Applied Mathematics Qualifying Exams

Ordinary Differential Equations

• Initial value problem—existence, uniqueness
• One-step methods for the numerical solution of the initial value problem—explicit and implicit schemes. Numerical solution of nonlinear equations
• Stability and phase plane analysis, bifurcations
• Applications such as population models, epidemiology
• Boundary value problems; finite difference methods—numerical solution of banded linear systems
• Sturm-Liouville systems

Fourier Series and Orthogonal Expansions, Discrete Fourier Series and Convolutions

The Heat Equation

• Heat flow, Fick’s law
• Separation of variables
• Fundamental solution from Fourier transforms; scale-invariance
• Smoothing effect, maximum principle
• Finite difference methods for heat equation in one dimension

Quasilinear First-Order Equations, Characteristics, Burger’s Equation

The Wave Equation

• Separation of variables
• 1D - d’Alembert’s formula, initial-boundary value problems
• 2D, 3D: method of spherical means, Hadamard’s method of descent
• Inhomogeneous equations via Duhamel’s principle
• Domain of influence/dependence, Huygen’s principle
• Conservation of energy

Distributions and the Delta Function; Green’s Functions and Fundamental Solutions

Laplace and Poisson Equations

• Separation of variables
• Green’s representation for solution to Dirichlet problem, Poisson integral
• Mean value inequality, strong and weak maximum principles, uniqueness for Dirichlet problem
• Dirichlet Principle
• Finite difference methods for Poisson equations
• A simple finite element method

Elementary Fluid Dynamics

• Derivation of equations of motion
• Vortex dynamics
• Conformal mappings and fluid flow

Dimensional Analysis and Scaling

Perturbation Theory for ODE’s, Asymptotic Methods

• Regular perturbation
• Asymptotic series
• Multiple scales, secular terms
• Boundary layers, matching
• Asymptotic methods for integrals; Stirling’s formula