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Implicit Boundary Value Problem

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Saeid
Saeid on 27 Feb 2020
Commented: Saeid on 28 Feb 2020
Is there a way in Matlab to use the bvp4c or bvp5c functions to solve a system consisting of boundary value and algebraic equations where the deifinction of at least one of the derivatives is implicit?
The system looks somewhat like this:
dY(1)/dx=a
dY(2)/dx=f(Y(1),Y(3))
g(Y(2),Y(3))*[dY(1)/dx]+h(Y(2),Y(3))=0
B.C.:
yb(1)=0
ya(2)=ya(3)=0
where a and K are constants and f, g, h and j are functions relating Y(1) and Y(3) to each other. Naturally, the boundary conditions are known.
  8 Comments
darova
darova on 27 Feb 2020
Please attach the functions

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Accepted Answer

J. Alex Lee
J. Alex Lee on 27 Feb 2020
Ok, so looks like what you are after is solving a DAE that has BCs at more than one boundary.
Docs say it is ode15s and ode23t that are capable of handling DAE (not ode15i; https://www.mathworks.com/help/matlab/math/solve-differential-algebraic-equations-daes.html)
It seems Matlab does not have built-in functionality to address directly (https://www.mathworks.com/matlabcentral/answers/478652-solving-bvp-for-dae-in-matlab)
This system seems simple enough to use a shooting strategy, I think. You have 3 different boundary locations, so you can treat the problem as if it were an IVP leaving 2 guesses for 2 out of 3 of the Y's at x=0 as guesses (not sure which ones, you are mixing a's, b's, and 1,2,3's), use either of the odeXX (with appropriate handling of the algebraic equation), and you will get some values for the Y's at the 2 other boundaries in the course of integration. Their deviations from your actual BCs will become residuals for a 2-variable root finding.

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