NOTE: you must also email me a 4 (alphanumeric) digit pin so you
can check your grades when I begin posting them (do not use any
string from your SSN!) 

(I will maintain a news page, where I will try to answer your 
questions that might be of interest to the rest of the class
---all identifying info will be removed)
------------------------------------------------------------------
<1> Please send me email with your name, major, and a list of all
math courses w. dates when you took them. I also would like a list
of all your engineering/physics or other courses of technical content,
again with dates. It would help me to teach a better class if you also
include with each math course a list of subsequent courses for which
it was useful (if none, then list nothing!). 
 If you have some comments about the class policy I would like to know
them too!
   My email is:      vageli@math.unm.edu
Thanks,
  Vageli
------------------------------------------------------------------
<2> 8/28/03: 8:30 office hour cancelled today; I'll be at the 1-2 o.h. 
------------------------------------------------------------------
<3> 9/03/03: 15:47 
Q: I got stuck on problem 1.4.3 and wondering if you could give me some hint.
Intuatively I treat the whole rod as two separate rods one at x
= 0 to x = 1, and the other one at x = 1 to x = 2 and find the equilibrium
temperature of each one, but I can't seem to find the relationship between
the equilibrium temperature of the two.

A: you must solve for the steady state temperature on each half of the rod.
  Then impose the BC at x=0 to the solution valid on 0<x<1, and the BC
at x=2 to the solution valid at x=2. Finally impose the "perfect thermal
contact" conditions at x=1. These relate the two solutions. One condition 
is that the temperature is continuous at x=1. The other is that the heat flux
out of the left equals the heat flux into the right piece. This condition
can also be found in the answer to problem 1.3.2.
------------------------------------------------------------------
<4> 9/03/03: 17:27 
NOTE: you must also email me a 4 (alphanumeric) digit PIN so you
can check your grades when I begin posting them (do not use any
string from your SSN!)
------------------------------------------------------------------
<5> 9/03/06: 23:50
Q:I need clarification.  I'm working on exercise 2.3.8, and part b) wants 
us to solve the 'time dependent problem.'  I understand from the text 
that separation of variables works for linear homogenous, which this is 
not.  do I regard u as u(t) only, and d2u/dx2 ==> 0?  This would be 
nice, because I did this, getting f(x)= e^(-at)=u(x,0), which is the 
term that is grafted on to the general form of the (homogeneous?) 
solution provided in the back.
I understand the principle of superposition, to use the initial 
condition t=0 to satisfy the general form.

A: The equation and BC are homogeneous, since u=0 satisfies them.
You can still use the same ansatz(=substitution), i.e.
   u(x,t) = X(x)T(t)
Plug this into the equation, perform indicated differentiations etc.
and divide both sides by XT. You get, just like before, an expression
involving a function only of t on the left equal to a function only 
of x on the right. Set them both equal to \lambda as before and you get
a problem that is like those we did earlier, except for an extra term
(which is constant, \alpha). The time dependent problem that results
is easy to solve, while the space dependent problem is still a "harmonic
oscillator" problem, only you must be careful when you solve it (to 
satisfy the BC, you want oscillating solutions, only now the constants
look a bit different-just do the algebra).
 You cannot do what you are suggesting and set d2u/dx2 ==> 0 because
then you would be looking for a function that is linear in x and vanishes
at two points, i.e. it is identically zero. 
------------------------------------------------------------------
<6> 9/10/06: 21:10
Q: For problem 2.4.1 (a,b,c) do you want us to go through the solution of 
the PDE as shown in the book, or just go head and use the solution form.

A: It is OK to use the solution in Sec. 2.4.1 in the book, and find the
  expansion coefficients for each given f(x) using formulas 2.4.23, 2.4.24
------------------------------------------------------------------
<6> 9/12/06: 10.10  HW2 solutions posted in (small!) PDF format.
------------------------------------------------------------------
<7> 9/14/06: 21.10  
Q. On problem 3.3.10, which asks me the even and odd parts of f(x) = x when
x>0, and f(x) = x^2, when x<0. When I plug in the formula fe(x) = 1/2
[(f(x) + f(-x)], I get fe(x)= 0 for both x>0 and x<0. I don't know how
does the answer on the back of the book come from. 

A. Here is an example with similar structure:
        / log(x+1), x>0
f(x) = <
        \ 1/(x-1), x<0
Now, for x>0, f(-x) = 1/(-x-1) (since -x<0)
     for x<0, f(-x) = log(-x+1)  (since -x>0)
So:  
                             / (1/2){log( x+1)+1/(-x-1)} , x>0
fe(x) = (1/2){f(x)+f(-x)} = <
                             \ (1/2){log(-x+1)+1/( x+1)} , x<0
and similarly for fo
------------------------------------------------------------------
<8> 9/14/06: 21.10  
Q. On problem 3.3.3 (a). I understand how to sketch the Fourier sine
series of f(x), but I don't understand what is the "sum of a finite
number of nonzero terms", so I don't know what to sketch. 

