| 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
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| 3 | Finite-difference methods and accuracy (PDF) |
Notes on difference approximations (PDF) IJulia notebook from the in-class demo
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| 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)
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| 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)
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| 9 | Separation of variables in cylindrical geometries: Bessel functions (PDF) |
Notes on Bessel functions and cylindrical problems (PDF) IJulia Bessel-function notebook
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| 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)
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| 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)
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| 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)
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| 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)
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| 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)
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| 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 |
