Linear Partial Differential Equations: Analysis and Numerics

Lecture Summaries

SES #	LECTURE SUMMARIES	HANDOUTS
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) sine-series Julia notebook
3	Finite-difference methods and accuracy (PDF)	Notes on difference approximations (PDF) IJulia notebook from the in-class demo
4	Discrete vs. continuous Laplacians: Symmetry and dot products (PDF)	No handouts
Optional	Julia Tutorial	Julia & IJulia Cheat-sheet (PDF)
5	Diagonalizability of infinite-dimensional Hermitian operators (PDF)	No handouts
6	Start with a truly discrete (finite-dimensional) system, and then derive the continuum PDE model as a limit or approximation (PDF)	Notes on from discrete to continuum (PDF)
7	Start in 1d with the “Sturm-Liouville operator”, generalize Sturm-Liouville operators to multiple dimensions (PDF)	Notes on elliptic operators (PDF - 1.2MB)
8	Music and wave equations, Separation of variables, in time and space (PDF)	18.303 and music (PDF) Music scales and intervals (PDF) Notes on separation of variables (PDF)
9	Separation of variables in cylindrical geometries: Bessel functions (PDF)	Notes on Bessel functions and cylindrical problems (PDF) IJulia Bessel-function notebook
10	General Dirichlet and Neumann boundary conditions (PDF)	No handouts
11	Multidimensional finite differences (PDF)	Notes on finite differences in 2(+) dimensions (PDF)
12	Kronecker products (PDF)	IJulia notebook lecture 10
13	The min-max theorem (PDF)	Notes on the min–max theorem (PDF)
14	Green’s functions with Dirichlet boundaries (PDF)	Introduction to Green’s functions (PDF) Notes on the 1d-Laplacian Green’s function (PDF)
15	Reciprocity and positivity of Green’s functions (PDF)	Notes on reciprocity and positivity of Green’s functions (PDF) Notes on delta functions and distributions (PDF)
16	Delta functions and distributions (PDF)	No handouts
17	Green’s function of ∇² in 3d for infinite space, the method of images (PDF)	No handouts
18	The method of images, interfaces, and surface integral equations (PDF)	No handouts
19	Green’s functions in inhomogeneous media: Integral equations and Born approximations (PDF)	Notes on Green’s functions in inhomogeneous media (PDF)
20	Dipole sources and approximations, Overview of time-dependent problems (PDF)	No handouts
21	Time-stepping and stability: Definitions, Lax equivalence (PDF)	No handouts
22	Von Neumann analysis and the heat equation (PDF)	No handouts
23	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)
24	Staggered discretizations of wave equations (PDF)	No handouts
25	Traveling waves: D’Alembert’s solution (PDF)	Notes on Fourier transforms, wave velocity, and dispersion (PDF - 2.1MB)
26	Group-velocity derivation and dispersion. Please see the related contents in Notes on Fourier transforms, wave velocity, and dispersion (PDF - 2.1MB)	No handouts
27	Material dispersion and convolutions. Please see the related contents in Notes on Fourier transforms, wave velocity, and dispersion (PDF - 2.1MB)	No handouts
28	General topic of waveguides, Superposition of modes, Evanescent modes (PDF)	No handouts
29	Waveguide modes, Reduced eigenproblem (PDF)	Notes on propagating and evanescent modes in waveguides (PDF)
30	Guidance, reflection, and refraction at interfaces between regions with different wave speeds (PDF)	No handouts
31	Numerical examples of total internal reflection (PDF)	Waveguide modes IJulia notebook Variational proof of slow-wave localization (PDF)
32	Perfectly matched layers (PML) (PDF)	Notes on perfectly matched layers (PDF)
33	Perturbation theory and Hellman-Feynman theorem (PDF)	No handouts
34	Finite element methods: Introduction (PDF)	No handouts
35	Galerkin discretization (PDF)	No handouts
36	Convergence proof for the finite-element method, Boundary conditions and the finite-element method (PDF)	No handouts
37	Finite-element software (PDF)	No handouts
38	Symmetry and linear PDEs (PDF)	No handouts