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Last updated:

December 24, 2022


Unlimited Duration


This course includes:

Unlimited Duration

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Unlimited Duration


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

About the instructor

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Massachusetts Institute of Technology
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