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.) The boundary conitions are
X(0)=0 and X'(L)=0.
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.)
Okay, so I have figured out how to solve for the c1, c2, etc. I can't figure out how to back-solve for the Lambda. Any advice.
Question, Page 546, Problem 16:
Plugging in the boundary conditions, I get a set of four equations involving A, B, C, D, beta and L. I'm not sure what to do from here to solve for F(x).
Is du/dt = 0 the only given initial condition for this problem? Also, I'm able to simplify it using the B.C.s and I.C. but I'm still not getting all of the constants, nor am I getting anything in terms of n. Please go over this in class; the example problem in the book is only 2nd order and is a lot easier than this one. Thank you!
Question, Page 546, Problem 9:
Using the f(x) given in the problem and plugging into formula (17) on page 544 gives me plots that look like waves (using Maple). However the ends are not fixed. How do I incorporate this into the equation?
