Definitions, Theorems, Examples, Topics for Asmar 2nd edition
Chapter 7: 7.6 only

 7.7 Heat Equation for a semi-infinite rod
====

 EXAMPLE 1. Semi-infinite rod
  u_t = c^2 u_xx = 0 on t>0, x > 0,
  u(x,0)=f(x)     Initial temp distribution along the rod.
  u(0,t) = 0      End x=0 held at 0 Celsius.

 SOLUTION. Take the FST across the PDE and BC. Let y(t)=FST[u(x,t)](w) 

  (d/dt) FST[u] = c^2 FST[u_xx] = -c^2w^2 FST[u] + 0 [see (14) page 438]
  y'(t) + c^2w^2 y(t) = 0, y(0)=FST[f], 
 Then
  y(t)=FST[f]exp(-c^2w^2t)
  u(x,t)=inverse sine transform of y(t)
        = sqrt(2/Pi) integral of FST[f]exp(-c^2w^2t)sin(wx), w=0..inf