A. Instead of the sum from 1 to infinity, only consider the sum from 
   1 to N.  The problem states that you can sketch at least the sum 
   of the first two terms in the series.
 Here is a Matlab program that sketches the sum of the square wave in part
 (b):
           /  1, 0<x<L/2
    f(x) = <
           \  0, L/2<x<L
               
                __
   Then, f(x) = >   A_n sin(n pi x/L) with A_n = (2/(n pi)) [1 - cos(n pi/2)]
                --
  and the sum ranges from 1 to infinity, so let's say you want to plot the
  sum of the first 3 terms. This sum is a function of x:
     
         f_3(x) = A_1 + A_2 cos (pi x/L) + A_3 cos(3 pi x/L) 
 etc.

    Then, the following script does the job for any N (below, N=20). 
%:::::::::::::::::::::::::::::::::::::::::::::::::::::
% script: Partial_sums 
% compute and plot partial sums of step function
   clear all;close all
   N = 20;
   L = 1; 
   x = linspace(0,1,201);
    sum = 0;
   for i = 1:N
       ipi = i*pi;
      sum = sum + 2/ipi*(1-cos(ipi/2))*sin(ipi*x/L); 
   end
   plot(x,sum)  
%:::::::::::::::::::::::::::::::::::::::::::::::::::::
------------------------------------------------------------------
<9> 09/21/03: 21:19
Q: I'm not sure if I'm on the right track for 3.4.6.

I believe the error made that is supposed to be detected is  that book 
assumes the second derivative of e^x is essentially the Fourier cosine 
series expansion of e^x.

The approach that I took to solve for An was to multiply both sides by 
cos (nPix/L) and integrate with respect to x over the interval from 0 
to L.  I integrated by parts twice - when I resulted in an integral 
that equaled An, I substituted in An and solved algebraically.  For Ao, 
I took the initial Fourier series expansion, multipled both sides by 
cos (nPix/L) and solved.  I'm not sure if I'm going down the right path 
(it seems harder than just using the existing equations for Ao and An).

A: The idea in this problem is to take advantage of the invariance of the
function under differentiation. So the second derivative is the same as
the original function (as is the first derivative). 
Apply formula 3.4.13 for the derivative of a sine series. So:
(1) differentiate the cosine series (is that OK? Why?)
(2) use formula 3.4.13 for the derivative of the sine series
(3) Now you have a new cosine series, which involves the original
coefficients A_n and is also a series for e^x.
 Equating the coefficients of this new series to those of the original cosine
series you get relationships that allow you to find the A_n. This is
equivalent to computing the integral of e^x cos x by performing two
integrations by part.


Q: For 3.4.12, in solving for the coefficients, do we need to multiply 
both sides of the equation by cos (nPix/L), or is it sufficient to 
recognize that for n=0, the only terms that contribute are the dAo/dt 
and e^-t terms; for n=3, the only terms that contribute are . . . ; for 
all other n the terms that contribute are . . .?

A: Yes, you could use a shortcut here; however I prefer that you do the
problem the long way, multiplying both sides by cos(n*pi*x/L) and
integrating, doing the whole calculation so you get used to the way these
calculations work out. Once you have walked this a few times, then it is  
OK to take the shortcuts and anticipate how they work out, but this is 
presumably the first time, so please be patient!
------------------------------------------------------------------
<10> 09/25/03: 18:14
   It was decided by popular referendum that the exam will be "take-home".
Therefore the exam I was intending as a "practise exam" is now the 
official Midterm I. It is due at the begining of class, next Tuesday.
If you missed class today, you can pick up a copy with Ms. Claudia Gans
(clouds@math.unm.edu) at the front office. She also has a copy of 
the solutions to Homework IV. Given the unexpected character of this,
if you are unable to make it to school until Tuesday's class (the officially
announced time of the exam), please contact me so we can make alternative 
arrangements.
 The rules of the situation are:
(1) Open book and notes.
(2) You are to work alone (I reserve the right of asking you to explain how
    you arrived at your results)!  
(3) The only time limit is that it all must be done by Tuesday 2pm.

Note that any similarity between the problems in the exam and the example of
heating of the earth by the sun is accidental, you are NOT expected to use
complex variables in the exam!
------------------------------------------------------------------
<11> 09/28/03: 03:20
Q: On question 3, the first condition you gave: delta u = 0. Is it a
condition to ensure that the mean value theorem applies to the problem,
or am I supposed to use this condition somewhere?

