By Transformation from the Black-Scholes differential equation to the diffusion equation - and back, we are able to transform the PDE $\frac{\partial V}{\partial t} +\frac{1}{2}\sigma^2S^2\frac{\partial ^2 V}{\partial S^2} +rS\frac{\partial V}{\partial S} - rV=0$ into a heat equation.
After I turn this equation into 2D by adding a term $S\frac{\partial V}{\partial J}$, we have $$\frac{\partial V}{\partial t} +\frac{1}{2}\sigma^2S^2\frac{\partial ^2 V}{\partial S^2} +rS\frac{\partial V}{\partial S} + S\frac{\partial V}{\partial J}- rV=0$$
Is there an algorithm that helps us transform this PDE into a heat equation, or at least eliminate the $S$ in the coefficient? (Since $S$ is a variable while the other coefficients are constant)
Hint:
Let $V=e^{rt}W$ ,
Then $\dfrac{\partial V}{\partial t}=e^{rt}\dfrac{\partial W}{\partial t}+re^{rt}W$
$\dfrac{\partial V}{\partial S}=e^{rt}\dfrac{\partial W}{\partial S}$
$\dfrac{\partial^2V}{\partial S^2}=e^{rt}\dfrac{\partial^2W}{\partial S^2}$
$\dfrac{\partial V}{\partial J}=e^{rt}\dfrac{\partial W}{\partial J}$
$\therefore e^{rt}\dfrac{\partial W}{\partial t}+re^{rt}W+\dfrac{\sigma^2S^2}{2}e^{rt}\dfrac{\partial^2W}{\partial S^2}+rSe^{rt}\dfrac{\partial W}{\partial S}+Se^{rt}\dfrac{\partial W}{\partial J}-re^{rt}W=0$
$\dfrac{\partial W}{\partial t}+\dfrac{\sigma^2S^2}{2}\dfrac{\partial^2W}{\partial S^2}+rS\dfrac{\partial W}{\partial S}+S\dfrac{\partial W}{\partial J}=0$