9.5 # 14

[Chad S]

This problem states: "at time t = 0 its two ends are insulated" Does
this mean the ends aren't insulated at time t = t? If so, I don't
think we have enough information to answer this problem. Also, would
you mind hinting at how to get the k value on this problem?

Thanks

[Chad S]

Forgive me, I just so the k value was given to us online. I am still
curious about the wording of the Neumann conditions, however.

[Chad S]

Or am I reading this completely wrong? If u(x,0) = 2x, 2x is an odd
functions, so can it even be Neumann?

[Professor Laugesen]

I don't have my book with me right now to check the wording, but if
the ends are said to be insulated then they should be assumed
insulated at all times. (It would be a horribly more complicated
problem if we changed the boundary conditions after some time!)

The function 2x is odd when the domain is all real numbers x, but here
I believe you are given this function only for 0<x<L. So you have the
freedom to extend it to be even on -L<x<L. Hence you need to find the
cosine series of the function 2x.