Chapter 12.7, Problem 1E

In Example 1 find the temperature u(x, t) when the left end of the rod is insulated.

The temperature in a rod of unit length in which there is heat transfer from its right boundary into a surrounding medium kept at a constant temperature zero is determined from

$\begin{array}{l}k\frac{{\partial }^{2}u}{\partial x^2}=\frac{\partial u}{\partial t},0<x<1,t>0\\ u\left(0,t\right)=0,\frac{\partial u}{\partial x}{|}_{x=1}=-hu\left(1,t\right),h>0,t>0\\ u\left(x,0\right)=0,0<x<1.\end{array}$

Solve for u(x, t).