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Advanced Analytical Methods in Science and Engineering

Fall 2015

Instructor: Homer Reid

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Course Overview

View this document in PDF form: Overview.pdf

Course Mechanics

Lectures: 11 AM MWF, E17-133
Exams, PSets, etc:
  • Approximately weekly problem sets.
  • Three journal-club reports.
  • Three in-class midterm exams.
  • No final exam.
  • 30% problem sets
  • 10% journal club reports
  • 20% exam 1
  • 20% exam 2
  • 20% exam 3
Office hours:

  • Homer:
    1. By appointment
    2. Fridays immediately after class
    3. Immediately after the last class preceding problem set due dates
  • Jan: Wednesday 4-5 PM, E17-301S

Course Objective

This course is designed to help you, as a human engineer or scientist, retain some semblance of usefulness in the rapidly approaching era of total supercomputer domination.

Now, lest there be any confusion, I hasten to add that I for one welcome our new parallel-processing overlords, with their ability to crunch even the most formidable sets of numbers. It's just that I submit---humbly, meekly, with all due respect---that, even in a world in which all problems can be solved numerically, human intuition and the analytical insight that fuels it might still have some role to play. The goal of 18.305 is to plant within you the seeds of this intuition and to strengthen the analytical muscles you'll need to exploit it.

For a more detailed survey of the sorts of problems we will consider---and the sorts of methods we will develop to solve them---see this Invitation to 18.305.

Course Outline

The content we will cover may be roughly subdivided into three units.

We will begin by challenging you to describe---without using a computer---the large-t behavior of the following ordinary differential equation (ODE), describing a simple harmonic oscillator with a weak nonlinear damping:

as well as a variety of other linear and nonlinear ODEs. In your previous studies they taught you how to solve simple ODEs analytically, and perhaps how to solve complicated ODEs numerically; but you might have noticed a certain shifty furtiveness on the part of your instructors, and that's because they were hiding from you the formidable arsenal of tools---lying intermediate between those extremes---that have been developed over the past several centuries to lend analytical insight into complicated systems. Thus the goal of our first unit is to fill you in on What They Didn't Tell You About ODEs. Proceeding in rough chronological order of invention, we will cover series solutions, singular points, and the Frobenius method (18th-19th centuries); symmetry analysis and the Lie-group approach to ODEs (19th-20th centuries); and perturbation methods, including WKB and boundary-layer techniques (20th century). Our discussion will culminate with a look at the renormalization-group technique, a 21st-century method inspired by theoretical physics that subsumes many of the earlier methods into a beautiful and powerful intellectual framework.

Meanwhile, at several times during our study of ODEs, we will encounter certain ornery mathematical beasts---impossible integrals, divergent sums, stochastic forcing functions, and more---that seem to be calling out for some sophisticated taming tools. This will set the stage for the second unit of our course, The Advanced Analyst's Toolkit. We will cover methods for approximating integrals (Feynman parameters, Schwinger parameters, stationary phase, steepest descent), methods for making sense of divergent, asymptotic, or slowly-converging sums (Borel summation, Pade approximants), and a number of other techniques that will be handy additions to your mental equipment shed as you tackle tough problems in your research career.

Finally, having mastered a variety of sophisticated mathematical tools, it will be time to apply them to some examples of real-world problems in science and engineering. This content will comprise our third unit, Applications to Elliptic PDEs. As a first example of an problem that you will find more tractable after the first two units of our course, we will consider electromagnetic radiation from a pointlike dipole radiator above a dielectric substrate, where your newfound ODE mastery and advanced analysts' tools will allow to glean insight from mathematical expressions you might previously have found inscrutable.

