The flow between two parallel rivers is shallow one$-$dimensional with prescribed hydraulic head ${}_0$ at $x=0$ and ${}_1$ at $x=L.$ The formation is unconfined with a horizontal base and conductivity $K.$ The infiltration intensity is $N.$

$($a$)$ Please derive the analytical solution for the hydraulic head distribution between $0$ and $L.$

$($b$)$ Draw the water table.

$($c$)$ Derive and draw the discharge rate between $0$ and $L.$

$($d$)$ Find the highest water table and its location.

Data for $($b$)-($d$)$ : $N={10}^{-9}\frac{m}{s},K={10}^{-6}\frac{m}{s},{}_0=12m,{}_1=10m,L=2000m.$

$($e$)$ Discuss the distribution patterns of the hydraulic head over the entire domain and the discharge rate at $x=0$ for a variable infiltration intensity, $N,$ i$.$e$.,$ considering $N$ from $0$ to a very large value.