Purpose

The purpose of this exam is to verify that the student has a sufficient degree of understanding of the basic principles and techniques that underly the theory of ordinary (ODE) and partial (PDE) differential equations as well as demonstrate sufficient competence in solving problems based on this understanding. The level of comprehension and competence demonstrated in this exam is expected to be a measure of the degree preparation of the student to pursue research at the PhD level in the areas of ODEs and PDEs. The syllabus for the comprehensive exam is detailed below. It is expected that the courses Math 612 and Math 614 along with their prerequisites cover the topics described in the syllabus for the comprehensive exam.

Syllabus: ODEs

• Topics from undergraduate ODEs: Separation of variables, (scalar) linear equations and integrating factors, variation of parameters.
• Theory of linear constant coefficient systems x' = Ax. Generalized eigenspaces. Matrix exponentials. Stable, unstable and center subspaces. Stability of equilibria. Characterization and phase diagrams of the two dimensional case.
• Converting higher order and non-autonomous systems of equations into first order systems of the form x' = f(x).
• Existence and uniqueness of solutions of initial value problems (IVPs) for general nonlinear systems x' = f(x), Picard iterations, continuous dependence on parameters and initial conditions. Gronwall's inequality.
• Stability of nonlinear systems: Liapunov and asymptotic stability. Liapunov functions.
• Local stability of equilibria of nonlinear systems via linearization. Hyperbolic and non-hyperbolic equilibria.
• Local stable, unstable and center manifolds.
• Periodic orbits. Invariant sets. Limit sets.
• Two dimensional systems and the Poincare-Bendixson theorem.

Suggested textbooks:

1. Morris W. Hirsch, Stephen Smale, Robert L. Devaney,  Differential equations, dynamical systems and an introduction to chaos.
2. James D. Meiss, Differential dynamical systems,
3. Lawrence Perko, Differential equations and dynamical systems.

Syllabus: PDEs

• General concepts: Well posedness, boundary and initial conditions, weak solutions, fundamental solutions and Dirac delta functions.
• First order quasilinear equations: method of characteristics, Hamilton-Jacobi equations, conservation laws: shock solutions.
• Laplace's equation: Maximum principle, mean-value properties, variational characterization, smoothness, energy methods.
• Heat equation: Maximum principle, regularity, energy methods, uniqueness.
• Wave equation: Energy method, speed of propagation, domains of dependence and influence, conservation, d'Alembert solution.
• Function spaces: Bilinear forms on functions spaces, Sobolev spaces H¹, H_0¹, and convergent sequences.

Suggested textbooks:

1. E. C. Zachmanoglou and Dale W. Thoe, Introduction to Partial Differential Equations with Applications.
2. Michael Renardy and Robert C. Rogers, Introduction to Partial Differential equations.
3. Robert McOwen, Partial Differential Equations: Methods and Applications.
4. Lawrence C. Evans, Partial Differential Equations.

Relevant courses and their catalog descriptions

MATH 612: Ordinary Differential Equations [3] Matrix exponentials, linear systems of equilibria, phase diagrams, non-linear systems, existence, uniqueness and dependence on initial data, stability by linearization and by Liapunov's direct method, limit sets and LaSalle's invariance principle, periodic orbits and self-sustained oscillations, Poincare Bendixon theory, Floquet theory, gradient systems, applications to mechanical systems and predator-prey problems. Prerequisite: MATH 225, MATH 301, MATH 302 or consent of instructor. MATH 614: Partial Differential Equations [3] Quasi-linear, first-order PDEs; conservation laws; the method of characteristics; discontinuous solutions and shock waves; linear second-order PDEs and their classification; maximum principles; elliptic PDEs; Sobolev spaces and existence of weak solutions; and regularity. Prerequisite: MATH 600 or consent of instructor.