How to solve a systems of ODE and Algebraic Equations

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.

[James Tursa]

Please show us the equations you are working with.

[jessupj]

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

Answer 1

[jessupj]

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

[Telema Harry]

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

Answer 2

[Telema Harry]

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)

[Alex Sha]

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.

[Nikhil Kapoor]

How to do this?