A two$-$dimensional square plate with unit sides is subject to the prescribed boundary conditions shown in the Figure. Using a regular grid:

a$.$ Calculate the temperature at the nine interior points using exact solution obtained in Section $4\mathrm{.}2$Separation of variables$$ by considering the first three non$-$zero terms of the infinite series.

b$.$ Calculate the temperature at the nine interior points using finite difference numerical methods.

c$.$ Compare the results from the above two methods

d$.$ Plot$/$sketch isotherms.

e$.$ BONUS POINTS: Calculate the temperature at the center point using exact solution obtained in Section $4\mathrm{.}2$Separation of variables$$ by programing calculation of the first $1000$ non$-$zero terms $($HINT: express sinh x as $($ex $-$ e $-$x $)/2=(1-$e $-2$x$)/2$e$-$x $)$ for computational stability $($no overflow$))$

f$.$ BONUS POINTS: Plot isotherms from e$.$