Here is a list of topics covered and all written assignments for the course as well as a dynamic listing of what was actually planned and covered in the lecture. Remember: Your work on assignments should be neatly presented---if you can't write legibly, then figure out a way to type it---and submitted on Canvas.
Date Chapter.Section/Topics or Page/Problems
T Aug 20 Lecture 1 Introduction/Review of ODE
Notes
Assignment 1 Pace: Thursday August 29, 2024 DUE Tuesday September 3, 2024
highlights/actual:
A first peek at PDE and a solution of the heat equation
A first peek at ODEs
Th 22 Lecture 2 Two point boundary value problems and Fourier series
Assignment 2 Pace: Thursday September 5, 2024 DUE Tuesday September 10, 2024
highlights/actual:
A second peek at PDE: Can you solve u_t = u_xx on (0,pi)X(0,infty) with u(0,t)=0=u(pi,t) for all t and
u(x,0) = x(pi-x)?
Existence and uniqueness for ODEs
C^k regularity and real analyticity (C^omega)
y' = y^2 finite time blow-up
T 27 Lecture 3 Fourier series (Chapter 3 in Haberman)
highlights/actual:
PDE: Visualization of heat flow, 1-D boundary conditions and heat dissipation through endpoints
Two point boundary value problems in ODE
Systems and second order ODE
Assignment 1 Problem 1 (continuity)
Th 29 Lecture 4 More Fourier series
Assignment 3 Pace: Thursday September 19, 2024 DUE Tuesday September 24, 2024
highlights/actual:
Assignment 1 Problem 6 open sets
Assignment 1 Problem 3 an initial value problem (complex eigenvalues)
PDE: higher dimensions, can you solve u_t = u_xx + u_yy on (0,L)X(0,M)X(0,infty) with
u(x,y,t) = 0 for (x,y) in boundary((0,L)X(0,M)) and
u(x,y,0) = sin(pi x/L) sin(pi y/M)?
Hint: Try u(x,y,t) = f(t) sin(pi x/L) sin(pi y/M)
Laplace's PDE. Solve u_xx + u_yy on (0,L)X(0,M) with u(x,y) = 0 for (x,y) in boundary((0,L)X(0,M))
T Sept 3 Lecture 5 Review
Assignment 4 Pace: Thursday September 26, 2024 DUE Tuesday October 1, 2024
highlights/actual: Assignment 1 Problem 10 equivalent two point boundary value problem with
homogeneous boundary conditions
Linear and affine
No substantial peek at PDE :(
power series solutions for ODE
X' = AX when A has complex eigenvalues (and complex eigenvectors)
Th 5 Lecture 6 Derivation of heat equation
Two point boundary value problems in ODE and Fourier series
T 10 Lecture 7
Th 12 Lecture 8 separation of variables (didn't do this)
mollification
derivation of the 1D heat equation (start)
Assignment 5 Pace: Thursday October 10, 2024 DUE Tuesday October 15, 2024
T 17 Lecture 9 derivation of the heat equation (continued)
average values for harmonic functions (cont.)---the divergence
Th 19 Lecture 10 average values for harmonic functions
measures of balls and boundaries of balls
T 24 Lecture 11 average values for harmonic functions (cont.)---the divergence (intro)
Th 26 rain day
T Oct 1 Lecture 13-1
measure and integration---when can you not integrate functions?
the divergence (definition) and the divergence theorem (statement)
Th 3 Lecture 14-1 proof of the divergence theorem (sketch)
T 8 Lecture 15-1 constructing a non-measurable set
Fourier coefficients (Problem 9 of Assignment 5)
Assignment 6 Pace: Thursday October 17, 2024 DUE Tuesday October 22, 2024
Th 10 Lecture 16-1 Fourier series solution of the 1D heat equation
T Oct 15 *Fall Break*
Th 17 Lecture 17-1 another solution of the 1D heat equation: infinite propagation speed
Assignment 7 Pace: Thursday October 31, 2024 DUE Tuesday November 5, 2024
T 22 Lecture 18-1 regularity of harmonic functions
more on mollification and integration
Assignment 8 Pace: Thursday November 7, 2024 DUE Tuesday November 12, 2024
Th 24 Lecture 19-1
T 29 Lecture 20-1
Th 31 Lecture 21-1
Assignment 9 Pace: Thursday November 21, 2024 DUE Tuesday December 3, 2024
T Nov 5 Lecture 22-1
Th 7 Lecture 23-1
T 12 Lecture 24-1
Th 14 Lecture 25-1
T 19 Lecture 26-1
Th 21 Lecture 27-1
T 26 Lecture 28-1
Th Nov 28 *Thanksgiving Holiday*
T Dec 3 Last class meeting
Final Assignment DUE Thursday December 12, 2024 (Scheduled Final Exam Time 11:20-2:10)
Problems 1-4 posted on Wednesday August 21
Date Chapter.Section/Topics or Page/Problems
T Aug 24 ODEs and PDEs
Didn't cover:
Introduction: Partial Differential Equations; The Heat Equation
See Lecture 2 notes (typed on main course page)
Assignment 1 (corrected version) Due Tuesday September 14, 2021
Assignment 1 (original version with errors in problems 9 and 10)
Th 26 Power Series and Fourier Series
Still haven't covered:
Derivation of the 1-D heat equation (see lecture 2 notes)
T 31 NOTE: On this day office hours start at 11:30 AM.
