Numerical methods for a system of coupled ODE

[Telemachus]

Hi there. I have to solve a system of coupled ordinary differential equations. I have some initial values, but in different points of the domain. The equations are all first order. Let's suppose the system looks like this:

$\displaystyle\frac{dy_1}{dz}=y_1+y_2+0.01$
$\displaystyle\frac{dy_2}{dz}=y_1+y_2+0.01$

with initial conditions: $y_1(0)=0, y_2(10)=0$

So, I use some discretization in z, and get some iteration scheme that looks like

$y_{1,n+1}=f_1(y_{1,n},y_{2,n})$
$y_{2,n+1}=f_2(y_{1,n},y_{2,n})$

I don't give exactly the recursion formula I've arrivesd I just simplified it to discuss the important aspects, $f_1$ and $f_2$ are just some functions. The thing is, that as you can see, the forward value depends in the current one for both solutions. So, the only way I can get a formula which I can solve is by setting the initial condition at the same point (lets say $y_1(0)=0$ and $y_2(0)=0$), otherwise, I lack the information necessary to update the values in both solutions.

Does anyone know where I can find some examples on how to solve coupled differential equations with arbitrary initial values?

Best regards.

 

[I like Serena]

Runge Kutta

 

[DrClaude]

What is the domain of z?

 

[Telemachus]

Hi, thank you both for your feedback, $z\in \{0,10\}$

Regarding runge kutta, I could use that, but how would it solve the issue with the boundary conditions?

 

[I like Serena]

Hi, thank you both for your feedback, $z\in \{0,10\}$

Regarding runge kutta, I could use that, but how would it solve the issue with the boundary conditions?

We can combine it with the Shooting method.