QUESTION $4$

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

Boundary conditions: $v(0,t)=v(1,t)=1,t>0$

Initial conditions: $v(x,0)=sinx,0x1$

Explicit Finite difference $($FD$)$ formula: $V_i^{m+1}=V_i^m+(V_{i+1}^m-2V_i^m+V_{i-1}^m)$

Use $x=0\mathrm{.}5$ and $t=0\mathrm{.}2$ and $=0\mathrm{.}8$

$4\mathrm{.}1$ Draw a fully labelled FD grid to determine the values of potential $(v)$ at $t=0\mathrm{.}2,0\mathrm{.}4,$

and $0\mathrm{.}6$

$4\mathrm{.}2$ Use the FD grid in Question $4\mathrm{.}1$ and calculate the potentials $($v$)$ at the selected

time intervals.

$4\mathrm{.}3$ Use Excel ${?}^{}$ to draw the distribution of the potential $(v^{})$ along the $x-$axis at $t=0,t=$

$0\mathrm{.}2$ and $t=0\mathrm{.}6.$ Also Draw the relationship between the potential $(v)$ and time at $x=$

$0\mathrm{.}5($Note provide a screenshots of the Excel sheet to show the graphs and the data

used$).$

$(7)$