Review of topics

• Week 1: General introduction/motivation. First order equations and the method of characteristics. Transport Equation. [T 1.1-1.2] [B 1.1-1.5, 3.2]
• Week 2: Method of characeristics for semilinear equations. Introduction to conservation laws and quasilinear equations. [T 1.2-1.3] [B 3.1-3.4]
• Week 3: Quasilinear equation (in particular Burger's equation), classification of second order equations. [T 1.3, 1.6] [B 3.4 1.3]
• Week 4: Separation of variables. The boundary value problem for the wave and heat equations on an interval. [T 3.1, 3.3] [B 5.2]
• Week 5: Heat kernel. Uniqueness of solutions to the heat and wave equations via energy methods. The Helmholtz equation in radial symmetry. [T 3.1, 3.3, 3.4] [B 5.3]
• Week 6: Helmholtz equation in radial and spherical symmetry. [T 3.4] [B 5.3, 5.4]
• Week 7: Heat and Wave equation on the real line. Fourier transform in 1d. [B 6.2] [T4.1, 4.2]
• Week 8: Heat and Wave equations on the real line via the Fourier transform. Dispersion and Wave propagation. [T.4.3-4.5]

T: Teschl, Partial Differential Equations
B: Borthwick, Introduction to Partial Differential Equations