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: For part a, I started by differentiating the two given un's and plugging them into formula 5 to prove that they're solutions. In part c, you need the fourier coefficients of f(theta) on the interval of [-pi, pi]. f(theta) is odd, so that simplifies things. I'm hoping that somebody else has the key to part b. I'm stuck there.
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?
