Syllabus for the Comprehensive Exam in Differential Equations

Part I: Partial Differential Equations

  1. Basic Material: Motivation and derivation of basic PDE; initial and boundary value problems; existence and uniqueness; classification of first- and second-order equations
  2. First-order Equations: Method of characteristics; transport equations; the Hamilton-Jacobi equations; conservation laws; weak solutions; solutions with shocks
  3. The Laplace and Poisson Equations: Maximum principles; mean value properties; regularity; Green's functions
  4. The Heat Equation: The fundamental solution; maximum principles; regularity; energy methods
  5. The Wave Equation: D'Alembert's formula; Kirchoff's and Poisson's formulas; domain of dependence; finite speed of propagation; Huygens' principle; energy methods
  6. Existence and Uniqueness: Well-posed PDE; characteristic and non-characteristic surfaces; power series; the Cauchy-Kowalevski and Holmgren theorems
  7. Techniques: Separation of variables and eigenfunction expansion; applications of the Fourier transform to PDE


Part II: Ordinary Differential Equations

  1. General Properties: Existence; uniqueness; dependence on parameters
  2. Asymptotic Properties: Stability; Lyapunov functions; attractors; limit sets; chain recurrence
  3. Poincaré-Bendixon Theory
  4. Linear Systems: Floquet theory; stability; nonlinear perturbations
  5. Mappings: Return mappings and time-t mappings
  6. Local Equilibria: The Hartman-Grobman theorem; stable/unstable/center manifolds; elementary local bifurcations


Suggested textbooks: Partial Differential Equations by Evans; Introduction to Partial Differential Equations by Folland; Ordinary Differential Equations with Applications by Chicone; Ordinary Differential Equations by Hale
Suggested courses: 6307 and 6341