There are various ways to solve nonlinear BVP numerically. If the BVP problem is  an ODE one can use a collocation method, Finite Difference method, Finite Element method and a shooting method. All all cases you will end up with a system of nonlinear algebraic equations that have to be solved. And Newton's method  at this point is normally your best choice, especially if you have a reasonable guess for the solution.

Here are some links where you can download some examples (using Mathematica  as the  software):

Spectral Collocation Method:

https://sites.google.com/site/ekayasolutionsdiffeqns/home/spectra/mathematica-notebooks

Shooting Method:

https://sites.google.com/site/ekayasolutionsdiffeqns/home/intermediate-odes/mathematica-n

Finte Difference Method:

https://sites.google.com/site/chemengwithmathematica/home/numerical-methods/mathematica-notebooks

If you do not wish to solve a system of non-linear algebraic equations, you can use shooting method where a two BVP is solved as an IVP, Adomian Decomposition method or He's method or use quasilinearization to form a system of linear equations. 

There are many numerical methods based on discretization to solve BVP of ODE or PDE type.