1. Dimensional Analysis
1.1 The Buckingham Pi Theorem
1.2 Exercises
2. Scaling in Differential Equations
2.1 Introduction
2.2 Scaling in Differential Equations
2.3 Scaling in Nonlinear ODE
2.4 Exercises
2.5 O (Big Oh); o (little oh)
2.6 Asymptotic Expansions
3. Perturbation in Differential Equations
3.1 An Introduction to Perturbation Methods
3.2 Regular Perturbation
3.3 Regular Perturbation in Differential Equations
3.4 The Poincaré–Lindstedt Method
3.5 Multiple Scaling
3.6 A Model for Weakly Nonlinear Standing Waves
3.7 Averaging Method
4. Boundary Layers in Singularly Perturbed Ordinary Differential Equations
4.1 Singular Perturbations and the Appearance of Boundary Layers
4.2 Equations with Variable Coefficients
4.3 Interior Boundary Layers
4.4 Existence and Location of Boundary and Interior Layers
4.5 Two Boundary Layers
5. Asymptotic Expansions in Integrals
5.1 Laplace-Type Integrals for Large λ
5.2 Asymptotic Approximation of an Exponentially Weighted Integral
5.3 Asymptotic Expansions via Integration by Parts
5.4 Laplace-Type Integrals with Power Prefactors and Endpoint or Interior Maxima
5.5 WKB Approximation for Singularly Perturbed Second-Order Equations
6. Systems of Ordinary Differential Equations
6.1 Preliminaries
6.2 Fundamental Matrix
6.3 x’ = Ax
6.4 x’ = A(t)x + g(t)
7. Stability of Linear Systems
7.1 Introduction to Stability Theory
7.2 Definitions and Examples
7.3 Stability of x’ = A(t)x
7.4 Stability of x’ = Ax
7.5 Autonomous Systems in the Plane
8. Stability and Bifurcations in Non-linear Systems
8.1 Bifurcations in Scalar Systems
8.2 Stability of Systems by Linearization
8.3 An SIR Epidemic Model
8.4 Limit Cycle
8.5 Lotka-Volterra Competition Model
8.6 Bifurcation in Planar Systems
8.7 Lyapunov Functions
8.8 Lyapunov Method
Appendix: Quick Review Of Ordinary Differential Equations
Biography
Youssef N. Raffoul is Professor and Graduate Program Director at the University of Dayton. He holds a Ph.D. from Southern Illinois University. He is the recipient of the University of Dayton College of Arts and Sciences’ Award for Outstanding Scholarship (twice), and the University of Dayton Alumni Award in Scholarship. Professor Raffoul has published over one hundred and sixty articles in prestigious journals in the area of functional differential equations, difference equations, and dynamical systems on time scales. He was honored by the Lebanese government with the Career in Science Award. The Archbishop of Lebanon awarded him the Lifetime Achievement Award. Most notably, he is the recipient of the Order of Merit, Silver Medal with Distinction and awarded by the president of Lebanon, General Aoun.
