How to solve 2nd order ODE inequality ?

Hello to all,

I am trying to numerically solve a 2nd order ODE inequality of the form : y"(x) + y'(x)*a(x) + y(x)*b(x) <= 0 ( a(x) and b(x) are spatially varying parameters). Also, my solution y(x) must be > 0 for all x.

It is possible to solve a similar problem in Matlab ( y"(x) + y'(x)*a(x) + y(x)*b(x) = 0 ) using ode solvers, however, I am uncapable of enforcing the above constraints (ODE<=0 and y(x)>0).

Are toolboxes like Yalmip useful in solving such problems?

Thanks in advance

Firas

[Hildo]

If this problem can be solved with Yalmip, you can use Yalmip + SEDUMi. But I don't understand with this problem can be solved con Yalmip or you are look for some other toolbox with similar declaration of the problem.

[Firas Mourad]

I was just wondering if such a problem can be solved using a toolbox like Yalmip. If not, then I welcome other suggestions :)

[Firas Mourad]

Just an update. I got a reply from a Yalmip forum : such problems cannot be sloved using Yalmip.

Answer 1

[Tamir Suliman]

lets assume that we have the equations:

y''+a*y'+b*y<=0 a , b are f(x) where x>0

let y(x)=Y1 and dy(x)/dx = Y2

dY1/dx= Y2 dY2/dx= -a*Y2-b*Y1

lets assume a =3 b =4 then the program code would be similar to

a=3;b=4;
syms y(x)
[V] = odeToVectorField(diff(y, 2) == -a*diff(y) -b* y);
M = matlabFunction(V,'vars', {'x','Y'})
sol = ode45(M,[0 20],[2 0]);
fplot(@(x)deval(sol,x,1), [0, 20])

if statement would be sufficient to add the constraints

[Firas Mourad]

Thank you Tamir for your reply.

Could you please elaborate more on how the if statement will be used?

[Tamir Suliman]

if sol > 0 then code please do some thing for me here

else if sol < 0 then code please do some thing for me else code

end

https://www.mathworks.com/help/matlab/ref/if.html