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I have a nonlinear system of algebraic equations with a nonlinear boundary condition of the form \begin{equation} f'''(x,t) - (f'(x,t))^3 - \alpha(t) f'(x,t) = 0, \end{equation} at $x =0$, where $\alpha(t)$ is constant in space but evolves in time. I use sided finite differences to approximate it and Newton's method to solve the nonlinear system approximated using finite differences. The solution does not converge when the mesh size decreases.

Do you suggest any other method to approximate the bc?

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    $\begingroup$ What is your original problem? How did you end up with boundary conditions that are differential equations from a system of algebraic equations? $\endgroup$ – nicoguaro Dec 15 '16 at 23:45
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    $\begingroup$ Also, what does $f'$ indicate -- a spatial derivative, or a temporal derivative? $\endgroup$ – Wolfgang Bangerth Dec 16 '16 at 5:02

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