Question $1$

Given $\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{t}}=\frac{de{l}^2}{del{t}^2},(0,t)=0,(4,t)=0$ for $t0,(x,0)=16-x^2$ for $(0,0)=0,x=1t=2sec(t)=2sec{}_i^{m+1}={}_i^m+0\mathrm{.}02({}_{i+1}^m-2{}_i^m+{}_{i-1}^m)0,(0,0)=0,x=1,$ and $t=2sec.$

$1\mathrm{.}1$ Draw a neat, labelled Finite Difference grid $(FD)$ that allows the calculation $of$ unknown potentials after $4$ seconds. Note: show the grid, node numbers, and boundary conditions $at$ each node.

$1\mathrm{.}2$ Explain that the given $is$ one$-$dimensional $(1D)$ problem, even the $FD$ grid $in$ Question $1\mathrm{.}1$ indicates $2D.$

$1\mathrm{.}3$ Use the explicit method $to$ determine the dependent variable, $,at$ node $1-1$ and time $(t)=2sec.$ Formula: ${}_i^{m+1}={}_i^m+0\mathrm{.}02({}_{i+1}^m-2{}_i^m+{}_{i-1}^m).$

$1\mathrm{.}4$ Determine $if$ the given $is$ a well$-$posed Partial Differential Equation problem and explain your answer.