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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.
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English [CC]
FREE
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Description
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 content
- 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
Instructor
Massachusetts Institute of Technology
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