Runge Kutta for 4 coupled differential equations

[UberG]

Hi,

I'm not a bright programmer , but I have to solve the fallowing equations:

$\frac{df}{dt} = \alpha f -\beta f + \theta g - (f+h)f$

$\frac{dg}{dt} = \psi f- \phi g$

$\frac{dh}{dt} = \xi f+ \mu h -\tau h + \epsilon w- (f+h)h$

$\frac{dw}{dt} = \nu h - \chi h$

Where $f(t)$ , $g(t)$ , $h(t)$ and $w(t)$ are only functions of $t$. Every reference I read for Runge Kutta 4th order Method mentions a function with more than 1 variable (i.e http://www.phy.davidson.edu/FacHome/dmb/py200/RungeKuttaMethod.htm).

My question: how can I implement the Runge-Kutta 4th order method for solve theses equations?

(OBS: I'm familiar with C and Python)

Thanks in advance

 

[DrClaude]

Every reference I read for Runge Kutta 4th order Method mentions a function with more than 1 variable (i.e http://www.phy.davidson.edu/FacHome/dmb/py200/RungeKuttaMethod.htm).

Don't confuse dependent and independent variables. In your case, you actually have four dependent variables $f,g,h,w$, and one independent $t$. The $y$ and $z$ in the link you gave play the role of, say $f$ and $g$ (if we were to ignore $h$ and $w$), and $x$ is $t$.

In C, I would suggest using GSL. You will basically only need to code a function taking current values of $y_i \in \{f,g,h,w\}$ and $t$ and returning the values $dy_i/dt$. The integrator will then take that and give you back solutions for subsequent times. It is also not very complicated to implement a simple RK4 integrator from scratch, see for instance Numerical Recipes in C (chapter 16).

 

[HallsofIvy]

I would do this as four separate "Runge-Kutta" solvers running simultaneously. That is, do a loop over the t values, calculating the next value of f, g, h, and w as functions of the previous values of all four.

 

[UberG]

Thanks for the responses !

 

[blue_raver22]

,

Hi,

The Runge-Kutta method is a numerical technique used to solve ordinary differential equations (ODEs). It can be used to solve systems of coupled ODEs, such as the four equations you have provided.

To implement the Runge-Kutta 4th order method for these equations, you will need to first rewrite them in a standard form. This means expressing them as a set of first-order differential equations, where each equation only has a single dependent variable.

For example, your first equation can be rewritten as:

$\frac{df}{dt} = \alpha f -\beta f + \theta g - (f+h)f$

$\frac{dg}{dt} = \psi f- \phi g$

$\frac{dh}{dt} = \xi f+ \mu h -\tau h + \epsilon w- (f+h)h$

$\frac{dw}{dt} = \nu h - \chi h$

This can be rewritten as:

$\frac{df}{dt} = \alpha f -\beta f + \theta g - f^2 - fh$

$\frac{dg}{dt} = \psi f- \phi g$

$\frac{dh}{dt} = \xi f+ \mu h -\tau h + \epsilon w- h^2 - fh$

$\frac{dw}{dt} = \nu h - \chi h$

Now, you can define new variables $x_1 = f$, $x_2 = g$, $x_3 = h$, and $x_4 = w$. This will give you a system of four first-order differential equations:

$\frac{dx_1}{dt} = \alpha x_1 -\beta x_1 + \theta x_2 - x_1^2 - x_1x_3$

$\frac{dx_2}{dt} = \psi x_1- \phi x_2$

$\frac{dx_3}{dt} = \xi x_1+ \mu x_3 -\tau x_3 + \epsilon x_4- x_3^2 - x_1x_3$

$\frac{dx_4}{dt} = \nu x_3 - \chi x_3$

Now, you can use the Runge-Kutta 4th order method