# Class schedule

*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.
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.