Solving Numerical System of Equations for $\lambda$ and $\mu$

[JohanL]

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

where

<br /> <br /> 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<br /> d\epsilon<br />
<br /> 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<br /> d\epsilon<br /> <br />

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

thank you.

 

[HallsofIvy]

Obviously, you can't solve a system of equations numerically if you have an unknown function (or unknown number) in it.

 

[JohanL]

Yepp,,,that is obvious but what's the best you can do?
Can you get a solution

<br /> <br /> \lambda = Af(\Phi)<br /> <br />

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

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