Problem $2$:

Determine the steady state temperature distribution in the region shown below. It has a thermal conductivity of $10\frac{W}{m}*K,$ and the dimension of $1m1m.$ The top and bottom surfaces are kept at constant temperature of $30C$ and $50C,$ respectively. The left surface is subjected to a laser heating, with a heat flux of $q^{\text{'}\text{'}}=10,000sin(y)[\frac{W}{m^2}],$ and the right surface is insulated. The system is divided into $33$ points. Node $7$ is the original point for both $x$ and $y$ axis $((0,0)$ point$).$

Give the temperature equation of the interior point $($point $5),$

Give the temperature equation of the points at four boundaries $(2,4,6,8),$

Give the temperature equation of the points at four corners $(1,3,7,9),$

Give the matrix eauation to solve the temperature distribution.