In the notes below I will refer to the textbook of Logan as
APDE
(Applied PDE) and to the textbook of Griffiths/Dold/Sylvester as
EPDE
(Essential PDE). Homework will be due in class
written neatly
and
stapled in order. Indicate your netid on the
solution in addition to your name.
For recitations I give a brief list below but for details see the
Recitation
Summary from J. Tong.
Review: Appendix A in APDE. Review eigenvalues/vectors and
solving linear systems of ODEs using matrix decompositions.
Reading: Chapters 1 and 2 in EPDE (first), and Sections
1.1, 2.1 and, if you have time, section 2.3 in APDE (second).
Homework 1 (due Tue 2/9, 40pts): Section 1.1, problems 3
(5pts), 5 (7.5pts), 6 (10pts), 7 (5pts), and 12(b+e, 2.5pts each,
no need to sketch the solution) in APDE. Problem 1.4 (7.5pts) in
EPDE.
Recitation 2/5: Cole-Hopf transform of the viscous
Burgers' equation, checking Green's function solution of heat
equation, dispersion relations, well-posedness, Problem 8 in APDE
(2/4 and 2/9) Lecture 4: Physical
Origin of PDEs: Conservation Laws
Review: Appendix C in EPDE, and multivariable calculus
including divergence, gradient, Laplacian, and Green's theorems.
Reading: Taught in this order: Sections 1.2 (also example
1.15) and 1.3 in APDE, opening of section 3.2 and Section
3.2.1 in EPDE, and lastly section 1.7 in APDE. If you are a
science major, especially physics, read all of chapter 3 in EPDE
and 1.4 in APDE; more advanced physics is in section 1.5 in APDE.
Homework 2 (due Tue 2/16, 45pts):
(1) The inviscid and viscous Burgers equation and the KdV equation
(see
Lecture 1) are all conservation
laws -- write down the flux function [7.5pts=2.5pts each].
(2) Section 1.2, problem 11 in APDE (see Example 1.15 for
guidance) [7.5pts].
(3) Section 1.2, problem 1 in APDE, listing
all steps
[10pts].
(4) Show that div(grad(u))=laplacian(u) in 2D and 3D [5pts]. Can
you show it to be true in any number of dimensions [5pts]?
(5) Section 1.7, problem 4 in APDE [10pts]
(6) [Extra Credit] Section 1.4, problem 12 in APDE [up to 10pts
extra]. Radially symmetric means that the solution u(r) is
known
(by symmetry arguments) to depend only on distance from the
origin, and not on the angles. Note that another way to do this is
to simply rewrite the heat equation you already know in
polar/spherical coordinates and then simplify [partial credit].
(2/11) Lecture 5: Classification
of Second-Order PDEs
Reading: Section 4.2 in EPDE is the basis of my lecture
but it simply states the results without deriving them. Instead,
I suggest also reading section 1.9 in APDE, where the
classification into hyperbolic, parabolic and elliptic is
justified from scratch. Section 4.3 in EPDE discussed
higher-order equations briefly.
(2/16) Lecture 6: First-order
PDEs: Characteristics
Reading: In this suggested order: Section 4.1 in EPDE and
section on "The Method of Characteristics" in Section 1.2 of APDE.
For more advanced students, characteristics in nonlinear equatins
are discussed in "Nonlinear Advection" in Section 1.2 of APDE, and
opening of section 9.3 in EPDE, and, for yet more advanced
material, section 9.3.1 in EPDE.
Homework 3 (due Tue 2/23, 50pts): All from Section 1.2 in
APDE: Problems 3 (7.5pts), 4 (7.5pts), 5 (15pts, 7.5pts each), and
7 (10pts, use solution of 3 to solve 7). For problem 7, draw a
picture of the characteristics showing you understand the hint
given in the problem formulation about why x>ct and x<ct are
different. For more advanced material, try problem 13 (up to 15pts
extra credit). From Section 1.9 in APDE: Problem 1 [10pts].
