Consider steady, two$-$dimensional, conduction with heat generation in the rectangular plate shown in

Figure. The boundaries of the domain are maintained at constant temperatures as indicated, that is the

left and right$-$walls OA and BC are at a temperature of $50\backslash$deg C$,$ the bottom wall OC is at a temperature of

$100\backslash$deg C$,$ and the top wall is at a temperature of $200\backslash$deg C$.$ There is a spatially dependent volumetric heat

generation given by$.$ q $=5000(1+($x$/$x$)+($y$/2$y$))$ W$/$m$^3.$ The grid spacing is uniform in each of the

directions and it is equal to $$x$=1$m and $$y $=0\mathrm{.}5$m$.$ The thermal conductivity of the plate is k $=10$ W$/($m

K$).$ Answer the following sub$-$questions:

$($a$)$ Using finite$-$difference method obtain an algebraic equation for temperature at any interior point.

$($b$)$ Obtain the discrete temperature equations for nodes $1,2,3$ and $4.$

$($c$)$ Using an initial guess of $400\backslash$deg C perform $4$ iterations of the Gauss$-$Seidel method and present your

results in a tabular form