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  • $\begingroup$ Well, I'd argue the problem is too localized. Anyway, several questions: 1. Are you sure the BVP itself correct? 2. Can you show us the original BVP in traditional notation so we can check? 3. Have you tried solving the BVP via other methods? If the answer is yes, do other methods succeed? 4. Have you tested the procedure you've implemented on other simpler problem? If the answer is yes, does it work well, or the issue persists? $\endgroup$ Commented Dec 3, 2023 at 1:35
  • $\begingroup$ Why not use a BVP method currently implemented in Mathematica? $\endgroup$ Commented Dec 3, 2023 at 5:00
  • $\begingroup$ @xzczd (1) Yes, in case of typos, I have checked it again. (2) See updated post. (3) No, relaxation method (based on finite-difference) for BVP is the only way I know (4) I have tested my setup for basic/textbook problems and it works as expected, so there shouldn't be any conceptual issue in the relaxation method. Thus, I believe it is because the problem I'm dealing with is just more complex (real situation) as opposed to textbook exercise where it is guaranteed to work smoothly. $\endgroup$ Commented Dec 3, 2023 at 7:31
  • $\begingroup$ @bbgodfrey If you mean this NDSolveBVP, it just shows the Shooting method which is mostly for boundary conditions which are both fixed, while the other is the Chasing method which is just a variation of Shooting method. I believe a derivative boundary condition is much harder to deal with and as I remember in some textbook it states that Shooting method is not good in dealing with this scenario and points to Relaxation method for BVP when dealing with a derivative boundary condition. $\endgroup$ Commented Dec 3, 2023 at 7:41
  • $\begingroup$ @mathemania The shooting method applied to second order or higher ODEs works well for any combination of Dirichlet and Neuman boundary conditions, and I am confident that I can solve your system of equations using it. By the way, please specify the two Dirichlet boundary conditions at b in your question. $\endgroup$ Commented Dec 3, 2023 at 13:49