Topic $2-2-$D Heat Conduction with IC $(40$ points$)$

Consider a square metal plate of side length L$=10$cm$.$ The temperature distribution u$($x$,$y$)$ in the plate

satisfies Laplace's equation, as the system has reached steady$-$state conditions:

$($del$^(2)$u$)/($delx$^(2))+($del$^(2)$u$)/($dely$^(2))=0.$

Boundary Conditions

The temperature along the left and right edges of the plate is held at $100\backslash$deg C :

u$(0,$y$)=100$ and u$($L$,$y$)=100$ for yL$.$

The temperature along the bottom edge of the plate is also $100\backslash$deg C :

u$($x$,0)=100$ for xL$.$

The temperature along the top edge of the plate is maintained at $100\backslash$deg C :

u$($x$,$L$)=100,$ for xL$.$

Initial Condition

Suppose that initially, the temperature distribution across the plate is non$-$uniform and given by:

u$($x$,$y$,0)=$f$($x$,$y$)=100+50$sin$((\backslash$pi x$)/($L$))$sin$((\backslash$pi y$)/($L$)).$

Questions

a$.$ Find the temperature distribution u$($x$,$y$,$t$)$ across the plate over time by solving the heat equation

with the given boundary and initial conditions.

b$.$ Interpret the solution by explaining how the temperature distribution evolves and approaches the

steady$-$state condition throughout the plate.