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