A: it means that the laplacian of u equals zero (i.e. u satisfies the laplace
   equation (Sec. 2.5) in the interior of the disk). It is up to you to figure
   if and how to apply the mean value theorem!
------------------------------------------------------------------
<12> 09/28/03: 03:20
Q: I'm not quite sure if I understand the wording on part B and C of Problem
4. Do you mean to draw the sine and cosine
expansions of the given function? and write the Fourier sine series and
Fourier cosine series of that function? and then express the coeffients
by their proper forms, but not compute the exact answer?

A: exactly. I could also have asked you to figure which coefficients vanish
given the symmetry of the function as drawn, but I didn't! All I want is
the form (i.e. sin(?), or cos(?), but leave the coefficients as a_n or b_n
etc.). I want you to draw the function correctly in the designated
interval though!
  I want to be sure you understand the periodicity intervals-there is a 
difference between the interval for the fourier series as opposed to those
for the sine and cosine series and I want you to understand what the
difference is.
------------------------------------------------------------------
<13> 09/28/03: 23:40
Q: I had a question on Problem #4.  For the Fourier series describing the 
> function, were the subscripts on the constants, a and b, supposed to 
> be k rather than n?  The summation is for k from 1 to infinity and the 
> scale on the cos and sin terms are k.

A: it should have been a_k, b_k
------------------------------------------------------------------
<14> 10/06/03;13:25
The 8:30-9:30 office hour is cancelled tomorrow (Tuesday 10/7) due to 
a meeting that I must attend. Please send your questions via email and 
I will answer them ASAP!
---------------------------------------------------------------------------
<15> 10/21/03;17:50
Q: i was looking for the matlab script on the email hotline, i didn't find it.
> where might i look?

A: Look at:
http://www.math.unm.edu/~vageli/courses/Ma312/zero.html
there are various matlab scripts there; explanation in the writeup
http://www.math.unm.edu/~vageli/courses/Ma375/IX/zero.pdf
but I would also try the simple fixed-point iteration method I presented
in class (I did not write a script for that, but here is a simple one:

==================================
x = zeros(10,1);
x(1) = 5*pi/4;
tol = .00001;
i = 1;
format long e
while abs(x(i) - tan(x(i))) >= tol
    x(i+1) = atan(x(i));
    i = i+1;
end
i
x(i)
==================================

It takes forever to converge, compared to the scripts on the web page;
I did not check if this is the right function for the homework! This
solves the equation
  x = tanx
for the smallest positive solution (you can change the tolerance if you want 
more accuracy).
---------------------------------------------------------------------------
<16> 10/21/03;23:14
Q: On problem 5.3.3 I try to find H(x) by dividing the second equation by 
the first equation but nothing really jump out, so do have any hints or 
tricks that maybe helpful for me.

A: multiply the first equation by H(x); you will get a new equation whose
   coefficients will contain H, as well as alpha, beta, gamma 
 Now look at the second equation. Expand out the derivative in the first
 term so that you can write it in a from similar to the first equation,
i.e. something times the second derivative + something times the first 
derivative etc. Now compare the coefficients of the two equations and try
to make them equal by choosing H(x) appropriately. That should give you the
forms for alpha(x), beta(x), gamma(x) and H(x).
---------------------------------------------------------------------------
<17> 10/21/03;23:14
Q: On problem 5.10.2 (I didn't quite understand the last part of this 
section where they talk about the unit length of the eigenfunction).  I 
would guess that we would use the Parseval's equality, where eigen 
function is cosine, plugin x for f then find alpha, but what about the 
weight function sigma?  would this cancel out somewhere?

A: Here you need to consider the eigenvalue problem for which cos(n*pi*x/L)
is the eigenfunction. That is a regular Sturm Liouville problem, and you can
figure out what the various functions and constants that appear in eq.5.10.14
are. Here, table 2.4.1 might help (p.69).
---------------------------------------------------------------------------
<18> 10/26/03;22:10
Q: I'm not sure what the book is asking for on 7.5.6. Do they want
something like equ. 7.5.7.
A: Yes!
---------------------------------------------------------------------------
<19> 10/26/03;22:10
Q: Should I consider a quarter-circular membrane has a periodic boundary
condition, since its angle is from 0 to half Pi?  If so, what would be
the boundary condition?  (Problem 7.7.3a)
A: 0-Pi/2 is not an interval over which we can demand periodicity. 
   The periodicity we impose on the interval 0-2*Pi follows from the fact 
   that theta=0 and theta=2*Pi correspond to the same point, but theta=0
   and theta=Pi/2 are different points. The boundary conditions at 0 and Pi/2
   can be defined as anything. However, if a function vanishes at these two
   places, then you can use the sine with an appropriate argument as your
   eigenfunction. This is equivalent to considering among all periodic 
   functions only those that happen to vanish at these two points. This
   way you can reuse some formulas from the periodic case and only consider
   Bessel functions J_m where m is an integer (which integers work?).
   In the general case you would have a wedge 0<=theta<= alpha, and you 
   would end up considering Bessel functions of a noninteger order.
---------------------------------------------------------------------------
<20> 11/01/03;23:30
Q: On problem 7.7.8, I got the same answer as the book did, except I also had
> a sin term in the answer.  What do I need to do to see that it is equal to
> zero?  I don't see how the initial condition depends on theta.

