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Problem $(1)$: Consider a sphere of radius $R.$ Initially at $t=0$ the temperature everywhere $0rR$ is zero. For $t>0$ the surface $r=R$ is maintained at a temperature $T_0.$ Show that the transient temperature distribution $T(t,r)$ within the sphere is given by

$T(t,r)=T_0+\frac{2T_{0}R}{r}{\displaystyle \underset{n=1}{\overset{}{}}}\frac{(-1{)}^n}{n}sin(\frac{nr}{R})exp(-\frac{n^2{}^2}{R^2}t)$;

where $$ is the thermal diffusivity of sphere material. Further show that the volumetric averaged temperature $($:$T$:$)$ is given by

$($:$T$:$)=T_0-\frac{6T_0}{{}^2}{\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{n^2}$exp$(-\frac{n^2{}^2}{R^2}t)$