In deed Navier-Stokes equations are not formally an elliptic PDE because of the advection term (convective acceleration). This incorporates some hyperbolic character to the equations. Under the assumptions of the Boundary Layer Theory, PDE are strictly parabolic and evolves in the axis parallel to the surface in the flow direction. This mean that an elliptic operator discretization in the transversal direction and a Runge-Kutta in the longitudinal direction (both of the same order) will be enough to solve reasonablely the problem. The reference above of Schlichting is a classic from 1951 (german) or 1955-1968 (english). A newer one is Frank M. White, Viscous Fluid Flow (2ed), McGraw-Hill, 1991. Other (more extended) is R. Kh. Zeytounian, Theory and Applications of Viscous Fluid Flows, Springer-Verlag, 2004. For numerical methods and solutions there is a lot. Any of them will be good.

As Andrés mentioned, the full N-S equations are parabolic. However, for the laminar-steady flow case, the equations become elliptical. This is true for the exact boundary layer equations. However they are very tough to solve, therefore approximate solutions are obtained instead. The classic approach is to perform a dimensional analysis of the equations, and neglect small terms in the exact equations. The resulting simplified approximate boundary layer equations are parabolic, and can be treated as a time-marching problem. I strongly recommend you to review White's book on viscous flow, as it has a very nice explanation of the equations, and even a numerical method to solve them using finite differences.

I was misunderstanding above in my explanation. When I said BL is parabolic, I refer to stationary cases only. Parabolic character for BL equations comes from longitudinal (parallel to surface) axis variable where the derivative is of first order. Elliptic character for BL equations comes from transversal second orden differential drivatives.

When you involve time in BL (transient problems), the PDE are also elliptic in transversal directions (perpendicular to surface) but the problem evolves simirlarly parabolic in time and longitudinal variable. Elliptic character depends on boundary conditions (y=0-->V=V_wall, Y=delta-->partial V / partial y=0 for all x), instead parabolic character depends on initial values time-space (t=0-->V=U_infity for all x and V=U_infty only at x=0 for all time). See transient Stokes problems for examples (e.g. I. G. Currie, Fundamental Mechanics of Fluids, 2ed, McGaw-Hill, 1993, Ch.7, Ch.9)

Full N-S equations are really parabolic-elliptic-hyperbolic (time-space-developing).

For spanish spoken people see also :

Andrés Granados, Mecánica y Termodinámica de Sistemas Materiales Continuos, USB, 2015, Cap.X, sec.3.1 y sec.4

The parabolized Navier Stokes equations are derived originally to solve the flow field in the boundary layer region. 

Your look into my contributions Boundary layer and Applications of Viscous flow, page  4 of my section in Researchgate, will be of help. Good luck with your study.

Thanks for everyone answer my question, sincerely.

All of yours are beneficial informations.

Thanks.