I need to solve a 4th order non-linear boundary value problem in the following form:
$$ \left\{\begin{matrix} x'_1 = f_1(t, x_1, x_2, x_3, x_4) & x_1(0) = a_1 \\ x'_2 = f_2(t, x_1, x_2, x_3, x_4) & x_2(0) = a_2 \\ x'_3 = f_3(t, x_1, x_2, x_3, x_4) & x_3(T) = b_1 \\ x'_4 = f_4(t, x_1, x_2, x_3, x_4) & x_4(T) = b_2 \end{matrix}\right. $$
If all the conditions are at the same point then I know what to do. But I have no idea how to deal with different points.
I've seen the solutions for BVP with two equations using the shooting method but unfortunately don't see how to generalize it to 4 equations.
So any ideas will be highly appreciated. Thanks.
EDIT. While I'm really interested in some general solution but here's the particular system which I need to solve now:
Here $\lambda$ is some small positive integer (not greater than $30$) and $\varepsilon$ is between $0.5$ and $0.001$.