PROBLEM $4(25$ points$)$

A hollow sphere $arb$ is initially at temperature $T=\frac{1}{r}.$ For times $t>0,$ the boundary surface

at $r=a$ is maintained at temperature $T_1,$ and the boundary surface at $r=b$ is maintained at

temperature $2T_1.$ In addition, the sphere is subjected to uniform internal energy generation $g_0$

$(\frac{W}{m^3}).$ Obtain an expression for the temperature distribution $T(r,t)$ in the sphere for times $t>$

$0.$