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Coupled Nonlinear Differential Equations

  1. May 22, 2012 #1
    Hey,

    I need your help to solve the following set of coupled differential equations numerically.

    dn(t,z)/dt=I^5(t,z)+I(t,z)*n(t,z)

    dI(t,z)/dz=I^5(t,z)-α(n(t,z))*I(t,z)

    where I(t,0)=I0*exp(-4ln2(t/Δt)^2) and n(t,0)=0 and n(-certrain time,z)=0. Some constant parameters I did not show here.
    α(n(t,z)) is just a parameter which depends linear on n(t,z)

    I tried to solve it with Mathematica and NDsolve.. since it works fine if I just solve the first equation without the z dependence. But Mathematica seems to be unhappy with the boundary conditions.

    Thanks!

    Thomas
     
  2. jcsd
  3. May 22, 2012 #2

    haruspex

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    There doesn't seem to be enough information. There's nothing to set how n varies with z or I with t. You said it was ok if you just took the first equation and treated z as constant, but then you have two unknown dependent variables, n and I, and only 1 equation.
     
  4. May 22, 2012 #3
    Thanks for your fast response.

    Yeah I know, but what I solved was
    dn(t)/dt=I^5(t)+I(t)*n(t)
    with a time dependent I(t).

    n depends on z only due to the fact that I is depending on z, there is no direct dependence. dI(t,z)/dt I only can give for the case I(t,0).. so I just adapt these coupled equations from a paper.

    Thank you

    Thomas
     
  5. May 23, 2012 #4
    This is why mathematica is not solving thsi.

    "where I(t,0)=I0*exp(-4ln2(t/Δt)^2)"

    are you sure this is so, what if Δt=0? then what?
     
  6. May 23, 2012 #5
    Δt is not zero. so I is just a gaussian distribution around Δt
     
  7. May 28, 2012 #6

    hunt_mat

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    Can't the method of characteristics be used here?
     
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