# Numerical solutions

1. Jan 24, 2005

### JohanL

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

where

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

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.

2. Jan 24, 2005

### HallsofIvy

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

3. Jan 24, 2005

### JohanL

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 $$\Phi$$´s or something

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