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Numerical methods for a system of coupled ODE

  1. Nov 22, 2016 #1
    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. Lets 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.
     
  2. jcsd
  3. Nov 22, 2016 #2

    I like Serena

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  4. Nov 22, 2016 #3

    DrClaude

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    What is the domain of z?
     
  5. Nov 22, 2016 #4
    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?
     
  6. Nov 22, 2016 #5

    I like Serena

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    We can combine it with the Shooting method.
     
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