercises
Chapter 6: Nonlinear systems
Abstract
6.1. Bifurcations in scalar systems
6.2. Stability of systems by linearization
6.3. An SIR epidemic model
6.4. Limit cycle
6.5. Lotka–Volterra competition model
6.6. Bifurcation in planar systems
6.7. Manifolds and Hartman–Grobman theorem
6.8. Exercises
Bibliography
Chapter 7: Lyapunov functions
Abstract
7.1. Lyapunov method
7.2. Global asymptotic stability
7.3. Instability
7.4. ω-limit set
7.5. Connection between eigenvalues and Lyapunov functions
7.6. Exponential stability
7.7. Exercises
Chapter 8: Delay differential equations
Abstract
8.1. Introduction
8.2. Method of steps
8.3. Existence and uniqueness
8.4. Stability using Lyapunov functions
8.5. Stability using fixed point theory
8.6. Exponential stability
8.7. Existence of positive periodic solutions
8.8. Exercises
Bibliography
Chapter 9: New variation of parameters
Abstract
9.1. Applications to ordinary differential equations
9.2. Applications to delay differential equations
9.3. Exercises
Bibliography
Bibliography
Index

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