18.305Advanced Analytical Methods in Science and EngineeringFall 2015Instructor: Homer Reid 
View this document in PDF form: Overview.pdf
Lectures: 11 AM MWF, E17133 Exams, PSets, etc:
 Approximately weekly problem sets.
 Three journalclub reports.
 Three inclass midterm exams.
 No final exam.
Grading:
 30% problem sets
 10% journal club reports
 20% exam 1
 20% exam 2
 20% exam 3
Office hours:
 Homer:
 By appointment
 Fridays immediately after class
 Immediately after the last class preceding problem set due dates
 Jan: Wednesday 45 PM, E17301S
Now, lest there be any confusion, I hasten to add that I for one welcome our new parallelprocessing overlords, with their ability to crunch even the most formidable sets of numbers. It's just that I submithumbly, meekly, with all due respectthat, 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 considerand the sorts of methods we will develop to solve themsee this Invitation to 18.305.
We will begin by challenging you to describewithout using a computerthe larget 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 toolslying intermediate between those extremesthat 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 (18th19th centuries); symmetry analysis and the Liegroup approach to ODEs (19th20th centuries); and perturbation methods, including WKB and boundarylayer techniques (20th century). Our discussion will culminate with a look at the renormalizationgroup technique, a 21stcentury 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 beastsimpossible integrals, divergent sums, stochastic forcing functions, and morethat 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 slowlyconverging 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 realworld 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.
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  

Local analysis of ODEs: Series expansions, singular points, Frobenius method. Applications: Hooke's atom, Kato cusps in electron wave functions.  

Symmetry analysis of ODEs, part 1: Lie groups and what they can do for you. Solution of nonlinear ODEs by Liegroup methods. Scaling and dimensional analysis as Lie symmetries. G. I. Taylor's famous backofenvelope calculation of nuclearbomb energy yield.  

Regular and singular perturbation theory: When is a small parameter not a small parameter? WKB theory, boundarylayer theory, method of multiple scales.  

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  

Methods for approximating integrals: Feynman parameters, Schwinger parameters, stationary phase, least descent  

Working with divergent, asymptotic, or slowlyconvergent sums: Borel summation, Pade approximants.  

Spectral theory: Eigenvalues and determinants of differential operators. GelfandYaglom method, heat kernels, zetafunction techniques.  

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.  

Scalar and vector solutions, dyadic Green's functions, contourintegral representations and analytic behavior.  

The method of complex images. Sommerfeld integrals.  

Spherical wave functions, Mie theory, ClausiusMossoti factors from Besselfunction manipulations.  
Mon 12/07  Midterm 3  
Wed 12/09  Tying up loose ends and looking ahead 
Date  Code 

10/08/2015  Invert4x4System.math 
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 
See the 18.305 PSet submission guidelines.
Date Due  Problem Set  Solutions  

9/21/2015  PSet 1  PSet 1 Solutions  
9/28/2015 

PSet 2 Solutions  
10/16/2015 

PSet 3 Solutions  
10/28/2015 

PSet 4 Solutions  





PSet 6 Solutions  


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.

An institution common to many fields of science and engineeringin both academic and corporate settingsis 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 researcherswhich 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 journalclub 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.)
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 