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## Created by: ## Last updated:

December 24, 2022

## Duration:

Unlimited Duration

FREE

## This course includes:

Unlimited Duration

## Certificate of completion

Unlimited Duration

### Description

This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat / diffusion, wave, and Poisson equations.

Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems. The Julia Language (a free, open-source environment) is introduced and used in homework for simple examples.

### Course Curriculum

• Overview of linear PDEs and analogies with matrix algebra Unlimited
• Poisson’s equation and eigenfunctions in 1d: Fourier sine series Unlimited
• Finite-difference methods and accuracy Unlimited
• Discrete vs. continuous Laplacians: Symmetry and dot products Unlimited
• Diagonalizability of infinite-dimensional Hermitian operators Unlimited
• Start in 1d with the “Sturm-Liouville operator”, Unlimited
• Music and wave equations, Separation of variables, in time and space Unlimited
• Separation of variables in cylindrical geometries: Bessel functions Unlimited
• General Dirichlet and Neumann boundary conditions Unlimited
• Multidimensional finite differences Unlimited
• Kronecker products Unlimited
• The min-max theorem Unlimited
• Green’s functions with Dirichlet boundaries Unlimited
• Reciprocity and positivity of Green’s functions Unlimited
• Delta functions and distributions Unlimited
• Green’s function of ∇² in 3d for infinite space, the method of images Unlimited
• The method of images, interfaces, and surface integral equations Unlimited
• Green’s functions in inhomogeneous media: Integral equations and Born approximations Unlimited
• Dipole sources and approximations, Overview of time-dependent problems Unlimited
• Time-stepping and stability: Definitions, Lax equivalence Unlimited
• Von Neumann analysis and the heat equation Unlimited
• Algebraic properties of wave equations and unitary time evolution, Conservation of energy in a stretched string Unlimited
• Staggered discretizations of wave equations Unlimited
• Traveling waves: D’Alembert’s solution Unlimited
• Notes on Fourier transforms, wave velocity, and dispersion Unlimited
• Notes on Fourier transforms, wave velocity, and dispersion Unlimited
• General topic of waveguides, Superposition of modes, Evanescent modes Unlimited
• Waveguide modes, Reduced eigenproblem Unlimited
• Guidance, reflection, and refraction at interfaces between regions with different wave speeds Unlimited
• Numerical examples of total internal reflection Unlimited
• Perfectly matched layers (PML) Unlimited
• Perturbation theory and Hellman-Feynman theorem Unlimited
• Finite element methods: Introduction Unlimited
• Galerkin discretization Unlimited
• Convergence proof for the finite-element method, Boundary conditions and the finite-element method Unlimited
• Finite-element software Unlimited
• Symmetry and linear PDEs Unlimited

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Students Massachusetts Institute of Technology