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Last updated:

December 24, 2022

Duration:

Unlimited Duration

FREE

This course includes:

Unlimited Duration

Badge on Completion

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 with a truly discrete (finite-dimensional) system, 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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Massachusetts Institute of Technology