Technique on solving non-linear ODE

[yogi10usa]

I have a set of non-linear ODE.
There are eight variables that depends on time (t)
4 of those are first order ODE
1 of those are first order non-linear ODE
3 of those are non-linear equations

The variables are
(t)

(p)
(ps)
(gs)
(gd)
(as)
(ad)
(hs)
(cs)

k1,k2,k3,k4,k5 are constants

The following initial condition are given at t=0
(p)
(gs)
(gd)
(as)
(ad)


The diff equations are

note: f(t) is a set a data defined from t =0 from t=1000

#1: d(p)/d(t) = (ps)-(p) + f(t)

#2: d(gs)/d(t) = (gd)-(gs)-(ps)-(p)-k3

#3: d(gd)/d(t) = (gd) - (gs)-k1

#4: d(gd)/d(t) =(gd)-(gs) -k2

#5: d(gd)/d(t) = (gd)-(gs)-k4


The non-linear equations are

#6: (ps) = ((hs)^2)/(cs)

#7: (cs) = ((as)-(hs))/2

#8: (hs) = (gs) - {(gs)^2 - k5*(as)*[2*(gs)-(as)]}^(1/2)

I figure there are two approaches so far

1st approach:)
I figure I can figure differentiate #6, #7, #8 and figure out their initial condition and use ODE45 or ODE23 to solve this system

However, #8 will be very messy...

2nd approach:)
I can solve for (ps) using #6, #7 and #8 and thus I have a system of
5 non-linear ODE. (#1 - #5) It is still very messy...


Any ideas in this kind of situation?

Also, what do I need to do for f(t) as it is given as a set of points and not a well-defined function.

Thanks so much for your help

 

[jvicens]

!

your first step would be to carefully analyze the set of equations and understand their physical meaning and implications. This will help you determine the best approach for solving them.

In this case, it seems like the second approach may be more feasible, as it reduces the number of equations to be solved and eliminates the need for differentiation. However, you will still need to use numerical methods such as ODE45 or ODE23 to solve the system of equations.

As for the data set f(t), you can interpolate the points to create a well-defined function that can be used in your equations. Alternatively, you can also use a piecewise function to approximate the data set.

It's also important to carefully choose the initial conditions for your equations, as they can greatly affect the behavior and stability of the system. You may need to do some sensitivity analysis to determine the best initial conditions for your specific problem.

Additionally, you may want to consider simplifying the equations if possible, by using assumptions or approximations that are appropriate for your problem. This can greatly reduce the complexity of the equations and make them more manageable for solving.

In summary, carefully analyzing the equations and understanding their physical implications, choosing appropriate initial conditions, and using numerical methods and simplifications can help you successfully solve this set of non-linear ODEs.

 

[tacman]

!


One possible technique for solving non-linear ODEs is the numerical method known as the Runge-Kutta method. This method involves approximating the solution at different time points through a series of iterations. It can handle systems of ODEs with both linear and non-linear components.

In your case, you could use the Runge-Kutta method to solve for the five first-order ODEs (#1-#5) and then use the results to solve for the remaining three non-linear equations (#6-#8). This would involve setting up a system of equations for each time point and using the previous time point's solution as the initial condition for the next time point.

As for the given set of data for f(t), you could use interpolation techniques to approximate a function that represents the data. This function can then be used in the ODEs as needed.

Overall, solving non-linear ODEs can be a challenging task and may require a combination of different techniques. It is important to carefully consider the nature of the equations and the available data before deciding on a specific approach.