An unconfined groundwater aquifer is fed by water seeping through the permeable ground directly above. The bottom of the aquifer is impermeable bedrock, which may have an additional confined aquifer running below it as shown in the below:

Image of Ground Water

Unconfined aquifers are the sources for wells used for drinking water and irrigation in most rural communities. The height of the water table in an unconfined groundwater aquifer is described by the following nonlinear ODE:

Unconfined groundwater nonlinear ODE

Where h is the height of the water table $($meters$),$ x is distance $($meters$)$ in the direction of groundwater flow, K is the hydraulic conductivity through the soil in meters$/$day $($e$.$g$.$ the ease at which water flows through the soil containing the aquifer$)$ and N is the infiltration rate $($meters$/$day$)$ of water entering the aquifer from the surface. A less realistic, simpler linear model has also been proposed:

Simpler Linear Model

In the simplified model h$-$bar is the average water table height.

This ODE is generally solved as a boundary value problem where the water table height h is known at two locations $($e$.$g$.$ from measurement wells$).$ Write a function that uses the shooting method to solve the linear water table models as follows. Your function should receive the following inputs $($in order$)$:

A column vector of x values at which the solution, h$($x$),$ is to be calculated.

A two$-$element vector of the boundary conditions, with the first element corresponding to x$(1)$ and the second element corresponding to x$($end$).$

A scalar input for the hydraulic permeability K$.$

A scalar input for the infiltration rate N$.$

Your function should return the following output

The values for h$($x$)$ corresponding to the input x vector obtained with the simplified linear ODE. Set the average water table height to be equal to the average of the boundary conditions.

Note: Use ode$45$ with default tolerances in your implementation of the shooting method.

An unconfined groundwater aquifer is fed by water seeping through the permeable ground directly above. The bottom of the aquifer is impermeable bedrock, under which there could be an

additional confined aquifer as illustrated in the figure below:

Unconfined aquifers are the sources for wells used for drinking water and irrigation in most rural communities. The height of the water table in an unconfined groundwater aquifer is described by

the following nonlinear ODE:

$\frac{d}{dx}(Kh\frac{dh}{dx})+N=0$

Where $h$ is the height of the water table $($meters$),x$ is distance $($meters$)$ in the direction of groundwater flow, $K$ is the hydraulic conductivity through the soil in meters$/$day $($e$.$g$.$ the ease at which

water flows through the soil containing the aquifer$)$ and $N$ is the infiltration rate $($meters$/$day$)$ of water entering the aquifer from the surface. A less realistic, simpler linear model has also been

proposed:

$\frac{K}{b}ar(h)\frac{d^{2}h}{d{x}^2}+N=0$