Materials:

Introduction: Chap 1

• What is a PDE?
• What is a solution of a PDE?
• Some commonly used PDE examples
• IVP and BVP
  
• Classification of PDEs
• Mathematical: order, linearity (quasi-linear),  constant/variable coefficient(s), homogeneity/source terms
  
• How to classify second order linear PDEs? hyperbolic, elliptic, parabolic (compare with quadratic curves)
• Physical: advection,  heat, wave, elliptic equations; Laplace/Poisson equations
• Well-posedness of PDES, existence, uniqueness, and sensitivity of PDES
  
• Solution techniques 
• Analytic: PDE--> ODE
• Approximate semi-analytic; numerical solutions
  

Advection and first order linear PDEs (1D)

• Advection equation: u_t + a u_x = 0
  
• Method of changing variables: a u_t + b u_x + c u = f(x,t) --> U' + c U = F(\xi,\eta), an ODE.
  
• Method of characteristics, homogeneous system,  a u_t + p(x,t) u_x = 0.
  
• General solutions: u(x,t) = f(x-at)
  
• Cauchy problem: u(x,t) = u_0(x,t)
  
• Bounded domain, where should we have a boundary condition?   ( \bar{x} - x = a (\bar{t}- t)) solve for bar{x} or \bar{t}

Wave equations (1D)

• General solution: u(x,t) = F(x-at) + G(x+ at)
  
• Solution to Cauchy problems: D'Alembert's formula: u(x,t) = ( f(x-at ) _ f(x+at))/2 + (\int_{x-at}^{x+at} g(s) ds )/(2a)
  
• Normal modes: sin (x pi x/L) sin (x pi c t);   sin (x pi x/L) cos (x pi c t) and solution to specific initial conditions.

Orthogonal functions and Sturm-Liouville  problems (basis for separation variables)

Series solution for PDEs on bounded domain in Cartesian coordinates

Series solution for PDEs on bounded domain in polar and spherical coordinates

Fourier transform and Laplace transforms

• Definition
• Application

Numerical solutions for PDEs on bounded domain