Differential Equations Topic List
For the Oral Qualifying Exam
For the oral qualifying exam in Differential Equations the candidate is to prepare a syllabus by selecting topics from the list below. The total amount of material on the syllabus should be roughly equal to that covered in a standard one-semester graduate course which has no other graduate course as a prerequisite. Once you have chosen your list discuss it with the professsor who will examine you.
Ordinary Differential Equations
- Fundamental existence theorems; uniqueness with the Lipschitz condition; the Gronwall inequality; continuation of solutions to the boundary; dependence on parameters and variational equations.
- Solution of linear systems of equations with constant coefficients; the exponent of a matrix; the Jordan canonical form and implications for the large time behavior of solutions; classification and phase portraits of 2 by 2 linear systems.
- The simplest numerical methods, order of accuracy.
- Equilibria; notion of stable and asymptotically stable equilibria; linearization about an equilibrium; stability of an equilibrium as a consequence of linearized stability; Liapunov functions and their implications for stability.
- The phase plane; limit cycles; the van der Pol equation; the Poincare-Bendixson Theorem (statement, not proof); the phase portrait of the pendulum equation; strange attractors and the Lorenz system; chaos; bifurcation of equilibria; Hopf bifurcation.
Partial Differential Equations
- Notion of well-posed problem; the classical examples (wave equation, heat equation, Laplace's equation); solution by Fourier series and Fourier transform.
- First-order equations; geometric interpretation of solutions; method of characteristics for linear and quasilinear equations; domain of dependence and influence; the simplest numerical methods for first-order linear hyperbolic equations; numerical stability and the CFL condition.
- The wave equation in one space dimension, explicit solution and energy conservation; solution in 3-D by spherical means and 2-D by descent; domain of dependence and influence for the wave equation.
- The heat equation in free space, fundamental solution, smoothing property; notion of similarity solutions; maximum principle; Duhamel's principle for nonhomogeneous problems.
- Two-point boundary value problems on an interval, and their Green's functions; Laplace's equation and Poisson's equation; the maximum principle and mean value property for harmonic functions; fundamental solutions of Laplace's equation; representation of solutions by boundary integrals; Dirichlet and Neumann problems; Green's functions (definition, half-space, disk).
- Notion of distributions, especially the delta function; distributional interpretation of fundamental solutions; weak derivatives; weak formulation of the Dirichlet problem in Hilbert space; eigenvalues of Laplacian in a bounded domain; solution of wave equation or heat equation in a bounded domain by eigenfunction expansions.