Syllabus

Course Meeting Times

Lectures: 3 sessions / week, 1 hour / session

Prerequisites

18.06 Linear Algebra, 18.700 Linear Algebra or equivalent.

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 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.

Julia programming language (a MATLAB®-like environment) is introduced and used in homework for simple examples. Julia is a high-level, high-performance dynamic language for technical computing, with syntax that is familiar to users of other technical computing environments. It provides a sophisticated compiler, distributed parallel execution, numerical accuracy, and an extensive mathematical function library.

Textbook

There is no required text for this course, though the following books are recommended:

Strang, Gilbert. Computational Science and Engineering. Wellesley-Cambridge Press, 2007. ISBN: 9780961408817.
(emphasizing more the numerical part of the course). More information, including online chapters, can be found on Prof. Strang’s CSE website.

Olver, Peter. Introduction to Partial Differential Equations. Springer, 2013. ISBN: 9783319020983. [Preview with Google Books] (free online book)

Requirements

There will be five problem sets and a mid-term exam. There is a final project instead of a final exam. Late problem sets are not accepted, however the lowest problem set score will be dropped at the end of the term.

Grading

ACTIVITIES PERCENTAGES
Assignments 45%
Midterm exam 25%
Final project 30%

Calendar

SES # TOPICS KEY DATES
L1 Overview of linear PDEs and analogies with matrix algebra  
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  
Optional Julia Tutorial
L5 Diagonalizability of infinite-dimensional Hermitian operators Problem set 1 due
L6 Start with a truly discrete (finite-dimensional) system, and then derive the continuum PDE model as a limit or approximation  
L7 Start in 1d with the “Sturm-Liouville operator”, generalize Sturm-Liouville operators to multiple dimensions  
L8 Music and wave equations, Separation of variables, in time and space Problem set 2 due
L9 Separation of variables in cylindrical geometries: Bessel functions  
L10 General Dirichlet and Neumann boundary conditions  
L11 Multidimensional finite differences  
L12 Kronecker products Problem set 3 due
L13 The min-max theorem  
L14 Green’s functions with Dirichlet boundaries  
L15 Reciprocity and positivity of Green’s functions  
L16 Delta functions and distributions  
L17 Green’s function of ∇2 in 3d for infinite space, the method of images Problem set 4 due
L18 The method of images, interfaces, and surface integral equations  
L19 Green’s functions in inhomogeneous media: Integral equations and Born approximations  
L20 Dipole sources and approximations, Overview of time-dependent problems  
L21 Time-stepping and stability: Definitions, Lax equivalence  
L22 Von Neumann analysis and the heat equation Problem set 5 due
L23 Algebraic properties of wave equations and unitary time evolution, Conservation of energy in a stretched string  
L24 Staggered discretizations of wave equations  
L25 Traveling waves: D’Alembert’s solution  
L26 Group-velocity derivation, Dispersion  
Midterm Exam
L27 Material dispersion and convolutions  
L28 General topic of waveguides, Superposition of modes, Evanescent modes  
L29 Waveguide modes, Reduced eigenproblem  
L30 Guidance, reflection, and refraction at interfaces between regions with different wave speeds  
L31 Numerical examples of total internal reflection  
L32 Perfectly matched layers (PML)  
L33 Perturbation theory and Hellman-Feynman theorem  
L34 Finite element methods: Introduction  
L35 Galerkin discretization  
L36 Convergence proof for the finite-element method, Boundary conditions and the finite-element method  
L37 Finite-element software  
L38 Symmetry and linear PDEs Final project due

Course Info

Learning Resource Types

assignment_turned_in Problem Sets with Solutions
grading Exams with Solutions