$(15$ points$)$ We are interested in replacing the standard natural boundary condition in the strong form

with a Robin boundary condition as follows.

Strong Form of Linear Steady State Heat Conduction: Given $f$:$,g$:${}_g,$:${}_h,$

$>0$ and $h$:${}_h,$ find $u\frac{\text{:}}{b}ar()$ such that

$q_{i,i}=f,in$

subject to

$u=gon{}_g$

$u-q_i{n}_i=hon{}_h$

Derive the weak form based on inclusion of the Robin boundary condition given by equation $(14).$ In

your final form, place all terms that are a function of the $u,$ the field being solved for, on the left hand

side of the equation and all known quantities on the right hand side.l