Math 199, Methods of Applied Mathematics II

Syllabus

1. Perturbation Methods

• Averaging
• WKBJ Methods
• Singular Perturbations, Matched Asymptotics
• Two-timimg

2. Elementary Nonlinear Evolution Equations

• Uniform Asymptotic Approximations to Fourier Transform Solutions of Linear PDE's
• Method of Characteristics
• Elliptic, Parabolic and Hyperbolic Equations
• Shock Formation
• Dispersive vs. Hyperbolic Equations
• Phase and Group Velocity
• Dispersive and Dissipative Systems, Amplitude Equations

3. Modulation Theory for Linear and Nonlinear Wave Equations

• Single-phase Averaging
• Whitham Equations
• Multiphase Solutions in Integrable and Near Integrable Evolution Equatiions

4. Green's Functions

• Sturm-Liouville Spectral Theory for ODE's
• Elements of Potential Theory, Laplace Equation in Two and Three Dimensions
• Applications in Electrostatics and Inviscid Hydrodynamics

5. Elementary Dynamical Systems Tools

• Invariant Manifolds, Linear and Nonlinear Systems
• Liapunov Functions
• Poincare Maps
• Nonlinear Oscillators
• Center Manifolds
• Examples of Global Bifurcations

Textbooks

Advanced Mathematical Methods for Scientists and Engineers, C. Bender and S. Orszag
Perturbation Methods in Applied Mathematics, J. Kevorkian and J.D. Cole
Linear and Nonlinear Waves, G.B. Whitham
Dynamics and Bifurcations, J.K.Hale and H. Kocak
Averaging Methods in Nonlinear Dynamical Systems, J.A. Sanders and F. Verhulst
Nonlinear Differential Equations and Dynamical Systems, F. Verhulst