AMAT416/516 Partial Differential Equations

MWF
9:20-10:15 ES 146

Instructor:
Professor Edward Thomas, ES 132F

Email : ethomas@albany.edu

Office
hours: TBA

Text:
“Applied PDEs with Fourier Series, etc”, Fifth Edition, by Richard Haberman

Topics include solution techniques for the
Heat and Wave Equations and Laplace’s Equation ( basic PDEs of Physics ), separation of
variables, Fourier series, orthogonal functions, Bessel functions and Legendre
polynomials. The focus will be on the formulation and solution of a variety of
concrete applications….we’ll emphasize computation rather than mathematical
theory. Our goal is to cover most of Chapters 1-4 and Chapter 7, with
additional topics as time permits.

Grading: There will be daily assignments
which will be collected, graded and promptly returned to you. These will count a significant portion of
your grade….at least one third and perhaps more. Since problems are time consuming,
good exam questions are hard to come by, so we will rely heavily on the
homework as a measure of progress. What we do about a midterm and final will be
determined a few weeks into the course.

The instructor is an
old man and can be cranky at times, so here are some ground rules:

> Please
get to class on time. Be in your seat, ready for action at 9:20.

> Make
sure you have the text on the first day of class. Not having it will
indicate a lack of preparation.

> Absolutely no cell phones or other electronic devices
will be used while class is in session. Turn them off and leave them off. Please don’t test me on this.

…………………………………………………………….

Assignments:

1(a) Find a bunch of solutions of the equation: u_{x }+
u_{y }=0 ( here the subscript denotes partial derivative)

(b) Find a PDE satisfied by the function: u(x,y) = x^{2 }y

c) ditto for u(x,y) =
e^{-y }sin(x)

2) a) compute** ****Δ****(** x^{2 }y z )
(I can’t get a “del” symbol to print so I had to use a delta
instead…sorry) b) compute Δ^{2 }( x + y
+ z ) c) find several

solutions of Δ^{2}u=0. d) Grad
students: If there is only one variable,

Δ^{2 }u=0
reduces to an ODE. What is the general solution of this equation?

3) Solve the steady state
heat equation for a thin rod of length L=42 ( inches) with u(0) =100, u(42) =
14 ( temps in degrees). What is the temperature at x = 36.8 ? at x=.03? at x=3
?Hint: when x =15, u (x) is approximately equal to 69.3 degrees.

4) page 51 2.3.1 (b), (c) and (f) and 2.3.2 (a) and (b)
( with lambda >0 )

5) page 52 # 2.3.5 (
orthogonality of the sines) plus, for you 516 students, compute the general
Fourier coefficient, B_{m }, for F(x) = x.

Now… in the Senility Department…. when I was
verifying that the Fourier series actually adds up to the initial temperature
f(x) = 100, I forgot to include the sine terms in the infinite series ( whata dope!).
As you will see next class, this makes the series add up to 100 exactly.

6) First, in the series
expansion 100=400/ Pi(
1 – 1/3 + 1/5…), compute the RHS going out to 7 terms and then 8 terms.

Second, take the full solution for u( x,t).
take L = Pi and take x
= Pi/2. Using the first two terms of the series, compute the temperature when t
= 1/10 and then when t = ½. ( the temp should drop considerably in that time)

By the way,
back about two lectures ago, we started taking
k=1 for simplicity. Thanks to Cody for pointing this out.

7) Take k = 1.Solve the
heat equation with insulated boundaries and initial temperature f(x) = ½ for
0< x <L/2 and 1 for L/2 < x < L. Expand f(x) in a cosine series,
write out the complete solution, and say what happens as x approaches infinity
( steady state)OOPS I should have said “as t approaches infinity”.

8) A handout on expanding
f(x) , -L<x<+L, in a Fourier sine /cosine series.I’ll leave a couple on
my door.

9) Another handout on
shifting boundary conditions around.
I’ll leave some copies on my door. Added Thursday AM: Couple of remarks
about this problem. First, it might help to rephrase things like this. Given a
function u(x,y) that satisfies Laplace’s equation, define a new function v(x,y)
= u( L-x,y). Then v also satisfies Laplace’s equation. ( When we say a function
satisfies Laplace’s equation, that means
it satisfies the equation for all values of the two independent
variables.) The second question is simply to figure out what the boundary
condition on the function u becomes for the function v.

10) We saw that the
temperature at the midpoint of our plate does not vary linearly across the
plate. Taking L=1 and using the first two terms of the series, estimate the
temp at the midpoint when H=1, then when H=5. Grad students, analyze the
temperature as a function of H as H goes to infinity.

On another tack, if you fix a value of
H, someone might want to plot the
temperature across the plate at height H/2 as a function of x…that would be
cool.

11) Fix L=H=1 and estimate
values of u(x,1/2) as x=.5,.6,.7, etc.
approaching x=1. ( You could use the average of the first two values of the
series as the estimator.) Then, in the
polar form of Laplace’s equation in a disk, substitute u(r,theta)=
G(r)Phi(theta) and separate variables as discussed in class.

[ I’m sorry that I can’t
get Greek letters to appear when I go to post these notes, so I have to write
“theta” , etc.)

