Syllabus

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. Studies 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 timestepping.

Topics: Diffusion, Laplace equation, Poisson, wave equation, separation of variables, Fourier series, Fourier transform, eigenvalue problems, Green's function, Heat Equation, Helmholtz equation, Sturm-Liouville Eigenvalue problems, quasilinear PDEs, Bessel functions.

Figure: Solution of the time-harmonic wave equation in an open waveguide. Credits: Bruno, Garza, Perez-Arancibia, 2017.

Prerequisites: linear algebra (18.06, 18.700, or equivalent). The assignments will involve computer programming in the language of your choice (Matlab recommended).

Reference material

Recomended textbook: Introduction to Partial Differential Equations by P. Olver (an electronic copy can be downloaded from the MIT library).

Material from previous years: Previous versions of the course taught by Prof. Steven Johnson: Fall 2016, Fall 2014, Fall 2010.

Who, when, and where

Date and Time: Tu-Th, 1:00-2:30pm, room 2-135.
Instructor: Carlos Pérez-Arancibia. Email: cperezar at mit dot edu
TA: Andrew Rzeznik. Email: rzeznik at mit dot edu.
Office hours: Carlos: M-W, 1:00-2:30pm, room 2-232B. Andrew Tu-Th 4:00-5:00pm, room 2-333C.

Evaluation

Grading: 45% homework, 25% mid-term (Mar. 22), 30% final project (May 17).

Homework: Problem sets are due in class on the due date. No late submissions will be accepted. The lowest pset score will be dropped at the end of the term. The homework problem sets will consist of both theoretical problems and numerical experiments.

Collaboration policy: You should think about the problems yourself before discussing them with others or googling them. Moreover, you must write up your solutions on your own. If you collaborate with other people in class, you must acknowledge your collaborators in your write-up.

Final project: You should consider a PDE, ideally in 2d, or possibly a numerical method not treated in class, and write a 5–10 page academic-style paper that includes:

Review: Why is this PDE/method important, what is its history, and what are the important publications and references? (A comprehensive bibliography is expected: not just the sources you happened to consult, but a complete set of sources you would recommend that a reader consult to learn a fuller picture.)

Analysis: What are the important general analytical properties? e.g. conservation laws, algebraic structure, nature of solutions (oscillatory, decaying, etcetera). Analytical solution of a simple problem.

Numerics: What numerical method do you use, and what are its convergence properties (and stability, for timestepping)? Implement the method (e.g. in Matlab or Julia) and demonstrate results for some test problems. Validate your solution (show that it converges in some known case).

You must submit a one-page proposal of your intended final-project topic, summarizing what you intend to do, by April 5.

Lecture material

• Lecture 1. See sections 11.1, 11.6 in P. Olver's textbook.
• Lecture 2. Fourier sine series examples (by Prof. S. Johnson). See also sections 3.1 and 3.2 in P. Olver's textbook. Matlab examples
• Lecture 3. Notes on finite differences (by Prof. S. Johnson). See also section 5.1 in P. Olver's textbook. Matlab examples
• Lecture 4. Handout. Notes on inner products, function spaces and Hermitian operators (by Prof. S. Johnson). See also section 9.1 in P. Olver's textbook. Matlab examples
• Lectures 5 and 6. See sections 9.2 and 9.4 in P. Olver's textbook.
• Lecture 7. See sections 9.5, 11.1 and 11.2 in P. Olver's textbook.

• Midterm
• Midterm solution

• Problem sets

• Problem set 1: due 02/22.
• Problem set 2: due 03/01.
• Problem set 3: due 03/08.
• Problem set 4: due 03/15.
• Problem set 5: due 04/12. Poisson 2D matlab example. Explicit and implicit heat equation examples.
• Problem set 6: due 04/19.
• Problem set 7: due 05/03.