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Due Monday 4/7
Read 0.1 PDEs and BVPs
Do 0.1.1 #6
0.1.2 #2
0.1.4 #1,2
0.1.5 #4
Read 0.2 Separation of Variables
(also read about cosh and sinh in your calc book, section 7.6 in Stewart
4th ed.)
Do 0.2.2 #4 - on cosh and sinh
0.2.3 #1 - separation
0.2.4 #1,3 -separated solutions
-answer to #3 in back of book is wrong
Read 0.3
Do #4,10-14
Re-read 0.3.3
Due Monday 4/14
Read 1.1
Do #7,12,13,14,15d,16d,17d
Read Ch. 1, sections 1,2 and 3 - scan the proofs
Do 1.1 #1
1.2 #16 (It's ok to use your result from #1)
1.3 #3 (hints: hints)
1.3 #10
1.3 #11 (hint:don't forget the constant of integration,
which you can get from your answer to 1.2 #16 c).
Then compare the result to that of 1.1 #1.)
1.4 #2,7 (Mimic Ex. 1.4.1)
While you're working with f(x) = x^2, you may as well plot it
and some series for it. In particular Use mathematica to obtain
the sine and the cosine series for x^2 on [0,1]. Then plot the
two series (thru several terms) on [-2,2].
Due Monday 4/21
Do 1.4 #5 (Parseval's Th. for x^2)
Read 1.5 in Pinsky
And read about centroids and moments in your Calc book.
Do
1.5 #3 (The answer is off by an extra n in the denominator.)
1.5 #9 (I get (m/6L)(1 + 2 cos(nPI/2) for alphan and alph0 = M/2L of
course)
1.5 #11 (I get 0 by employing anti-symmetry a lot, and alpha0 = M/2L of course)
Read 1.6 esp. 1.6.5 and 1.6.6
1.6 #1,
1.6 #5 (and for lambda = 0, phi = Ax is a solution if L = 1).
On #5, also use Mathematica to plot the transcendental eq and obtain
the first
two or three nonzero eigenvalues. Let L = 1.
1.6 #13,14,15
hints:#13 (correction: there should be a derivative of the right hand side.)
#14. Actually what's desired here is to redo the orthogonality.
proof using the operator notation, i.e. the SL-eq becomes
L P = - l r P (where L is the op. P=phi, r=rho, l = lambda.
outline: let P1 and P2 be two sols
L P1 = - l1 r P1
L P2 = - l2 r P2
mult by P2 and P2 resp, subtract, integrate, apply BCs,...
Due Monday 4/28
Read 2.1, 2.2, 2.3
Do 2.1 #3, 2.2 #2>
Read 2.4.
Do #14, 15, 16.
In 14, that should be an F, not an f, on the right;
and in #16 there should be no derivatives on the right.
Read 2.5
Do 2.5 #3,4,5
Due Monday 5/11
Read 3.1, 3.2 (skim 3.2.3-3.2.6 for now)
Do 3.1 #18, 19
Do 3.2
#2 (see #1)
#4 (hint: you might need #3)
#6 ( replace rho by x or vice-versa, use the
nonzero a(2n)'s
#8 ok. except for sign error
#10 (this is as bad as it looks; I found it was easiest to
start with (1/2)J(m-1) - J'(m), combining them notice that
the first term is zero, then do a shift on m, and viola!)
Read 3.3
Do #2
#9 (only the time phi-dependent part)
Due Monday 5/19
Use the Ritz method to obtain an upper bound to
the first eigenvalue of
y" + (Lambda) y = 0
y(0) = y(1) = 0
using trial functions
y1 = x (1-x)
y2 = x (1-x) (1-2x)
Use Mathematica; parrot the ex. on the handout.
Do 4.2 #2
#4, #5 (one way is by induction and using (4.2.18);
you may as well do #4 and #5 together.)
#6, #7
(you can do these generally using the definition of derivative from calc I;
or induction and the recurrence relation;
or by Rodrigues' formula.)
#8 (Big Hint = Leibnitz's Rule for the nth derivative
of a product. See Calc I again. Write
(s^2-1)^k = (s-1)^k (s+1)^k
differentiate k times using Leibnitz's rule,
and note all the stuff that's 0 at 1.)
#9 (follows from 4,5 and 8)
#11
#12 (see Ex. 4.2.4)
Due Monday 5/26
Read 5.1.1-3, 5.2.1-4 (except Th. 5.4), 5.3.1, 5.3.6
Do 5.1
#1,
#4,The ans. may be off by a sign
#11,
#13,
#14, use the def. of the trans. and a change of variable
#15,
#16,
#17,
#18
Worksheet on Cosine Transform
Due Monday 6/2
Do skipped problem from the exam.
Read Ch. 8, esp. 8.1.1 and 8.1.2
Work the problem on Green's functions from the handout.
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maintained by:
R. W. Nau: rnau@carleton.edu
Last Modified: