Find u(t, x) solving the ivp on the half-line for the diffusion equation:

ut - uxx = 0 (t> 0; x in r)
ux(t,0) = sin(t)
u(0,x) = x

i can do this problem for the most part but i'm struggling to get rid of the non-zero neumann boundary condition. (i know the heat kernel/fundamental solution and odd/even reflection technique).

A steel cylinder with a moveable piston on top is filled with helium (He) gas. The force that the piston exerts on the gas is constant, but the volume inside the cylinder doubles, pushing the piston up.

Which of the following answers correctly states the cause for the change described in the scenario?

1. minimum

2. 6400

3. eight

hope this !