Question $1$: $(12)$ A sheet pile cutoff wall is driven $10\mathrm{.}0m$ into permeable sediments and the water level on one side is maintained $($by pumping$)$ at the sediment surface. On the other side of the sheet pile the water level is $10\mathrm{.}0m$ above the sediment surface. The boundary conditions on the left, right and bottom of the flow region shown below indicate impermeable $($no$-$flow$)$ boundaries.

The PDE that describes the steady$-$state distribution of hydraulic head is the Laplace equation:

$\frac{de{l}^{2}h}{del{x}^2}+\frac{de{l}^{2}h}{del{y}^2}=0$

Using only $2^{nd}-$order finite difference equations $($e$.$g$.O(x^2)),$ solve for the distribution of hydraulic head, $h,$ in the saturated sediments. Use $x=y=10\mathrm{.}0m.$ Include a defining sketch and write out $($by hand$)$ the finite difference equation for every node at which the hydraulic head must be calculated. You can then solve the system of simultaneous equations in a spreadsheet using an appropriate method $-$ in the printout of the spreadsheet be sure to clearly indicate the solution, i$.$e$.h,$ for every node.

Hint: For location $(4,2),$ which will be the node at the tip of the sheet pile if you have node $(1,1)$ in the lower left corner, there can be no flow in the vertical $($y$)$ direction. For that particular location$/$node the appropriate equation is: $\frac{de{l}^{2}h}{del{x}^2}=0.$