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Numerical solutions

  1. Jan 24, 2005 #1

    e^{-2 \lambda}(2r \lambda_r -1) + 1 = 8 \pi r^2G_\Phi(r,\mu)
    e^{-2 \lambda}(2r \mu_r +1) - 1 = 8 \pi r^2H_\Phi(r,\mu)




    G_\Phi(r,\mu) = \frac{2\pi}{r^2}\int_{1}^{\infty}\int_{0}^{r^2(\epsilon^2-1)} \Phi(e^{\mu(r)\epsilon,L}) \frac{\epsilon}{\sqrt{\epsilon^2-1-L/r^2}}dL
    H_\Phi(r,\mu) = \frac{2\pi}{r^2}\int_{1}^{\infty}\int_{0}^{r^2(\epsilon^2-1)} \Phi(e^{\mu(r)\epsilon,L}) \frac{\epsilon}{\sqrt{\epsilon^2-1-L/r^2}}dL


    If you have a system like this and want to solve it numerically for [tex]\lambda [/tex] and [tex] \mu [/tex] how do you deal with the function [tex] \Phi [/tex]. I mean: It can be any function...i have never solved a system like that before.

    thank you.
  2. jcsd
  3. Jan 24, 2005 #2


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    Staff Emeritus
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    Obviously, you can't solve a system of equations numerically if you have an unknown function (or unknown number) in it.
  4. Jan 24, 2005 #3
    Yepp,,,that is obvious but whats the best you can do?
    Can you get a solution


    \lambda = Af(\Phi)


    Where A is the numerical solution and f is some function.
    And then you can plot lambda for the most probable [tex]\Phi[/tex]´s or something

    Whats the best method to attack non-linear system of equations like this one?
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