In the notes below I will refer to the textbook of Strauss as just
PDE,
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.
I have asked NYU ITS to
record my lectures and post them on NYUClasses
(click the Panopto tool in the menu on the left).
(1/28, 1/30, 2/4, 2/6) Introduction to PDEs
Review: Appendix A in APDE. Review eigenvalues/vectors and
solving linear systems of ODEs using matrix decompositions.
(1/28) Lecture 1: What is a PDE and what
can you do with it?
Reading: Sections 1.1 in PDE ; Chapter 1 in EPDE; Section 1.1
in APDE.
(1/30) Lecture 2: Boundary and Initial
Conditions
Reading: Sections 1.4 in PDE; Chapter 2 in EPDE; and
Section 2.1 in APDE.
(2/4 and 2/6) Lecture 3: Linear PDEs
Reading: Sections 1.6 in PDE ; Chapters 2 in EPDE; and
Section section 2.3 in APDE.
Review: Appendix C in EPDE, and multivariable calculus including
divergence, gradient, Laplacian, and Green's theorems. We will
not
cover boundary conditions in dimensions larger than one (pages 13-15).
Reading: The PDE book does not really use the phrase
"conservation laws" but they are there -- Example 1 in Section 1.3 derives
the advection eq. from conservation, and Example 4 in Section 1.3 derives
the heat equation; The notes follow Sections 1.2 (also example 1.15) and
1.3 and section 1.7 in APDE; opening of section 3.2 and
Section 3.2.1 in EPDE.
Note: 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. Derivations of the wave equation for a string and a drum are
given in Examples 2 and 3 in section 1.3 of PDE, while Example 7 talks
about the Schrodinger equation (quantum mechanics).
Review: Methods for solving simple ODEs, especially first
order linear equations (integrating factors) and separable equations, also
second-order constant-coefficient equations (see Appendix A of APDE).
Reading: In this suggested order: Section 1.2 in PDE; Section 4.1 in
EPDE and section on "The Method of Characteristics" in Section 1.2 of
APDE.
Reading: Section 1.6 in PDE. 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.
(Due Tue March 3rd) Homework 3: First
Order PDEs and classification
Reading: Section 2.1 in PDE; 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.
Reading: Section 14.1 in PDE. Characteristics in nonlinear
equations 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. This is more advanced material and will not be
included in exams. We will
not cover page 10.
Reading: Section 2.4 in PDE; Section 2.1 in APDE.
Reading: Section 3.3 in PDE; Section 2.5 of APDE; Section 3.3 in
PDE discusses Duhamel's principle for the diffusion equation, and section
3.4 for the wave equation. The
addendum to
my notes give another way to think about this, which I will use in
class to motivate the more formal derivation.
Here is also a summary of
differences
between advection/waves and diffusion.
Reading: Sections 2.3 and 2.5 in PDE; 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.
Reading: Distributions are discussed in section 12.1 in PDE, but
not particularly clearly in my opinion. The delta "function" 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.
Note: Distributions will not appear in the midterm and are advanced
material for your own benefit.
Spring break (and Covid-19 outbreak)
I will not follow the textbooks here. Instead of directly starting with
Fourier series, I will start by reviewing key concepts from abstract
linear algebra, with a key emphasis on concepts that will generalize to
function spaces, as we will need to solve PDEs. Appendix B in EPDE has a
(somewhat overly specific) review of linear algebra.
(3/26, NYU Classes) Midterm
Reading: I suggest starting from Section 4.1 in PDE and also
Section 4.1 in APDE; a similar process is followed in 8.1 in EPDE. After
this lecture you should be able to go through Section 4.2 in PDE
(Neumann BCs) and for more advanced material also Seection 4.3 (Robin
BCs). The beginning of section 4.1 in PDE and Example 4.5 in APDE
covers the separation of variables for the wave equation, which you
should go over (will also be covered in recitation). Note that
both APDE/EPDE assume the Fourier series has been seen by the reader --
we will learn about Fourier series next but first we will motivate why
we need to do it. A number of useful examples are worked out in Section
4.1 of APDE. Example 4.2 in APDE is a must for everyone, and more
advanced students should study Remark 4.3.
Reading: My lecture will not directly follow any book. Having
reviewed some abstract linear algebra, we will explain the concept of
function spaces and orthogonal functions, see Section 5.3 in PDE and
maybe also Section 5.3 in EPDE. Then we will cover Section 5.1 in PDE
(see also 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 (important for advanced students).
Reading: The mechanics of Fourier series are covered in Section 5.1
in PDE, but more insight is gained by using complex exponentials as in the
last subsection of section 5.2 in PDE on "The Complex Form". 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 the Olver book for a more general
treatment of operators and their eigenspaces.
We will briefly discuss how to convert inhomogeneous BCs into Laplace
equations, which we will study later. We already discussed how to handle
sources in unbounded domains (recall Duhamel's principle) but here we
will cover bounded domains.
Reading: Inhomogeneous BCs are covered in section 5.6 in PDE.
Sources and inhomogeneous BCs for the heat equation are discussed in
Section 4.7 of APDE, and also in Section 8.1.2 of EPDE (in more
generality).
Reading: These are only briefly mentioned in Section 11.4 in PDE.
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.
Reading: My lecture follows more or less Section 8.3 of EPDE, which
is also (crammed) in Section 6.2 of PDE. 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.
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.
This focuses on pre-midterm material: method of characteristics (change of coordinates) for
first-order PDEs and wave equations, heat equation on the whole real line.
Practice problems are suggested..
(skipped) 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.
(skipped) 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.
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.
Reading: None; we will need this to do numerical methods for
solving the heat equation in an interval later on.
IMPORTANT: Obtain access to MATLAB
for NYU as it will be required for the last
homework.
Maple is not available freely everywhere at NYU but is
available at
Courant.
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.
(
Optional and not graded)
Homework 7:
Numerical Methods for PDEs
(5/14, 10-11:50am) Final Exam