Line 10: Line 10:
 
Initial Conditions: u(x,0)= f(x), which can be seen from the diagram as
 
Initial Conditions: u(x,0)= f(x), which can be seen from the diagram as
  
f(x) = x - 1/4 for 1/4 < x < 1/2 and f(x) = -x + 3/4 for 1/2 < x < 3/4
+
f(x) = x - 1/4 for 1/4 < x < 1/2 and f(x) = -x + 3/4 for 1/2 < x < 3/4.
  
so you'll have to split up the integral when calculating A_n.
+
(Actually, the initial shape is supposed to be k=.01 times this, but
 +
that just puts the same  k  in front of the series for the solution.)
 +
 
 +
You'll have to split up the integral when calculating A_n.
 
And yes, the last Initial Condition is
 
And yes, the last Initial Condition is
  
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Question, Page 547, Problem 15:
 
Question, Page 547, Problem 15:
  
How do we show that the constant = B^4 without any boundary conditions to work with?
+
How do we show that the constant is beta^4 without any boundary conditions
 +
to work with?
  
 
Answer.  The beta to the fourth power is just a way to name the positive
 
Answer.  The beta to the fourth power is just a way to name the positive
 
constant lambda to make the solutions easier to write.  There will also
 
constant lambda to make the solutions easier to write.  There will also
be the cases lambda=0 and lambda negative (minus beta to the fourth) to
+
be the cases lambda=0 and lambda negative (= minus beta to the fourth) to
 
deal with.  You won't use boundary conditions to eliminate solutions
 
deal with.  You won't use boundary conditions to eliminate solutions
until problem 15.
+
until problem 16.
  
 
Question, Page 552, Problem 5:
 
Question, Page 552, Problem 5:
  
How do we show Pn?  I think I understand that this is part of calculating lambdas using the Sturm-Louiville, but I haven't been able to figure it out.
+
How do we show p_n?  I think I understand that this is part of calculating lambdas using the Sturm-Louiville, but I haven't been able to figure it out.
 +
 
 +
Answer:  The p_n come from the boundary conditions.  The problem is similar
 +
to the Sturm-Liouville problem on Exam 2.
  
  
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What about f(x) for this problem?  I am really having a hard time identifying the f(x) for these problems.  (Actually, this entire section in general)  Does anybody know of a good reference for example problems?
 
What about f(x) for this problem?  I am really having a hard time identifying the f(x) for these problems.  (Actually, this entire section in general)  Does anybody know of a good reference for example problems?
 +
 +
Answer:  There is no f(x) in this problem.  (You can see that initial conditions
 +
are set in the next problem, which is a continuation of the whole thing.
  
 
[[2010 MA 527 Bell|Back to the MA 527 start page]]  
 
[[2010 MA 527 Bell|Back to the MA 527 start page]]  

Revision as of 05:56, 28 November 2010

Homework 13 collaboration area

Question, Page 546, Problem 9:

Are the Boundary Conditions u(1/4,t) = 0 and u(3/4,t)=0 or u(0,t)=1/4 and u(L,t)=3/4? Also, can anyone help explain what f(x) is in this problem? I think g(x) = 0, but I'm not sure about f(x)?

Answer: Boundary Conditions: u(0,t) = 0, u(L,t) = 0. In this problem, L=1.

Initial Conditions: u(x,0)= f(x), which can be seen from the diagram as

f(x) = x - 1/4 for 1/4 < x < 1/2 and f(x) = -x + 3/4 for 1/2 < x < 3/4.

(Actually, the initial shape is supposed to be k=.01 times this, but that just puts the same k in front of the series for the solution.)

You'll have to split up the integral when calculating A_n. And yes, the last Initial Condition is

d(u)/dt(x,0) = g(x) = 0.

You'll get an ugly integral evaluation but most terms cancel and it leaves you with 3 sine terms that then go into the fourier series.


Question, Page 547, Problem 15:

How do we show that the constant is beta^4 without any boundary conditions to work with?

Answer. The beta to the fourth power is just a way to name the positive constant lambda to make the solutions easier to write. There will also be the cases lambda=0 and lambda negative (= minus beta to the fourth) to deal with. You won't use boundary conditions to eliminate solutions until problem 16.

Question, Page 552, Problem 5:

How do we show p_n? I think I understand that this is part of calculating lambdas using the Sturm-Louiville, but I haven't been able to figure it out.

Answer: The p_n come from the boundary conditions. The problem is similar to the Sturm-Liouville problem on Exam 2.


Question, Page 548, Problem 16:

What about f(x) for this problem? I am really having a hard time identifying the f(x) for these problems. (Actually, this entire section in general) Does anybody know of a good reference for example problems?

Answer: There is no f(x) in this problem. (You can see that initial conditions are set in the next problem, which is a continuation of the whole thing.

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