In this section, we will analyze the stability conditions of the Crank-Nicholson scheme for the generalized groundwater flow equation.
Let . Here is the approximate solution at the point (, ) and, in addition, and the function is chosen to be
Then the function can be expressed in Fourier series as follows:
It was established by Chen et al.  that
Observe that for all , , in addition, according to the problem in point, the storativity S, and transmissivity T are positive constants. Then the following properties of the coefficients , , , and can be established:
It is customary in groundwater investigations to choose a point on the centerline of the pumped borehole as a reference for the observations, and therefore neither the drawdown nor its derivatives will vanish at the origin, as required. In such situations where the distribution of the piezometric head in the aquifer is a decreasing function of the distance from the borehole, the expression . Under this situation, the error committed while approximating the solution of the generalized groundwater flow equation with the Crank-Nicholson scheme can be presented as follows:
If we assume that in equation (4.1) can be put in the delta-exponential form as follows:
where ϕ is a real spatial wave number, new replacing the above equation (4.5b) into (4.5a), we obtain
Equation (4.6) can be written in the following form:
Our next concern here is to show that for all , the solution of equation (4.7) satisfies the following condition:
To achieve this, we make use of the recurrence technique on the natural number k.
For and remembering that , are positive for all , we obtain
Assuming that for the property is verified, we get
Making use of the triangular inequality we obtain
Using the recurrence hypothesis, we have
which this completes the proof.