$(3)$ A $2-$m long metal rod with uniform cross$-$sectional area is initially heated to a temperature of $200C.$ The bar is well$-$insulated except at its $2$ ends. At time $t=0,$ the $2$ ends are suddenly immersed in an ice bath at $0C.$ If the relevant partial differential equation $($PDE$)$ describing the problem is

$\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}=\frac{de{l}^{2}T}{del{x}^2},\{\begin{array}{c}\text{:}[00]\end{array}$

where temperature $T$ is a function of time $t$ and position $x,$ and thermal diffusivity $$ can be treated as a constant.

$($i$)$ Write down the appropriate initial$/$boundary conditions.

$($ii$)$ Determine the Fourier series solution for the temperature profile $T$ along the bar as a function of time $t$ and position $x.$

$($iii$)$ Estimate the time needed for the temperature at the centre of the rod to drop to $25C$ if the bar is made of silver