PROBLEM $1(25$ points$)$

A $3-$D cube $0xa,0ya,0za$ is initially at temperature $T=xyz+T_{}.$ For times $t>0,$

the boundaries at $x=0,y=0,$ and $z=0$ are insulated, and the boundaries at $x=a,y=a,$ and $z=$

a all dissipate heat by convection into an environment with fluid temperature $T_{}$ and with

identical heat transfer coefficients $h$ at all three boundaries. Obtain an expression for the

temperature distribution $T(x,y,z,t)$ in the region.