Homework 14 collaboration area
Question Page 560, Problem 11:
Are the boundary conditions for this problem u(0,t)=U1 and u(L,t)=U2 or u'(0,t)=U1 and u'(L,t)=U2? I think it is the u' option, but I'm not sure why?
Answer: The problem says that it is fixed at U1 and U2 for all time. I interpret this as X(0)=U1 and X(L)=U2, and X'(0)=X'(L)=0 since they are not changing at those points. Apply these to the X equation after you separate your variables. The T equation will come out the same as before.
RESPONSE: I don't think that X'(0)=X'(L)=0 is correct as problem 10 only mentions that the temperatures are fixed. Making the gradients zero means that the ends are insulated. Now we can have a fixed temperature even if flux is non-zero (if flux in is same as flux out). The part that i am confused about is that while solving X equation we will have non-zero solutions for all values of lambda .. so do we solve three general cases according to lambda values. But the solution at the book's back only mentions a single solution !!.
Question Page 585, Prob 6:
Can anyone provide some direction on how to start this problem? I'm not really sure how to get started on it.
Answer: I'm not sure about part b, but for part a, differentiate the two given solutions and plug them into formula five to verify that they're solutions. In part c, you're given f(theta), and to solve, you find the fourier coefficients and plug them into (20). Since f(theta) is odd, this is pretty simple.
Question Page 568, Prob 2:
What are the limits of integration for A(p) and B(p) in this problem. I think it is -infinity to infinity, but I'm not sure if this is correct.
Answer: see page 508 for a Fourier Integral refresher. You are right that it is -inf to inf.
Question Page 562, Prob 31: I am confused regarding the boundary conditions as the problem says that the faces are insulated. However the sides are at 0. So should the B.Cs be X(0)=0, X(24)=0,Y(0)=0 or X'(0)=0, X'(24)=0, Y'(0)=0?
Question Page 560, Prob 7: I am having an issue getting the solution in the back of the book. When I evaluate the integral for Bn using integration by parts, I get Bn = 4/(n^2 pi*2) * sin(n*pi/L). For even values of n, Bn is zero, so I'm not sure where the second term in the answer in the book comes from. Anybody know what I'm doing wrong?
