Derive Greens function 1D heat equation with space dependent material parameters

Can someone confirm the correctness of my derivation or see where the mistake is? I want to derive the Greens function for the 1D heat equation with (‘general’) SPACE DEPENDENT parameters in infinite space and zero initial condition:

$ frac{partial T}{partial t} = frac{1}{rho(x) c(x)}frac{partial }{partial x}( k(x) frac{partial T}{partial x}) + delta(x-x’) delta(t)$

In particular I’m looking for a solution where the infinite domain is divided into subdomains which contain constant material parameters. So the material parameters are actually block like functions. Let’s take a simple example: for $x < a: rho_1 c_1, k_1, alpha_1$ and for $x > a: rho_2 c_2, k_2, alpha_2$.

My derivation is as follows:

1. Define a transformation: $ k(x) frac{partial T}{partial x} = frac{partial T}{partial y} $ or: $ frac{partial y}{partial x} = frac{1}{k(x)} $

This gives:

$ frac{partial T}{partial t} = frac{1}{rho(x) c(x)}frac{partial }{partial x }( k(x) frac{partial T}{partial x}) + delta(x-x’) delta(t)$

$ frac{partial T}{partial t} = frac{1}{rho(x) c(x)}frac{partial }{partial y}( frac{partial T}{partial y}) frac{partial y}{partial x} + delta(x-x’) delta(t)$

$ frac{partial T}{partial t} = frac{1}{rho(x) c(x) k(x)} frac{partial^2 T}{partial y^2} + delta(x-x’) delta(t)$

$ frac{partial T}{partial t} = frac{alpha(x)}{k^2(x)} frac{partial^2 T}{partial y^2} + delta(x-x’) delta(t)$

2. Following transformation:

$ frac{partial T}{partial t} = frac{alpha(x)}{k^2(x)} frac{partial^2 T}{partial y^2} + delta(x-x’) delta(t)$

$ frac{partial T}{partial t} = frac{alpha(x)}{k^2(x)} frac{partial^2 T}{partial z^2} (frac{partial z}{partial y})^2 + delta(x-x’) delta(t) = frac{partial^2 T}{partial z^2} + delta(x-x’) delta(t)$

Where $(frac{partial z}{partial y})^2 = frac{k^2(x)}{alpha(x)} $ or : $ frac{partial z}{partial y} = frac{k(x)}{sqrt{alpha(x)}} $

3. Do Laplace transform wrt time and a Fourier transform wrt z (This is one of the things I doubt about):

$s hat{T} = -omega^2 hat{T} + int_{-infty}^{infty} delta(x-x’) e^{-j omega z} dz$

We have:
$ frac{partial z}{partial y} = frac{k(x)}{sqrt{alpha(x)} } $

$ frac{partial z}{partial x} frac{partial x}{partial y} = frac{k(x)}{sqrt{alpha(x)} } $

$ frac{partial z}{partial x} k(x) = frac{k(x)}{sqrt{alpha(x)} } $

$ frac{partial z}{partial x} = frac{1}{sqrt{alpha(x)} } $

$ dz = frac{1}{sqrt{alpha(x)} }dx $

So this gives:
$hat{T} = frac{1}{s + omega^2}int_{-infty}^{infty} delta(x-x’) e^{-j omega z} frac{1}{sqrt{alpha (x)}}dx$

$hat{T} = frac{1}{s + omega^2} e^{-j omega z(x’)} frac{1}{sqrt{alpha (x’)}}$

4. This looks like the Greens function transform wrt t and x with constant parameters:

$T = frac{exp{ -frac{(z(x) – z(x’))^2}{4t} }}{sqrt{4 pi t alpha(x’)}}$

Where: $ z(x) = int frac{1}{sqrt{alpha(x)} }dx $

Is this derivation and result correct?