This is an archived course. A more recent version may be available at ocw.mit.edu.
Lectures: 3x / week, 1 hour / session
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 including operator adjoints and eigenproblems, series solutions, Green's functions, and separation of variables.
Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems, including stability and convergence analysis and implicit/explicit time-stepping.
There is no required text for this course, though the following book is recommended (emphasized more the numerical part of the course).
Strang, Gilbert. Computational Science and Engineering. Wellesley, MA: Wellesley-Cambridge Press, 2007. ISBN: 9780961408817. More information, including on-line chapters, can be found on Prof. Strang's CSE Web site.
Late problem sets are not accepted, however the lowest problem set score will be dropped.
| ACTIVITIES | PERCENTAGES |
|---|---|
| Assignments | 50% |
| Midterm exam | 20% |
| Final exam | 30% |
| SES # | TOPICS | KEY DATES |
|---|---|---|
| L1 | Overview of linear PDEs and analogies with matrix algebra | Problem set 1 out |
| L2 | Poisson's equation and eigenfunctions in 1d: Fourier sine series | |
| L3 | Finite-difference methods and accuracy | |
| L4 | Discrete vs. continuous Laplacians: symmetry and dot products |
Problem set 1 due Problem set 2 out |
| L5 | Hilbert spaces and adjoints for differential operators | |
| L6 | Algebraic properties of the 1d Laplacian: consequences for Poisson, heat, and wave equations | |
| L7 | Laplacians in higher dimensions, and general Dirichlet and Neumann boundary conditions |
Problem set 2 due Problem set 3 out |
| L8 | Separation of variables, in time and space | |
| L9 | Separation of variables in cylindrical geometries: Bessel functions | |
| L10 | Multidimensional finite differences and Kronecker products |
Problem set 3 due Problem set 4 out |
| L11 | Rayleigh quotients, the min-max theorem, and estimating the first few eigenfunctions | |
| L12 | Green's functions and inverse operators | |
| L13 |
Green's function of the 1d Laplacian Reciprocity |
Problem set 4 due Problem set 5 out |
| L14 | Delta functions and distributions I | |
| L15 |
Delta functions and distributions II Green's functions via delta functions | |
| L16 | Green's function of the 3d Laplacian |
Problem set 5 due Problem set 6 out |
| L17 | The method of images, interfaces, and surface integral equations | |
| L18 | Green's functions in inhomogeneous media: Integral equations and Born approximations |
Problem set 6 due Problem set 7 out |
| L19 |
Dipole sources and approximations Overview of time-dependent problems | |
| L20 | Time-stepping and stability: Definitions, Lax equivalence | |
| L21 | Von Neumann analysis and the heat equation | |
| L22 |
Explicit and implicit timestepping, and Crank-Nicolson schemes Wave equations in first-order form | |
| L23 |
Algebraic properties of wave equations and unitary time evolution Conservation of energy in a stretched string | |
| Midterm exam | ||
| L24 |
Wave equations in higher dimensions D'Alembert's solution and planewaves |
Problem set 7 due Problem set 8 out |
| L25 | Staggered discretizations of wave equations | |
| L26 |
Wave propagation examples Phase and group velocity via Fourier analysis |
Problem set 8 due Problem set 9 out |
| L27 |
Group velocity dispersion Waveguides with hard walls | |
| L28 | Reflection and refraction, evanescent waves, and dispersion relations |
Problem set 9 due Problem set 10 out |
| L29 | Waveguide eigenproblems | |
| L30 | Maxwell's equations | |
| L31 | Numerical simulation of Maxwell's equations: computational electromagnetism | |
| L32 | Wave equations in frequency domain: Helmoltz equations and Green's functions | Problem set 10 due |
| L33 | Perfectly matched layers (PML) | |
| L34 |
PML in the time domain Finite element methods: introduction | |
| L35 | Galerkin methods | |
| L36 | Tent functions and recovering finite-difference methods from the Galerkin approach | |
| L37 | Symmetry and linear PDEs | |
| Final exam | ||
MIT OpenCourseWare makes the materials used in the teaching of almost all of MIT's subjects available on the Web, free of charge. With more than 2,200 courses available, OCW is delivering on the promise of open sharing of knowledge. Learn more »
© 2001–2015
Massachusetts Institute of Technology
Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use.
Subscribe to the OCW Newsletter
