A thin, homogeneous bar of thermal diffusivity $9$ cm$^2/$s and length $2$ cm with insulated sides has its left end maintained at temperature zero, while its right end is perfectly insulated. The bar has an initial temperature f$($x$)=$ x$^2$ for $0<=$ x $<=2.$Determine the temperature distribution in the bar. What is the lim $($t $->\backslash$infty $)$ u$($x$,$t$)?$ What is the temperature of the bar at x $=1$ cm at time t $=20$ seconds?