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`Extracted from the article `*Propagation of vertical
hydraulic fracture* by P. R. Nordgren (Society of Petroleum Engineers
Journal 12, 4, 306-314, 1972)

The underlying mathematical problem is a partial differential
equation with moving boundary. Let be the width of the crack
at position *x* and time *t*. Let *L*(*t*) be the length of the crack at time
*t*, and be the time required for the crack to reach length *x*.
Then satisfies the partial differential equation:

A standard algorithm uses a discretization in *x* and
*t* to form finite difference approximations to the partial derivatives in
the previous equation. At each time step *t*_{j}, one must solve a nonlinear
system of equations for
The solution
provided by the numerical methods dealing with these nonlinear systems is
not very satisfactory since in many cases problems arise when the program
attempts to take the square root of a big negative number^{}.