Nonlinearity and boundary conditions

Collocation approach: replace functions by vectors, derivatives by differentiation matrices and use quasi-Newton method for nonlinear system.

$$\mathbf{f(u)} = \left[\begin{array}{c} \mathbf{E}(\mathbf{D}_{xx} \mathbf{u} - \mathbf{r(u)})\\ g_1(u_0, u_0')\\ g_2(u_n, u_n') \end{array}\right] = \mathbf{0}.$$

• Parameter continuation: technique to initialize method; solve at one value of the parameter and then use it as initialization for next value (ex: step function with $\epsilon$)