Note: The slides and topics for future lectures are tentative.
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.
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]
Poisson equation for the scalar potential.
Fundamental solutions of the Laplace equation.
Dirichlet problem for the Laplace equation.
Monday, January 17 [slides]
Method of electrostatic images.
5-point finite differencing.
Tuesday, January 18 [slides]
Mean value property and maximum principle for the discrete Laplacian.
Thursday, January 20 [slides]
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.
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.
Tuesday, February 1 [slides]
Wave equation in 3D and 2D.
Method of descent.
Thursday, February 3 [slides]
Alternative derivation of Poisson's formula.
Domain of dependence.
Monday, February 7 [slides]
Heat equation in 1D.
Tuesday, February 8 [slides]
Connection to the Laplace equation
Finite differences for the heat equation.
Thursday, February 10 [slides]
Discrete maximum principle.
Classification of second order PDEs.
Monday, February 14 [slides]
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.
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.
Physical units and scaling.
Monday, March 7 [slides]
Neumann boundary conditions.
Fourier cosine series.
Tuesday, March 8 [slides]
Inhomogeneous boundary conditions.
Thursday, March 10 [slides] [hw3] - (Midterm due)
Inhomogeneous heat equation with reaction term.
Inhomogeneous wave equation.
Monday, March 14 [slides]
Heat and wave equations.
Tuesday, March 15 [slides]
Harmonic functions on the disk.
Thursday, March 17 [slides]
Laplace eigenproblem on the disk.
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
Exterior Dirichlet problem.
Tuesday, March 29 [slides]
Some cylindrical and spherical problems.
Thursday, March 31
Monday, April 4
Associated Legendre equation.
Associated Legendre functions.
Tuesday, April 5
Thursday, April 7 - (Assignment 4 due)
Spherical harmonics continued.
Friday, April 8