I'm trying to re-solve the governing equations in hydraulic fracturing modeling
$$ \frac{\partial q}{\partial x} + \frac{2hC}{\sqrt{t-\tau(x)}} + \frac{\partial A}{\partial t} = 0 , \qquad 0<x<L \tag{A-1} $$ $$ q(0,t) = q_i \tag{A-2} $$
as instructed step by step in a paper*. In the Appendix A, the author stated that
$$\begin{aligned} &\frac{\Delta t_m}{2\, \Delta x} \big(q_{i+\frac12}^{m+1} - q_{i-\frac12}^{m+1} + q_{i+\frac12}^{m} - q_{i-\frac12}^{m}\big) \\ &\quad + 4hC\big(\sqrt{t_{m+1}-\tau_i} - \sqrt{t_{m}-\tau_i}\big) \\ &\quad + \frac{\pi h}{4}\big(W_i^{m+1} - W_i^{m}\big) = 0\end{aligned} \tag{A-4}$$
[...]
$$W_i^0=0 \tag{A-6}$$
A finite-difference analog of Eq. 5 is
$$q_{i+\frac12} = \frac{\pi G}{256(1-\nu)\mu\Delta x} \big[ (W_{i+1}^m)^4 - (W_i^m)^4\big] \tag{A-7}$$
The result of substituting Eq. A-7 in Eq. A-4 is a nonlinear system of equations for $W_i^{m+1}$ with $W_i^m$ considered known. In order to solve this system we write
$$W_i^{m+1}=W_i^m+\Delta W_i^m \tag{A-8}$$
and take
$$(W_i^{m+1})^4 \simeq (W_i^m)^4 + 4(W_i^m)^3 \Delta W_i^m \tag{A-9}$$
On substitution of Eqs. A-6, A-8 and A-9 into Eq. A-4, we obtain a linear tridiagonal system, which is easily solved for $\Delta W_i^m$ at $m=1,2 \dots$ .
Could anyone please help me at this step? How to get the linear tridiagonal system from the above equations? I guess that Eq. A-7 is meant instead of Eq. A-6 in this last sentence. Moreover, it seems that exponent index $m$ is missing in Eq. A-7.
*R.P. Nordgren, "Propagation of a Vertical Hydraulic Fracture", Society of Petroleum Engineers Journal 12(04), 306-314, 1972. doi:10.2118/3009-PA
You are right, the writing is quite misleading when one is not used to it. Using A-7 and A-9, let us rewrite the first term of A-4 as \begin{aligned} q_{i+\frac12}^{m+1} - q_{i-\frac12}^{m+1} + q_{i+\frac12}^m - q_{i-\frac12}^m &= \frac{\pi G}{256(1-\nu)\mu\Delta x} \big[ (W_{i+1}^{m+1})^4 - 2(W_{i}^{m+1})^4 + (W_{i-1}^{m+1})^4 \\ &\phantom{ = }\qquad + (W_{i+1}^m)^4 - 2(W_{i}^m)^4 + (W_{i-1}^m)^4\big] \\ &\simeq \frac{\pi G}{128(1-\nu)\mu\Delta x} \big[ (W_{i+1}^{m})^4 + 2(W_{i+1}^{m})^3 \Delta W_{i+1}^m \\ &\phantom{ = }\qquad - 2(W_{i}^{m})^4 - 4(W_{i}^{m})^3\Delta W_{i}^m \\ &\phantom{ = }\qquad + (W_{i-1}^{m})^4 + 2(W_{i-1}^{m})^3\Delta W_{i-1}^m \big] \end{aligned} where $W_i^{m+1} - W_i^{m} = \Delta W_{i}^m$. Note that this equation is linear with respect to the increments $\Delta W_{i-1}^m$, $\Delta W_{i}^m$ and $\Delta W_{i+1}^m$. When injecting it in A-4, a linear tridiagonal system in terms of the vector of increments $(\Delta W_i^m)$ is obtained.