I have been using differentiation matrices to approximate solutions. However my matrices are defined on $x\in[a,b]$ which varies between $[-1,1], [0, 1], [-\pi,\pi]$ for different equations and which are not necessarily in terms of my physical space $y\in[c,d]$. I wish to find the scaling factors from $[a,b] \to [c,d]$ for finite, infinite and semi-infinite domains for first and second order derivatives.
I have tried to do this using the chain rule using a map $g: [a,b]\to[c,d]$:
$\frac{du}{dy} = \frac{dx}{dy}\frac{du}{dx} = \frac{1}{g'(x)}\frac{du}{dx}$
and
$\frac{d^2u}{dy^2} = \frac{d^2x}{dy^2}\frac{du}{dx} + (\frac{dx}{dy})^2\frac{d^2u}{dx^2}= ?\frac{du}{dx} + (\frac{1}{g'(x)})^2\frac{d^2u}{dx^2}.$
For a finite domain $[c,d]$ I find $\frac{d^2x}{dy^2}=0$ but what should it be for when the maps for $x\in[-1,1]$ are
$g(x) = \frac{4x}{\sqrt{1-x^2}},$
$g(x) = \frac{x+1}{1-x}$?
and
$g(x) = \frac{x-1}{x+1}$?