Lecture Summaries

1 Overview of linear PDEs and analogies with matrix algebra (PDF) A few important PDEs (PDF)
2 Poisson's equation and eigenfunctions in 1d: Fourier sine series (PDF) Fourier sine series examples (PDF)
3 Finite-difference methods and accuracy (PDF) Notes on difference approximations (PDF)
4 Discrete vs. continuous Laplacians: symmetry and dot products (PDF) MATLAB cheat sheet (This resource may not render correctly in a screen reader.PDF)
5 Hilbert spaces and adjoints for differential operators (PDF) Notes on function spaces, Hermitian operators, and Fourier series (PDF)
6 Algebraic properties of the 1d Laplacian: consequences for Poisson, heat, and wave equations (PDF)  <no handouts>
7 Laplacians in higher dimensions, and general Dirichlet and Neumann boundary conditions (PDF)  <no handouts>
8 Separation of variables, in time and space (PDF)  <no handouts>
9 Separation of variables in cylindrical geometries: Bessel functions (PDF)  <no handouts>
10 Multidimensional finite differences and Kronecker products (PDF)  <no handouts>
11 Rayleigh quotients, the min-max theorem, and estimating the first few eigenfunctions (PDF)  <no handouts>
12 Green's functions and inverse operators (PDF)  <no handouts>

Green's function of the 1d Laplacian

Reciprocity (PDF)

Notes on the 1d-Laplacian Green's function (PDF)
14 Delta functions and distributions I (PDF) Notes on delta functions and distributions (PDF)

Delta functions and distributions II

Green's functions via delta functions (PDF)

 <no handouts>
16 Green's function of the 3d Laplacian (PDF)  <no handouts>
17 The method of images, interfaces, and surface integral equations (PDF)  <no handouts>
18 Green's functions in inhomogeneous media: Integral equations and Born approximations (PDF) Notes on green's functions in inhomogeneous media (PDF)

Dipole sources and approximations

Overview of time-dependent problems (PDF)

 <no handouts>
20 Time-stepping and stability: Definitions, Lax equivalence (PDF)  <no handouts>
21 Von Neumann analysis and the heat equation (PDF)  <no handouts>

Explicit and implicit timestepping, and Crank-Nicolson schemes

Wave equations in first-order form(PDF)

Matlab demo animheat.m (M)

Algebraic properties of wave equations and unitary time evolution

Conservation of energy in a stretched string (PDF)

Notes on the algebraic structure of wave equations (PDF)

Wave equations in higher dimensions

D'Alembert's solution and planewaves (PDF)

 <no handouts>
25 Staggered discretizations of wave equations (PDF)  <no handouts>

Wave propagation examples

Phase and group velocity via Fourier analysis (PDF)

Wave equation animation animwave.m (M)

Group velocity dispersion

Waveguides with hard walls (PDF)

 <no handouts>
28 Reflection and refraction, evanescent waves, and dispersion relations (PDF)  <no handouts>
29 Waveguide eigenproblems (PDF)  <no handouts>
30 Maxwell's equations (PDF)  <no handouts>
31 Numerical simulation of Maxwell's equations: computational electromagnetism (PDF)  <no handouts>
32 Wave equations in frequency domain: Helmoltz equations and Green's functions (PDF)  <no handouts>
33 Perfectly matched layers (PML) (PDF)  <no handouts>

PML in the time domain

Finite element methods: introduction (PDF)

 <no handouts>
35 Galerkin methods (PDF)  <no handouts>
36 Tent functions and recovering finite-difference methods from the Galerkin approach (PDF)  <no handouts>
37 Symmetry and linear PDEs (PDF)  <no handouts>