Recitation 2/12 and 2/19: Solution of (1.15) on page 16 in
APDE, and then examples 1.9 and 1.11 in APDE. Example 4.2 in
section 4.1 in EPDE. Problem 12 in Section 1.2 of APDE. Solving a
geneal hyperbolic equation with constant coefficients is done in
these
notes from J. Tong for
the solution of Problem 4.9 in EPDE.
(2/18) Lecture 7: The Wave
Equation
Reading: The two main textbooks do not have a single
section where the wave equation is discussed concisely, it is a
bit spread out. See Section 2.2 in APDE, and example 4.5 in EPDE,
and if you want more details, look at section 2.4 in the
optional
textbook of Olver available on SpringerLink. My lecture is
based on the optional book of Strauss but most books follow a
similar presentation.
For homework see next lecture.
(2/23 and 2/25) Lecture 8 and Lecture 9: The Diffusion Equation
Reading:
Lecture 8: Section 2.1 in APDE describes how to solve the Cauchy
problem for the diffusion equation. My lecture notes also borrow a
couple of pieces from the optional book of Strauss. Here is a
summary of
differences between
advection/waves and diffusion. Homework is included in HW4
above.
Lecture 9: Chapter 7 in EPDE. Section 2.3 in APDE
(especially example 2.3). The maximum principle for the Laplace
equation (similar to the heat equation) is derived in Theorem 1.23
in Section 1.8 of APDE.
Homework 4 (due Tue 3/1, 45pts): From EPDE: Problem 4.7
[15pts, 7.5pts for each of the two parts of the problem], 4.13
[10pts] (also for your own benefit do 4.9 for more general change
of variables). From APDE: Problem 6 in Section 2.2 [use
d'Alambert's formula, 10pts]. Graphing the solution is 5pts extra
credit. From APDE: Problem 1 in Section 2.1 [10pts].
Recitation 2/26: Problem 13 in section 1.2 of APDE.
Solution of heat equation starting from a Gaussian.
Characteristics of the inviscid Burger's equation (shocks).
(3/1) Lecture 10: Duhamel's
Principle
Reading: Section 2.5 of APDE
Homework 5 (due Tue 3/8, 15pts): All from Section 2.5 of
APDE:
1) Problem 2 but replace sin(x) by exp(-|x|) [5pts, do as many of
the integrals as you can]
2) Problem 3, see the last page of the lecture for the answer but
give all steps of the derivation [10pts]
(3/3) Lecture 11: The Delta
"Function"
Reading: The delta function is not properly discussed in
the textbooks we are using, but it is explained very well in
section 6.1 in the
optional
textbook of Olver freely available to you on SpringerLink.
My lecture notes are based on this textbook but with some
additional insights regarding Duhamel's principle.
Recitation 3/4: Problem 4.7 in EPDE - solving a more
general hyperbolic equation. Advection equations with a right hand
side and Duhamel's principle, with and without time invariance.
Delta function as a limit of sequences of functions.
(3/8) Review and (3/10)
Midterm
Spring break!
Reading: I suggest starting from Section 4.1 in APDE,
and a similar process is followed in 8.1 in EPDE. Note that both
of these assume the Fourier series has been seen by the reader
-- we will learn about it next but first I wanted to motivate
why we need to do it.
A number of useful examples are worked out in Section 4.1 of
APDE. Example 4.2 is a must for everyone, and more advanced
students should study Remark 4.3. Example 4.5 in of APDE
covers the separation of variables for the wave equation which
will be done in recitation as well.
Recitation 3/25: Separation of variables for heat
equation with homogeneous Neumann BCs, as well as periodic BCs.
Also wave equation with homogeneous Dirichlet conditions.