Class Schedule

This is a rough schedule of the content we will cover and the weeks in which we will cover it. The midterm dates will not change, and the material will be covered in roughly the order listed below, but we may spend slightly more or less time on certain topics than is budgeted below.
Date Topic
Wed 9/09 Invitation to 18.305
Unit 1: What they didn't tell you about ODEs
Fri 9/11 Review: What they told you about ODEs
Mon 9/14
Wed 9/16
Fri 9/18
Local analysis of ODEs: Series expansions, singular points, Frobenius method. Applications: Hooke's atom, Kato cusps in electron wave functions.
Mon 9/21
Wed 9/23
Fri 9/25
Symmetry analysis of ODEs, part 1: Lie groups and what they can do for you. Solution of nonlinear ODEs by Lie-group methods. Scaling and dimensional analysis as Lie symmetries. G. I. Taylor's famous back-of-envelope calculation of nuclear-bomb energy yield.
Mon 9/28
Wed 9/30
Fri 10/02
Regular and singular perturbation theory: When is a small parameter not a small parameter? WKB theory, boundary-layer theory, method of multiple scales.
Mon 10/05
Wed 10/07
Fri 10/09
Symmetry analysis of ODEs, part 2: The renormalization group and the modern approach to asymptotic analysis.
Tue 10/13 Midterm 1
Unit 2: The Advanced Analyst's Toolkit
Wed 10/14
Fri 10/16
Methods for approximating integrals: Feynman parameters, Schwinger parameters, stationary phase, least descent
Mon 10/19
Wed 10/21
Fri 10/23
Working with divergent, asymptotic, or slowly-convergent sums: Borel summation, Pade approximants.
Mon 10/26
Wed 10/28
Fri 10/30
Spectral theory: Eigenvalues and determinants of differential operators. Gelfand-Yaglom method, heat kernels, zeta-function techniques.
Mon 11/2
Wed 11/4
Fri 11/6
Introduction to stochastic calculus. Just enough differential geometry to be dangerous.
Mon 11/09 Midterm 2
Unit 3: Applications to elliptic PDEs
Fri 11/13 The stalwarts: Laplace, Helmholtz, Schrodinger. Wiggles, wiggle budgets, and unwiggles. Life in unusual coordinate systems.
Mon 11/16
Wed 11/18
Fri 11/20
Scalar and vector solutions, dyadic Green's functions, contour-integral representations and analytic behavior.
Mon 11/23
Wed 11/25
The method of complex images. Sommerfeld integrals.
Mon 11/30
Wed 12/02
Fri 12/04
Spherical wave functions, Mie theory, Clausius-Mossoti factors from Bessel-function manipulations.
Mon 12/07 Midterm 3
Wed 12/09 Tying up loose ends and looking ahead

Lecture Notes

Last Updated Topic
09/09/2015 Invitation to 18.305
09/11/2015 Review: What they told you about ODEs
09/23/2015 Local solutions to ODEs
09/18/2015 Smoothness of Functions: The distinction between analytic and C functions
10/05/2015 The exponential ansatz
10/13/2015 Boundary Value Problems and Green's Functions
10/13/2015 The WKB Approximation
10/07/2015 Eigenvalue Perturbation Theory
10/21/2015 More On Green's Functions
10/27/2015 The Method of Multiple Scales
10/27/2015 Renormalization Group Approach to Nonlinear Oscillators
11/05/2015 Boundary-Layer Problems via Classical and RG Techniques
11/10/2015 Mathieu Stability via Classical and RG Techniques
11/25/2015 Power Tools for Definite Integrals
12/05/2015 Power Tools for Summation of Series

Mathematica Codes

Date Code
10/08/2015 Invert4x4System.math

Julia Codes

Date Code
11/05/2015 IntegrateODE.jl
11/05/2015 BLProblems.jl
11/05/2015 MathieuStability.jl
12/04/2015 Integrals.jl
12/04/2015 Sums.jl

Problem Sets

See the 18.305 PSet submission guidelines.

Date Due Problem Set Solutions
9/21/2015 PSet 1 PSet 1 Solutions
PSet 2
Supplemental notes on PSet 2
PSet 2 Solutions
PSet 3
PSet 3 Solutions
PSet 4
PSet 4 Solutions
11/06/2015 (problems 1 and extra credit)
11/13/2015 (problem 2)
PSet 5
PSet 6
PSet 6 Solutions
PSet 7

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There is no assigned textbook for this course, but you may find the following references helpful. Most of these may be accessed electronically (in some cases requiring MIT certificates); in cases where that is not possible, the book is on reserve in the Hayden library.

  • Bender and Orszag, Advanced Mathematical Methods for Scientists and Engineers.

    A book co-authored by an MIT physics professor and used as the textbook for past semesters of 18.305. Available electronically (with MIT certificates) from the link above.

  • Cheng, Advanced analytic methods in applied mathematics, science, and engineering.

    A book authored by an MIT mathematics professor and used as the textbook for prior semesters of 18.305. On reserve at the Hayden library.

About the Journal-Club Reports

An institution common to many fields of science and engineering---in both academic and corporate settings---is the journal club, a regular or irregular sequence of informal talks in which members of a research group take turns describing to their colleagues a research paper (typically from an academic journal) that the presenter happened to find interesting.

In a typical journal club meeting, a group of researchers---which might be a university research group, including students, postdocs, and professors, or which might be a corporate R&D group, including junior and senior engineers, staff scientists, etc.---gather, perhaps over lunch, to look over the paper in question while one of their group members summarizes it and delivers a synopsys of its most interesting points. Journal club presentations are generally characterized by the following features:

The goal of the journal-club assignments in 18.305 is to mimic the experience of an actual journal club by giving you an opportunity to see how the topics and methods covered in our course are used in the actual practice of your favorite field of science or engineering.

For each assignment, you will do the following:

Note: Make sure your submitted writeup includes a link to an online version of the paper you read so that we can peruse it while reading your report. (If you write out your report by hand, you may email us the link.)

Due dates for journal-club reports

Report Due date
JC report 1 Friday 10/16/2015
JC report 2 Friday 11/13/2015
JC report 3 Wednesday 12/09/2015

18.305 Fall 2015 Main course page