Solve a set of ODEs with given initial conditions and final conditions

I am trying to solve a set of ODEs in MATLAB. I have four different ODEs for four variables and solving them should be easy if we had initial conditions for all the variables. But that's where things get tricky, I have initial conditions for three variables but final condition for the fourth variable. All the traditional solvers are designed to handle IVPs only. What can be the most optimised approach to handle it since I might have to solve the equation multiple times.

Answer 1

[Jan]

If you have some initial and some final values, this is not an IVP (initial value problem), but a BVP (boundary value problem). A single oder mutliple shooting appraoch is working then.

There is no "most optimized" approach and you can write a PhD about a solver for a specific problem.

See https://www.mathworks.com/help/matlab/boundary-value-problems.html

Answer 2

[Bjorn Gustavsson]

To me this seems to be a problem that the shooting-method would be suited for. That will require you to solve the problem multiple times, so hopefully it is not too stiff ODEs or too long period of integration. Just set up something like a Newton-Raphson method for testing improved initial guesses for your fourth variable.

HTH