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How to solve a systems of ODE and Algebraic Equations

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Telema Harry
Telema Harry on 28 Jan 2021
Commented: Alex Sha on 29 Jan 2021
I have a system of 3 nonlinear ODE and 2 nonlinear algebraic equations.
Please how can I solve these systems of equation.
ODE 45 can easily solve the ODE part. However, I don't know how to combine the solution from ODE45 and the algebraic equations.
Thank you.
  2 Comments
jessupj
jessupj on 28 Jan 2021
it sounds like what you're after is "how to solve a DAE" if the algebraic eqations constrain the solutions of the ODE part https://www.mathworks.com/help/matlab/math/solve-differential-algebraic-equations-daes.html
otherwise, if the algebraic equations aren't constraints (ie. they determine diagnostic variables), you probably want to solve the ODE and then solve the algebraic equations 'offline' using e.g. fsolve

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Accepted Answer

jessupj
jessupj on 28 Jan 2021
Edited: jessupj on 28 Jan 2021
looks like you've got a non-autonomous DAE.
with u=x(4) and y = x(5), you'd have:
dx(1) = -wh.*x(1) + wh.* x(5)
dx(2) = -wl.*x(2) + A.*sin(w.*t).* wl.*(x(5) - x(1))
dx(3) = K.*x(2)
0 = x(3) + A.* sin(w.*t) - x(4)
0 = 25 - (5 - x(4) ).^2 - x(5) % = 25 - (25 -10*x4 + x4^2) -x5 = x4*(10 -x4)-x5
and check this old post:
https://www.mathworks.com/matlabcentral/answers/360710-how-to-solve-a-set-of-odes-and-a-nonlinear-equation
  1 Comment
Telema Harry
Telema Harry on 29 Jan 2021
Thank you for your help @jessupj and @James Tursa.
I have been able to run the scripts but there is something I am not doing correctly.
I want to be able to plot my x(1) , x(2), x(3), x(4),, x(5) againt time. But I have not been able to figure out how to do that.
% x(1) = eta (n)
% x(2) = psi (E) - that is the second differential equation
% x(3) = Uhat
% x(4) = u
% x(5) = J
tspan = 0:4;
iCon = [0; 0; 0 ; 0; 0];
M = [1 0 0 0 0; 0 1 0 0 0; 0 0 0 0 0; 0 0 0 0 0; 0 0 0 0 0];
options = odeset('Mass',M,'RelTol',1e-4,'AbsTol',[1e-6 1e-10 1e-6 1e-6 1e-6]);
[t,y] = ode15s(@ESCdae,tspan,iCon)
plot (t,y)
function out = ESCdae(t,x)
f = 10;
A = 0.2;
w = 2*pi*f;
wh = 0.8;
wl = 0.2;
K = 5;
out = [-wh.*x(1) + wh.* x(5)
-wl.*x(2) + A.*sin(w.*t).* wl.*(x(5) - x(1))
K.*x(2)
x(3) + A.* sin(w.* t) - x(4)
x(4).*(10 - x(4)) - x(5)
]
end

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More Answers (1)

Telema Harry
Telema Harry on 28 Jan 2021
Edited: Telema Harry on 28 Jan 2021
Thank you James for the feedback. See the sample equations.
f = 10;
A = 0.2;
w = 2*pi*f;
wh = 0.8;
wl = 0.2;
y = 25 - (5 - u).^2
dx1dt = -wh.*x(1) + wh.* y
dx2dt = -wl.*x(2) + A.*sin(w.*t).* wl.*(y - x(1))
dx3dt = K.*x(2) % the solution to this equation gives x(3)
u = x(3) + A.* sin(w.*t)
  1 Comment
Alex Sha
Alex Sha on 29 Jan 2021
Hi, since: u = x(3) + A.* sin(w.*t) and y = 25 - (5 - u).^2, so y = 25 - (5 - ( x(3) + A.* sin(w.*t))).^2, substitute y into dx1dt and dx2dt, then pure ODE functions will be formed.

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