It is easy to solve non linear problems with initial condition.Feeling it difficult to solve non linear problems with two boundary conditions.Any guidance or direction will be appreciated.
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.
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