Reading: After reviewing some abstract linear algebra,
we will explain the concept of function spaces and orthogonal
functions, see Section 5.3 in EPDE for a quick start. Then we
will cover Chapter 3 in APDE, focusing on Fourier Series. If we
have time, I will also explain the Fourier transform on the
whole real line, discussed in Section 2.7 in APDE.
Homework 6 (due Tue 4/5, 80pts):
(1) Prove by direct integration that the complex exponential
functions exp(i*n*pi*x)/sqrt(2) on the interval [-1,1] are
orthonormal [5pts].
(2) Problems 1 in Section 4.1 of APDE [20 pts].
(3) Problem 2 in Section 4.1 of APDE [30pts = 7.5+10+7.5+5 for
parts a-d respectively]. [Hints: For problem 2, use a
computational tool to make the plots (and explain what you
used); this is good practice to prepare for our venturing into
numerical solutions of PDEs. Some nice plots (you do not have to
do as well) are shown in Figs. 8.2 and 8.3 in EPDE, make sure to
examine and understand those before doing the homework.]
(4) Problem 8.4 in EPDE [25 pts].
Recitation 4/1: Convergence in L_2 norm, heat equation
with spherical symmetry (Section 4.5 in APDE), first look at
Laplace's equation on a disk.
Reading: The basic theorems about convergence of Fourier
series are given in Section 3.2 of APDE, in particular the section
on Convergence. But a much more detailed and more pedagogical
discussion can be found in section 3.5 in the
optional
textbook of Olver freely available to you on SpringerLink.
Advanced students can also consult Chapter 9 of this book for a
more general treatment of operators and their eigenspaces.
Reading: My lecture follows more or less Chapter 5 of EPDE,
although several of the sections there were covered earlier --
this lecture focuses on sections 5.3.1, 5.3.2 and 5.4. In APDE, SL
problems are discussed in Section 4.2 and 4.3. Both texts discuss
weighted SL problems, which we will not do in class, but advanced
students should examine that. Note that solving SL problems on
paper is often hard and we will do it using computers later on.
Homework 7 (due Tue 4/19, 60pts):
1) Problem 1 in Section 4.2 of APDE [10pts].
2) Problem 5.4(b) in EPDE [7.5pts].
3) Problem 5.28 in EPDE [15pts, take a look also at Example 4.13
and Problem 3 in 4.2 of APDE].
4) Problem 6 in Section 4.3 of APDE [take a look also problem 5.8
in EPDE to get you started) [12.5pts].
5) Problem 4 in Section 4.3 of APDE, but only the second part
(find eigenvalues and eigenvectors) [15pts. Hint: The SL two-point
BVP you will get is the s-called Cauchy-Euler equation, discussed
in Appendix A of APDE. It's solutions are power law functions].
Reading: My lecture follows more or less Section 8.3 of
EPDE. Some more advanced material can be found under "General
Results for Laplace's Equation" in section 4.4 of APDE (the
example in section 4.5 will be covered in recitation), and for
more advanced material on the Poisson equation see Section 4.8 in
APDE.
Recitation 4/8: Section 4.5 in APDE, separation of
variables for the Laplace equation in a wedge (section of a
circle).
Reading: Sources and BCs for the heat equation is discussed
in Section 4.7 of APDE, and also in Section 8.1.2 of EPDE (in more
generality).
Homework 8 (due Thur 4/28, 60pts + 20pts extra credit):
1) Problem 1 in Section 4.4 of APDE [15pts, look at example 8.12
in EPDE].
2) Problem 3 in Section 4.7 of APDE [20pts = 10pts for each half
(find the formula and then show it is...)].
3) Problem 5 in Section 4.7 of APDE [15pts]. Here Q and u_0 are
constants.
4) Problem 8.13 in EPDE. Here it is OK to use the solution (4.70)
from APDE, but also use computer software to plot u(0,t) by using
a certain number of terms in the infinite series (explain what you
did) [10pts].
