Using ode45 to solve system of ODEs with some initial conditions and some terminal conditions

I have a system of ODEs with , and with a mix of initial and terminal conditions on x. Specifically, for the first 8 components of x, we have the initial conditons, , while for the last 8 components, we have terminal conditions, . Mathematically, this should be enough to guarantee uniqueness of the solution, and I know that if you ever have only terminal or only initial conditions, ode45 handles it no problem.

The first thought would be to just separate the first 8 and last 8 components and just solve 2 system of ODEs, one with terminal conditions and the other with initial conditions, but unfortunately the equations governing both the first 8 components and the last 8 all involve all 16 components, so separating them out isn't possible.

Basically, I want to know if theres a way to corral ode45 into solving a system of ODEs such as this one, since I couldn't find a built-in way to do it when searching, or if there's some mathematical trick to where I can transform the system to only have initial conditions, or something like that.

Any help is appreciated!

Answer 1

[Torsten]

This is a boundary-value problem.

Use BVP4C or BVP5C instead of ODE45.

[Christian]

Awesome, reading the docs looks like that would work, thanks for the fast reply. The MATLAB docs and YouTube videos I tried to watch on bvp4c and bvp5c have only had very basic examples, so I've been having a hard time figuring out all the parameters that bvp4c and bvp5c require but which ode45 does not - specifically the boundary condition function, the bvpinit parameter, and the region parameter, which I guess is pretty much everything new. In my case, is piecewise smooth on 5 different regions, separated by , so the region parameter should be 5 as far as I can tell. But would you mind showing me an example of how I should go about coding f, the boundary conditon function, and the bvpinit to account for the regions and so that bvp4c or bvp5c would accept it? Thanks in advance again

[Torsten]

Here is an example that should fit your needs:

https://de.mathworks.com/help/matlab/math/solve-bvp-with-multiple-boundary-conditions.html