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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 $0.001$.

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  • $\begingroup$ The shooting method can work here, too. You basically need to search a two-dimensional grid of values for the $x_1'(0)$ and $x_2'(0)$, and try to get the final values to land on the desired values. These can be pretty computationally intensive algorithms, just to warn you. $\endgroup$ – Adrian Keister Dec 12 '18 at 21:59
  • $\begingroup$ Do you need to roll your own or can you use the BVP solver of your preferred numerical software? bvp4c in matlab, solve_bvp in python scipy,... $\endgroup$ – LutzL Dec 12 '18 at 22:13
  • $\begingroup$ @LutzL I need something from matlab preferably. Will look now at bvp4c. $\endgroup$ – Igor Dec 12 '18 at 22:15
  • $\begingroup$ @LutzL: Not entirely sure. Might depend on the nature of the $f_i$. Definitely, if they're stiff, Igor should use a specialized solver for stiff DE's. $\endgroup$ – Adrian Keister Dec 12 '18 at 22:15
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    $\begingroup$ @Igor: no problem. It doesn't look stiff to me, but I'm not an expert. $\endgroup$ – Adrian Keister Dec 13 '18 at 0:31

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