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 C's: u(0,t) = 0, u(L,t) = 0
Initial C's: 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 so you'll have to split up the integral when calculating An. 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. Happy Thanksgiving.
Question, Page 547, Problem 15:
How do we show that the constant = B^4 without any boundary conditions to work with?
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.
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?