A: There is a sine term as well; the book only gives the form of the 
eigenfunction in the answer. There is nothing to select the cosine and rule 
out the sine, so they are both present.
---------------------------------------------------------------------------
<21> 11/01/03;23:30
Q: Also, for 7.7.12a, I can't find the answer in the back.  Could you post the
> answer?

A: This problem is tricky to do completely; the easy answer is
 y = C1 (1/x^2+...) + C2 (x^3 + ...)

but in reality the form of the answer (and the expressions one must
substitute to determine the form of y at all orders) is found as follows
(the "Frobenius theory of the behavior of the solution of a DE near a
regular singular point", i.e. where for the ODE
    y'' + p(x) y' + g(x) y = 0
the coefficient p(x) has a simple pole and g(x) a double pole at most).

(1) First substitute the series
y = x^s Sum (ak*x^k) into the DE

(2) When you solve for s, you find two values, s1 and s2 (here 3 and -2).
In the general case, if the two values s1 and s2 are such that
s1-s2 is not equal to an integer,
you can turn the crank and determine two series solutions to arbitrary
order.
(3) However, when the two values s1, s2 are such that
   s1 - s2 = n, an integer, then it turns out that only one solution is
guaranteed:
 y1 = x^s1 SUM (ak*x^k)

If you try

 y2 = x^s2 SUM (bk*x^k)

you may find that the process does not work (at some point you will get an
equation that cannot be solved).
(4) In that case, what works for the second solution, y2, is

 y2 = x^s2 SUM (bk*x^k)  + C*y1*lnx

with C some constant to be determined (which is also the case for the ak, bk).

In some special cases, like problem 7.7.12e it works out that C=0 and you
get  two solutions without the logarithm, but in 7.7.12a the logarithm does
come in.
 The book is not asking for this kind of detail though, just for the
leading terms near x=0 which you discover by setting y ~ x^s and solving for s.
---------------------------------------------------------------------------
<22> Mon Dec  1 11:43:45 MST 2003 
Q: I couldn't figure out problem 8.3.7, the one with a
      boundary condition U(L,t) = t.  Wouldn't you give some
      hints on finding r(x,t)?

A: the answer given for this problem is derived like in 8.2.4;  
the problem is converted to one with
homogeneous BC, by u = v + r, where r(x,t) solves non-homo. BC like
r(x,t) = t*x/L + a(x) with a(0)=0, a(L) = 0 where the function
a(x) ensures r(x,t) satisfies the heat equation:
r_t = r_{xx}
i.e.
x/L = a_xx
=> a(x) = A*x^3 + B*x^2 + C*x + D, with the constants
determined by DE and BC for a(x).
---------------------------------------------------------------------------
<23> Mon Dec  1 11:43:45 MST 2003 
There is a slight change in the assignment due Friday. The corrected
assignment is:
Set 9 (due 12/4) 8.3(1c, 7), 8.4.1b, 8.6(3a, 5, 6, 9ab)
(I ommitted case 8.6.9c)
---------------------------------------------------------------------------
<23> Wed Dec  3 15:42:12 MST 2003
> I'm confused about 8.6.9 (a). Do I need to explain the physics behind it?

A: write
Lap*u = div grad u    
and apply the divergence theorem
---------------------------------------------------------------------------
<24> Fri Dec 12 21:09:05 MST 2003
You can find a printable list of the problems in Test3 at
http://www.math.unm.edu/~vageli/courses/Ma312/solutions/exam3_312.pdf
---------------------------------------------------------------------------
<25> Sun Dec 14 21:33:32 MST 2003
There was an error in the list as given originally: the correct list is:
 Exam Part III (Ch. 7.10, 8.1-6, 9.1-3) (due in my office by 14:30, T 12/16)
  < (1) 7.10.10c, (2) 8.3.1b, (3) 8.3.6,        (4) 8.4.2,
    (5) 8.5.3,   (6) 8.5.6,  (7) 9.3.11ab-12b, (8) 9.3.23-24  >
NOTE: 7.10.10c is the problem I intended to assign here, not 7.10.9c as was
      written down!
Also, 9.3.11ab+12b is one problem (do not do 9.3.12a!)
same applies for 9.3.23-24.
For problem 8.5.3b, all I want you to do is figure out the form of the solution
when beta = 0 (no damping).
---------------------------------------------------------------------------