1-D heat conduction
Assignment 2 Due Tuesday September 28, 2021
Th Sept 2 higher spatial dimension
Separation of Variables (Chapter 2).
T 7 Separation of Variables
Separation of Variables (continued)
Assignment 3 = Exam 1 Due Tuesday October 12, 2021
Note that this assignment is being posted over a month in advance of the due date,
and we have covered almost all the material needed for completion of this assignment
in lecture already. Suggestion: Assignment 1 is due in a week. After you get that
one done, pick up the pace and get Assignment 2 done a little early, so you have
plenty of time to get Assignment 3 done before Fall Break. Of course, if you
procrastinate, you can spend the first four or so days of Fall Break doing Assignment 3.
Th 9 See the mathematica notebook posted on the main page concerning graphical representation
of solutions
Lecture 6 (?)
Transport equations
The divergence and the divergence theorem
T 14 Lecture 7 Intrinsic Mathematics
Lecture 7 Intrinsic Mathematics
Th 16 Lecture 8 Heat conduction on a ring
T 21 Lecture 9 Calculus of Variations
Th 23 Lecture 10 Calculus of Variations (continued) last page
Assignment 4 Due October 26, 2021
T 28 Lecture 11 Calculus of Variations: Dirichlet energy
Assignment 2 Problem 5: Polar coordinates and the chain rule
Th 30 Lecture 12 Directional derivatives, Laplace's PDE on a disk last page
T Oct 5 Lecture 13 The mean value property and the maximum principle(s)
Assignment 5 Due November 9, 2021
Th 7 Lecture 14 Open sets and other mathematical sounding words (topology)
T Oct 12 Fall Break
Th 14 Lecture 15 Chapter 3 (of Haberman) Fourier series
T 19 Lecture 16 Fourier's theorem (pointwise convergence) last page
and eigenfunction expansion
Th 21 Lecture 17 Riesz-Fischer theorem (L^2 convergence)
and clean up of various topics:
(a) Lecture 16/Chapter 3
(b) Initial condition in eigenfunction expansion
(c) Weak maximum principle (to do list)
T 26 Lecture 18 Wave equation: separation of variables/superposition
Th 28 Lecture 19 Wave equation: derivation
T Nov 2 Lecture 20 Hanging slinky and river crossing
Assignment 6 = Exam 2 Due November 23, 2021
Th 4 Lecture 21
Trig identity/induction for Gibb's phenomenon
river crossing problem (calculus of variations)
d'Alembert's solution/method of characteristics
T 9 Lecture 22 d'Alembert's solution and Streamlines last page
Th 11 Lecture 23 Streamlines contined
Assignment 7 Due December 7, 2021
Assignment 8 = Final Exam Due December 13, 2021
T 16 Lecture 24 Transverse Oscillations and Sturm-Liouville Theory
Th 18 Lecture 25 Sturm-Liouville Theory and Bessel Functions
T 23 Lecture 26 Transverse Vibrations
1-D
2-D rectangle
2-D disk
Th Nov 25 Holiday
T 30 Lecture 27 Transverse Oscillations of a Hanging Chain
Th Dec 2 Lecture 28 Free Day
Assignment 7 Problem 1
Problem 2
1-D Internal Oscillations
(motion of a hanging slinky/spring in gravity)
T 7 Last class meeting
Lecture 29
M 13 Final Exam (11:20-2:10)