Partial Differential Equations Preliminary Exam Syllabus

Formula-sheet (PDF) for the exam.

Texts

  1. M Shearer and R Levy (2015) Partial Differential Equations (Chapters 1-9)
  2. LC Evans (1997) Partial Differential Equations (Chapters 1-2)
  3. RB Guenther & JW Lee, Partial Differential Equations of Mathematical Physics (Chapters 1-6, 8)
  4. R Haberman, Applied Partial Differential Equations with Fourier Series and Boundary Value Problems (Chapters 2-5, 7-10, 12)

Syllabus

  1. Method of Characteristics for quasilinear first order equations
    • Existence and uniqueness theorems
    • Solution techniques
    • Shocks/Rankine-Hugoniot Condition
       
  2. Wave Equation
    • D’Alembert’s Solution
    • Duhamel’s Principle
    • Energy Methods and Uniqueness
    • Two and three space dimensions
       
  3. Heat Equation
    • Fundamental Solution
    • Energy Methods and Uniqueness
    • Maximum Principle
    • Duhamel’s Principle
       
  4. Laplace’s and Poisson’s Equation
    • Fundamental Solutions
    • Strong and Weak Maximum Principle
    • Mean Value Theorem
    • Energy Methods and Uniqueness
    • Green's functions, method of images
       
  5. Separation of variables/Fourier Series
    • Sturm-Liouville Theory
    • Solutions to Heat, Wave, and Poisson’s Equation
    • Convergence properties of Fourier Series
    • Fourier transform methods