Q$3.$

Determine the transient temperature distribution in a one$-$dimensional

$($I$-$D$)$ solid with a thermal diffusivity $=2$ if the initial temperature in

the solid is $1\mathrm{.}0$ and if at all subsequent times, the temperature of the

left side is held at $0$ while the right side is held at $T_0=2.$ The distance

between the two walls is $L=1$ meter.

The governing differential equation is the $1-$D heat equation

$\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}=\frac{de{l}^{2}T}{del{x}^2}$

a$)$ Using the second order backward difference in time and explicit central difference

in space for the numerical approximation, write the $1-$D heat equation.

b$)$ What is the truncation error of the approximation that resulted from a$).$

c$)$ Write a MATLAB function to perform the calculation of the approximation that

resulted from a$)($FTCS$).$

d$)$ Divide the domain into equidistance $($equal distance$)5$ points and calculate $x$

$(\frac{2}{35})$

e$)$ Assume that the stability condition is $\frac{t}{x^2}0\mathrm{.}25,$ calculate $t$

f$)$ Fill the following table $($the green boxes$)$ to solve the $1-$D heat equation

g$)$ Comment on the results $($is the result accurate or not and why?$).$ and how to impro

it$?$