Unsteady-state heating of a slab (Laplace transform method) 

(a) Re-solve the problem in Example 12.1-2 by using the Laplace transform, and obtain the result in Eq. 12.1-31. 

(b) Note that the series in Eq. 12.1-31 does not converge rapidly at short times. By inverting the Laplace transform in a way different from that in (a), obtain a different series that is rapidly convergent for small times.

(c) Show how the first term in the series in (b) is related to the "short contact time" solution of Example 12.1-1.