I am having a bit of difficulty in trying to understand a paper. The paper uses spectral method to solve for an eigenvalue that comes from a system of coupled ODEs. I will write out only one equation now, because it is enough to get to the crux of my question(s).
The equation is
$V[r] = \frac{e^{-(\nu[r] +\lambda[r])}}{\epsilon[r] + p[r]} *\biggr[ (\epsilon[r] + p[r])( e^{\nu[r] +\lambda[r]})r W[r] \biggr]'$
I carry out the derivative and get
(Eq1) $V = \biggr[ \frac{\epsilon' +p'}{\epsilon + p} + r(\nu'+\lambda') +1 \biggr] W + r W'$
Now according to the paper I should be able to expand equilibrium quantities $(\epsilon ,p ,\nu ,\lambda$) of the system as Chebyshev Polynomials of the form
$B[r] = \Sigma_{i=0}^{\infty}b_i T_i[y] - \frac{1}{2} b_0 $, where $T_i[y]$ are the polynomials. I know how to get the $b_i$ using code I wrote in Mathematica. Also $y = 2(r/R) -1$, and the domain of $r$ is $(0,R)$.
The paper also states that the functions ($V,W$) can be expanded as $F[r] = (\frac{r}{R})^l \Sigma_{i=0}^{\infty}f_i T_i[y] - \frac{1}{2} f_0 $, and that in general a term like $B[r]F[r]$ can be expressed as
$B[r]F[r] = \frac{1}{2} (\frac{r}{R})^l \Sigma_{i=0}^{\infty} \pi_i T_i[y] - \frac{1}{2} \pi _0 $
where $\pi_i = \Sigma_{j=0}^{\infty}[b_{i+j} + \Theta(j-1)b_{|i-1|} ] f_l $ and $\Theta(k) = 0$ for $k<0$ and equals 1 for $k\geq 0$.
With all that being said, lets say I make the following equilibrium functions
$\frac{\epsilon' +p'}{\epsilon + p} = B_1[r] $ and $ r(\nu'+\lambda') = B_2[r]$, Then Eq1 becomes
(Eq2) $ \bigg((\frac{r}{R})^l \Sigma_{i=0}^{\infty}v_i T_i[y] - \frac{1}{2} v_0 \bigg) = \biggr[ B_1[r] + B_2[r] +1 \biggr] \biggr( (\frac{r}{R})^l \Sigma_{i=0}^{\infty}w_i T_i[y] - \frac{1}{2} w_0 \biggr) + r W' $.
Question1: What do I do with the $(\frac{r}{R})^l $terms? The polynomials are functions of $[y]$ so how can I even have an expansion like $B[r]= (\frac{r}{R})^l$ X function of [y]? Also it seems like I can just divide them out on each side of the equation, so what was the point of introduction that term? I mean, according to the paper this term is supposed to impose the boundary condition that $V,W$ go to zero as $r$ goes to zero.
*Question2:*How am I supposed to deal with the $r$ in the $r*W'$ term. The paper gives a description of how to handle derivative terms, but what about the $r$ itself. Am I supposed to treat it like an equilibrium value and use the rule for terms like $B[r]F[r]$ or should I express this $r$ in terms of $y$. Or should I do something else altogether?