5) Extra Credit: Problem 8.6 in EPDE. [20pts extra credit]
Recitation 4/15: Heat equation on a semi-axes
(x>0,t>0) with Neumann and Dirichlet conditions using the
reflection principle. Problem 9 in section 4.7 of APDE (wave
equation with inhomogeneous data and a soruce term).
(4/14) Solving ODEs using Maple & Matlab
Before we try to solve PDEs using Maple & Matlab we need to
review how to solve ODEs numerically. I will rely on notes from
my ODE class for this, also see these brief notes on Euler's method
(which should be familar to most everyone already).
(1) Using Maple (symbolic algebra tool with some numerical
abilities, similar are Mathematica and
Sage)
Here is a Maple script (execute 'xmaple
ODE_Maple.mw'). Here is
a
PDF and an
html
version.
(2) Using Matlab (interpreted language for numerical computing,
similar in some ways to using scipy+numpy in python but generally
easier to use for beginners)
Here is a Matlab script
ODE_Matlab.m
(run matlab and then execute 'ODE_Matlab') which solves the
pendulum equation studied using Maple above, both using built-in
methods and from scratch using Euler's method. And here is a more
advanced script
PredatorPrey.m
to solve the
Lotka-Volterra
equations.
(4/19 and 4/21) Solving PDEs using Maple
Here is a Maple worksheet (
Maple,
PDF,
html) showcasing some
of the power (and limitations) of the PDETools package, focusing
on
analytical solutions and the facilities for
plotting/animating those solutions. One of the problems I try to
solve there using Maple is
Problem
9 in section 4.7 of APDE, which was done in recitation.
And here is a Maple worksheet (
Maple,
PDF,
html) showing how to
solve PDEs
numerically with Maple (better done with
Matlab), at this point without really knowing how Maple does this
beyond the basic idea that there is a space and time step size
that need to be set correctly.
We start by simply using Matlab to evaluate and plot the
analytical series solution we got in class and from Maple. Just
run
HeatAnalytical from the
Matlab command line.
In the code
HeatNumerical we
obtain the same solution numerically using the
Discrete Sine Transform (DST), as
implemented in the Matlab PDE toolbox function
dst.
The main difference with the analytical code above is that now we
obtain the Fourier coefficients not by doing an analytical
integral but rather by doing a discrete sum. That is, we have now
completely converted the problem from a PDE to a system of ODEs,
which here is trivial to solve since the ODEs are uncoupled and
simple linear first-order ODEs that have a solution
exp(-lambda^2*t) that we know and can use; no time stepping like
Euler's method is needed here. The method we have implemented here
is called a
spectral method and is in fact the best method
there is for solving a linear PDE with simple boundary conditions.
Note that for periodic solutions the DST is replaced by the
Fast
Fourier Transform (FFT), which is why you will see calls to
fft and ifft in the example below.
As an example of what a real "state-of-the-art" code to solve a
nonlinear PDE may look like, here is a
pseudospectral
code to solve the
KdV
equation (u_t+uu_x+u_xxx=0) written by A. K. Kassam and L.
N. Trefethen with some small changes by me. It illustrates soliton
solutions but you can easily change the initial condition as
shown. It uses a fancy "exponential integrator" method for time
stepping a system of ODEs in time. Here is some
explanation
of the method used here and a
simpler code (but less
accurate!) that solves the last problem in the Maple worksheet
NumericalSolutions (
PDF,
html) from the previous
lecture and makes the animation that we couldn't get in Maple.
This focuses on pre-midterm material: method of characteristics
for first-order PDEs and wave equations. Practice problems are
suggested. For one of them, recall that solving a geneal
hyperbolic equation with constant coefficients is done in these
notes from J. Tong for the
solution of Problem 4.9 in EPDE.
In the second part of the review we focus on post-midterm
material, notably, method of separation of variables for
second-order PDEs in a finite domain with Dirichlet, Neumann or
periodic boundary conditions. Focus should be on the heat and
Poisson equations.
(5/12, 10-11:50am) Final Exam