12) Grads: Verify that the
eigenvalues for the disk are lambda = n^{2} assuming perfect thermal
contact along the seam theta = +/- Pi

Everybody: Verify that the
solutions for G(r) are a r^{n }+ b r^{-n }if r is not equal to
0, and c + d ln(r) if r = 0. Suppose we are given the boundary function
f(theta) =0 for –Pi < theta<-Pi/2,
150^{o} from there to+ Pi/2, 0 from there to +Pi. Calculate the
temperature values at the origin and at r=a/2, theta = 0, Pi/2. Pi, 3Pi/2.

13) Compute the Fourier ( sine and cosine) series
for the function f(x) = 0 for –L<x<0, 15 for 0<x<L ( I hope I
remembered that 15 right. In particular check the jump condition.

Secondly, compute the Fourier series for the
even extension of f(x) =sin(Pi*x/L) on the interval –Pi < x < Pi. Then
take L=1 and add up the first four nonzero terms at x =1/2.

Grad special) Grad students:
compute the integral of sin(nPix/L) times cos( m Pix/L) on the interval
[0,L]Oops…I might have said to take L=1 in this problem. If you want to do it
with L=1, go ahead.

14) **The
Wave Equation** In “plucked string”,
show that the second initial condition implies that all Bn = 0. Then you want
to expand f(x) in a sine series, so your next step is to compute the formula
for f….. go ahead and do this.

This will set you up to compute An. But,
you may assume ( as a gift) that all the even Ans are 0 ( Grads prove this) and
the formula for the odd An is what I gave in class.

Actually, let me change that formula:

A_{n} = 4/L _{ }times the integral from 0 to L/2 times(
2h/L)times x sin(nPi/L)x dx

Finally write out the
formula for u(x,t) …you only need to write the first few terms so the pattern
is obvious.

( It goes without saying
that you didn’t want to miss class on this particular day.)

15) page 142 # 4.4.3( b)

16) Make the change of
variables z = sqrt( lambda) times r to obtain Bessel’s equation in the form
7.7.25 on page 299. Be explicit about how the chain rule comes into the
computation.

17) Find the first 5 terms
of the power series expansion for J_{0 }(z), **showing the details of the computation**. Grads do this plus the same
thing for J_{1 }(z)

18) Draw the nodal
patterns for : m =3, n=2, sine; m=1,n=3, cosine; and m=4,n=2, sine

EXAM I
Wednesday October 23

19) A variety of easy
problems to get comfy with complex variables ( you kinda had to be there) plus
516 students be ready to present a derivation of Euler’s Formula based on
Maclaurin series.

20)
i) Verify the orthogonality relationship for complex exponentials ( page 129 in
the text) ii) compute c_{m }for
the step function given in class and iii) grad students be ready to prove that
if f(x) is real then, in the complex Fourier series for f, we must have c_{-n
} equal to the conjugate of c_{n}

21) page 447 # 10.3.6 plus Grad Students be prepared to
present the shift formula #10.3.5

22) page 447 # 10.3.1 and 10.3.3 plus 516 students do number
10.3.7 as discussed in class. One among you will be asked to present this.

23) Two problems…first show directly that exp( -1wx) times
exp(-kw^{2}t) solves the infinite heat equation ( where w = omega);
second, find the solution of the infinite heat equation with initial
temperature distribution f(x) = 0 if abs(x) > 5, 3 if abs(x) <5. Grade
this solution on a scale of 1 to 10 , in order of increasing usefulness.

24) Two grad student presentation opportunities plus for
everyone’s enjoyment: starting with the formula for the inverse Fourier
transform of exp( -bw^{2} ) ,derive
the formula for the Fourier transform of exp(- ax^{2
}) where, of course, a,b and w
stand for alpha, beta and omega.

25)
A handout with three problems relating our
solution of the heat equation on the infinite rod to the error function.

26)
Solve the heat equation in an infinite rod with k = 16 and initial temperature:
f(x) = 15 if x<0 and 0 if x>0.

Estimate
the values of u(x,t) at x = 1 and 2 for t = 1, 5 and 10. It’s certainly OK to use a program to access
the error function…just make it clear what you’re doing.

Also, some folks asked about the little PDE
book that I like, now out of print. It’s “ An Introduction to PDEs for Science
Students” by G. Stephenson.

27)
page 541 # 12.3.1 Take c=1 and do a series of “stop action” pictures of the
traveling wave solution with t = 0, ¼, ½, ¾. 1. 2

Plus,
solve the wave equation with f(x) = exp(-x^{2}) and g(x) =0. Stop
action sketch the solution for t=1,2,3,4…avoiding the pitfall.

28) Draw a stop action sequence of the solution
of the struck string problem with g(x) = 1 for x>h or <-h and x for
–h<x<h.

Plus 516 students, be prepared to present how
you get from the wave equation in x,t coordinates to the wave equation in xi
and eta coordinates and then how you get from that simple equation to the
solution form F(x+ct) + G (x-ct)

29) Give a stop action sketch of the
solution of the semi-infinite wave equation with g(x)=0 and f(x) = 1 if 4 < x
< 5, f(x) =0 otherwise. Take c =1 if you want
and indicate time values