Consider the slab on the left. It is infinitely large along $2$ dimensions. For the finite dimension, the $1$D heat equation in dimensionless form is given as $($$,=0)=00$ S $-08$ Y$($$,=$ L$)=00$s$1$ and t$0$ where Y is the dimensionless temperature, s is the dimensionless position, and t is the dimensionless time. a$)$ Use separation of variables to find the general solution Y $($t$,$s$),$ which can satisfy the boundary condition Y$($t$,$ s $=0)=$ Y$($t$,$ s $=$ L$)=0.$ You do not need to apply initial conditions. b$)$ Assume the general solution of the $1$D heat equation subject to the stated boundary conditions is Y$($$,)=00$ n$=1$n sin$($$)$ e$-$n$^2$ Apply the initial condition Y$($t $=0,$ s$)=$ F sin$($as$)$ where F is a constant and state the solution to the $1$D heat equation.