Materials

Introduction: 1.1-1.2

• What is a differential equation?
• What is a partial differential equation (PDE)?
• What is a solution to a PDE?
  
• Classification of PDE, examples
  
• Order (first, second, ...)
• Linear, quasi-linear, non-linear
• constant coefficient, variable coefficient
• Homogeneous or not
• Elliptic, parabolic, and hyperbolic
• 
  
• Some particular PDES

• Advection equation (one way wave equation, transport equation)
• Wave equation
• Heat equations
• Laplace equation
• Poisson equation
• Navier-Stokes equations (system)
• Maxwell equations

• Initial and boundary conditions (IVP and BVP)
  

Solution to one-dimensional (1D) advection equation: 2.1-2.7

• General solution (how to derive it?)  change variable; method of characteristics
  
• Solution of the initial value problem (formula for the Cauchy problem)
  
• Solution to  boundary value problem problem (not in the book)
• Where should we define a boundary condition
• The solution is piecewise (line equation, two regions)
• Solution to variable coefficient, method of characteristics

Solution to one-dimensional (1D) wave equation: 3.1-3.2

• General solution (formula,  how to derive it? how to use it)  change variable.
• How many initial conditions? 
• normal modes.
• Superposition
  
• Solution to some of IVP/BVP for certain initial conditions.

ODE Review: A1-A2

• Solution to first order linear ODE (homogeneous and non-homogeneous)
• Solution to n-th order constant coefficient homogeneous linear ODE

Sturm-Liouville eigenvalue problem 4.1-4.6

• Orthogonal functions:
  
• L^2(a,b) linear space,
  
• inner product, norm, distance, orthogonality, weighted inner product, normalized orthogonal set
  
• How do we know functions are orthogonal?
• Are {1, cos(mx), sin(nx) } are orthogonal? Yes,  in any 2 pi interval; otherwise they are not!
• Expansion of a function in terms of an orthogonal set.
• Sturm-Liouville theory and related eigenvalue problem
• Two point boundary value problems of second order linear ODEs
• Associated eigenvalue problems: Find a scalar and functions that satisfy the ODE and BC.

Fourier Series and expansions, 5.1-5.5.

• Periodic functions, piecewise functions, 2.1.
  
• period, piecewise continuous, smooth functions
• integration of periodic functions
• {sin (mx), cos (mx) }, floor functions , sawtooth, triangular wave.
• Expand f(x) in terms of Fourier series, when does it converge? Converge to what? Gibbs phenomenon. 2.2
  
• Partial sum; Maple code.
• Fourier expansion in (-pi, pi), (-L, L), (0, L), (a,b); ha;f range expnasion
• Parseval's identity and application
• Uniform convergence, how to judge. The M-test.

Series solution of PDEs in Cartesian coordinates, 6.1-6.5

• 1D wave equation, solve the S-L problem for space first, \sum sin(n pi x/ L) ( b_n cos( n pi c t/L) + b_n^* sin( n pi c t/L) )
• Different boundary condition, non-homogeneous BC
• 1D Heat equations \sum sin(n pi x/ L)  b_n exp( -(n pi c/L)^2  t) ; different BC, non-homogeneous BC
• 2D Laplace equation on rectangular domain. Decompose up to 4 problems, \sum sin(n pi x/ L) ( b_n cosh( n pi c t/L) + b_n^* sinh( n pi c t/L) )

Series solution of PDEs in polar coordinates, 6.6-6.7

• Solve Laplace equation on a circle by separation variables. Solve for theta variables and use the period condition; then solve for r, the solution should be bounded at r=0 (no boundary condition is defined at r=0). How to solve the Euler equation (ODE), indicial equation, solution |x|^{\alpha}, |x|^{\alpha}log(|x|), ...

Numerical solutions of PDEs, 8.1-8.4