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

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?

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