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I want to solve this equation

$$ \partial_{\rho}^{2}\phi+\frac{3}{\rho}\partial_{\rho}\phi+\left(\frac{M^{2}}{(1+\rho^{2})^{2}}-\frac{l(l+2)}{\rho^{2}}\right)\phi=0 $$

numerically.

I know that this equation can be transformed into the hypergeometric equation through the transformation $$ \phi(\rho) = \rho^l (1+\rho^2)^{-\alpha} P(\rho) $$ (in which $P$ is some function) whose exact solution is the well known function see here $$ _2 F_1(a,b;c;\rho) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{\rho^n}{n!}. $$

The crucial characteristic of this function is that if $a$ or $b$ are negative integers, then the series is finite.

However, I'm interested in exploring a numerical solution for this equation and I would like to know how to obtain numerically the finite series solutions.

Any idea?

Thanks.

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  • 3
    $\begingroup$ You might want to try Math SE. Take a look at the non-linear example here: en.wikipedia.org/wiki/… $\endgroup$ – user6972 Nov 25 '13 at 18:00
  • $\begingroup$ You might be able to identify the form/solution with this book books.google.com/… $\endgroup$ – user6972 Nov 25 '13 at 18:06

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