Tuesday, January 4 [slides]
Organization of the course. Basic concepts: PDEs, ODEs, functions, domain and range, continuity, differentiability, directional derivatives, solution.Thursday, January 6 [slides]
Examples of simple first order ODEs. Linear algebra review. Finite differences.Monday, January 10 [slides]
Initial and boundary value problems in 1D. Discrete and continuous Green's functions. Laplacian in 1D. Initial and boundary conditions.Tuesday, January 11 [slides] [hw1]
Poisson equation in 1D. Green's functions in 1D. Introduction to electrostatics.Thursday, January 13 [slides]
Gauss' law. Poisson equation for the scalar potential. Fundamental solutions of the Laplace equation. Dirichlet problem for the Laplace equation.Monday, January 17 [slides]
Dirichlet problem. Method of electrostatic images. 5-point finite differencing.Tuesday, January 18 [slides]
Finite differences. Matlab implementation. Mean value property and maximum principle for the discrete Laplacian.Thursday, January 20 [slides]
Dirichlet energy. Dirichlet principle. Uniqueness. Maximum principle.Monday, January 24 [slides]
Advection in 1D. Linear hyperbolic systems in 1D.Tuesday, January 25 [slides]
Wave equation in 1D. Initial value (Cauchy) problem for the wave equation. D'Alambert's solution.Thursday, January 27 [slides] - (Assignment 1 due)
Examples of waves in 1D. Domain of dependence. Range of influence. Finite speed of propagation.Monday, January 31 [slides] [hw2]
Advection equation with variable coefficients. Method of characteristics. Linear hyperbolic systems. Finite differences. Courant-Friedrichs-Lewy condition.Tuesday, February 1 [slides]
Maxwell's equations. Wave equation in 3D and 2D. Poisson's formula. Huygens' principle. Method of descent.Thursday, February 3 [slides]
Spherical waves. Alternative derivation of Poisson's formula. Energy. Domain of dependence. Uniqueness.Monday, February 7 [slides]
Heat equation in 1D. Fundamental solution. Cauchy problem.Tuesday, February 8 [slides]
Convection-diffusion equation. Steady state. Connection to the Laplace equation Finite differences for the heat equation. Explicit schemes.Thursday, February 10 [slides]
Implicit schemes. Discrete maximum principle. Classification of second order PDEs.Monday, February 14 [slides]
Matlab basics.Tuesday, February 15 [slides]
Solving matrix-vector linear equations by using eigenvector bases (which is equivalent to diagonalization). Poisson boundary value problem on interval. Fourier sine series.Thursday, February 17 [slides] - (Assignment 2 due)
Orthogonality and Pythagorean theorem. Bessel's inequality. Optimality of the truncated series.February 21–25 - Study week
Monday, February 28 [slides]
Introduction to separation of variables. Heat and wave equations on interval, with homogeneous Dirichlet boundary conditions. Laplace boundary value problem on a rectangle.Tuesday, March 1 [slides]
General discussions on PDE theory. Examples of ill-posed problems.Thursday, March 3 [slides]
Separation of variables for heat and wave equations. Boundary conditions. Physical units and scaling.Monday, March 7 [slides]
Neumann boundary conditions. Fourier cosine series.Tuesday, March 8 [slides]
Fourier series. Ring problems. Inhomogeneous boundary conditions.Thursday, March 10 [slides] [hw3] - (Midterm due)
Inhomogeneous heat equation with reaction term. Inhomogeneous wave equation. Damped waves. Poisson problem.Monday, March 14 [slides]
Rectangular problems. Laplace eigenproblem. Poisson problem. Heat and wave equations.Tuesday, March 15 [slides]
Harmonic functions on the disk. Poisson kernel.Thursday, March 17 [slides]
Laplace eigenproblem on the disk. Bessel's equation. Bessel functions.Monday, March 21 [slides]
Properties of the Bessel functions. Eigenfunctions of the Laplacian on the disk.Tuesday, March 22 [slides] [hw4]
Eigenfunction expansions on the disk. Problems on the disk.Thursday, March 24 - (Assignment 3 due)
Review of separation of variables in a simple setting. Radial problems on the disk.Monday, March 28
Half disk. Exterior Dirichlet problem.Tuesday, March 29 [slides]
Some cylindrical and spherical problems.Thursday, March 31
Legendre equation. Legendre polynomials.Monday, April 4
Associated Legendre equation. Associated Legendre functions.Tuesday, April 5
Spherical harmonics.Thursday, April 7 - (Assignment 4 due)
Spherical harmonics continued.Friday, April 8
Review.