In this section, we will analyze the stability conditions of the Crank-Nicholson scheme for the generalized groundwater flow equation.

Let {\zeta}_{l}^{k}={\mathrm{\Phi}}_{l}^{k}-{\mathrm{\Theta}}_{l}^{k}. Here {\mathrm{\Theta}}_{l}^{k} is the approximate solution at the point ({x}_{l},{t}_{k}) (k=1,2,\dots ,N, l=1,2,\dots ,M-1) and, in addition, {\zeta}^{k}={[{\zeta}_{1}^{k},{\zeta}_{2}^{k},\dots ,{\zeta}_{M-1}^{k}]}^{T} and the function {\zeta}^{k}(x) is chosen to be

{\zeta}^{k}(x)=\{\begin{array}{cc}{\zeta}_{l}^{k}\hfill & \text{if}{x}_{l}-\frac{h}{2}x\le {x}_{l}+\frac{h}{2},l=1,2,\dots ,M-1,\hfill \\ 0\hfill & \text{if}L-\frac{h}{2}x\le L.\hfill \end{array}

(4.1)

Then the function {\zeta}^{k}(x) can be expressed in Fourier series as follows:

\begin{array}{r}{\zeta}^{k}(x)=\sum _{m=-\mathrm{\infty}}^{m=\mathrm{\infty}}{\delta}_{m}(m)exp[2i\pi mk/L],\\ {\delta}_{k}(x)=\frac{1}{L}{\int}_{0}^{L}{\rho}^{k}(x)exp\left[\frac{2i\pi mx}{L}\right]\phantom{\rule{0.2em}{0ex}}dx.\end{array}

(4.2)

It was established by Chen *et al.* [19] that

{\parallel {\rho}^{2}\parallel}_{2}^{2}=\sum _{m=-\mathrm{\infty}}^{m=\mathrm{\infty}}{\parallel {\delta}_{k}(m)\parallel}^{2}.

(4.3)

Observe that for all k,l\ge 1, 0\le 1-{\alpha}_{l}^{k+1}<1, in addition, according to the problem in point, the storativity *S*, and transmissivity *T* are positive constants. Then the following properties of the coefficients {T}_{l}^{k+1}, {G}_{l}^{k+1}, {\lambda}_{j}^{l,k+1}, and {b}_{l}^{k+1} can be established:

\begin{array}{rl}1.& {G}_{l}^{k+1},{T}_{l}^{k+1}\phantom{\rule{1em}{0ex}}\text{are positive for all}l=1,2,\dots ,M-1;\\ 2.& 0{\lambda}_{j}^{l,k}\le {\lambda}_{j-1}^{l,k}\le 1\phantom{\rule{1em}{0ex}}\text{for all}l=1,2,\dots ,M-1;\\ 3.& 0\le {b}_{j}^{l,k}\le 1,\phantom{\rule{1em}{0ex}}\sum _{j=0}^{k-1}{b}_{j+1}^{l,k+1}=1-{\lambda}_{k}^{l,k+1}\phantom{\rule{1em}{0ex}}\text{for all}l=1,2,\dots ,M-1.\end{array}

(4.4)

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 \frac{1}{{r}_{l}}\to 0. 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:

\begin{array}{rcl}{\zeta}_{l}^{k+1}(1+2{T}_{l}^{k+1})& =& {\zeta}_{l+1}^{k+1}\left({T}_{l}^{k+1}\right)+{\zeta}_{l-1}^{k+1}\left({T}_{l}^{k+1}\right)+{\zeta}_{l+1}^{k}\left({T}_{l}^{k+1}\right)+{\zeta}_{l}^{k}(1+2{T}_{l}^{k+1})\\ +\sum _{j=1}^{k}({\zeta}_{l}^{k+1-j}-{\zeta}_{l}^{k-j}){\lambda}_{j}^{l,k+1}{G}_{l}^{k+1}.\end{array}

(4.5a)

If we assume that {\zeta}_{l}^{k} in equation (4.1) can be put in the delta-exponential form as follows:

{\zeta}_{l}^{k}={\delta}_{k}exp[i\varphi lk],

(4.5b)

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:

\begin{array}{r}{\delta}_{1}=\frac{[1-4{T}_{l}^{1}{sin}^{2}(\frac{\varphi h}{2})]{\delta}_{0}}{[1+4{T}_{l}^{1}{sin}^{2}(\frac{\varphi h}{2})]},\\ {\delta}_{k+1}=\frac{[1-4{T}_{l}^{1+k}{sin}^{2}(\frac{\varphi h}{2})-{e}_{1}^{l,k+1}]{\delta}_{k}+{\sum}_{j=0}^{k-1}{\lambda}_{j+1}^{l,k+1}{\delta}_{k-j}+{\lambda}_{k}^{l,k+1}{\delta}_{0}}{[1+4{T}_{l}^{1+k}{sin}^{2}(\frac{\varphi h}{2})]}.\end{array}

(4.7)

Our next concern here is to show that for all k=1,2,\dots ,N-1, the solution of equation (4.7) satisfies the following condition:

|{\delta}_{k}|<|{\delta}_{0}|.

To achieve this, we make use of the recurrence technique on the natural number *k*.

For k=1 and remembering that {d}_{l}^{k+1}, {b}_{l}^{k+1} are positive for all l=1,2,\dots ,M-1, we obtain

\frac{|{\delta}_{1}|}{|{\delta}_{0}|}=\left|\frac{[1-4{T}_{l}^{1}{sin}^{2}(\frac{\varphi h}{2})]}{[1+4{T}_{l}^{1}{sin}^{2}(\frac{\varphi h}{2})]}\right|<1.

(4.8)

Assuming that for m=2,3,\dots ,k the property is verified, we get

|{\delta}_{k+1}|=\left|\frac{[1-4{T}_{l}^{1+k}{sin}^{2}(\frac{\varphi h}{2})-{e}_{1}^{l,k+1}]{\delta}_{k}+{\sum}_{j=0}^{k-1}{\lambda}_{j+1}^{l,k+1}{\delta}_{k-j}+{\lambda}_{k}^{l,k+1}{\delta}_{0}}{[1+4{T}_{l}^{1+k}{sin}^{2}(\frac{\varphi h}{2})]}\right|.

(4.9)

Making use of the triangular inequality we obtain

|{\delta}_{k+1}|\le \frac{|1-4{T}_{l}^{1+k}{sin}^{2}(\frac{\varphi h}{2})-{e}_{1}^{l,k+1}||{\delta}_{k}|+|{\sum}_{j=0}^{k-1}{p}_{j+1}^{l,k+1}{\delta}_{k-j}|+|{e}_{k}^{l,k+1}{\delta}_{0}|}{|1+4{T}_{l}^{1+k}{sin}^{2}(\frac{\varphi h}{2})|}.

(4.10)

Using the recurrence hypothesis, we have

which this completes the proof.