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:

> 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.

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Assignments:

1(a) Find a bunch of solutions of the equation: ux + uy =0 ( here the subscript denotes partial derivative)

(b) Find a PDE satisfied by the function: u(x,y) = x2 y

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

2) a) compute Δ( x2 y z )  (I can’t get a “del” symbol to print so I had to use a delta instead…sorry)  b) compute Δ2 ( x + y + z )   c) find several

solutions of  Δ2u=0.   d) Grad students: If there is only one variable,

Δ2 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, Bm , 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 = n2 assuming perfect thermal contact along the seam theta = +/- Pi

Everybody: Verify that the solutions for G(r) are  a rn  + 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, 150o 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:

An  = 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 J0 (z), showing the details of the computation. Grads do this plus the same thing for J1 (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 cm 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 cn

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(-kw2t) 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( -bw2 ) ,derive the formula for the Fourier transform of exp(- ax2 ) 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(-x2) 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