I know that one of the methods of solving the black scholes PDE given by : $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$
is to transform it into the heat equation and then using the explicit euler FTCS scheme.
I was wondering if you could directly use the explicit euler scheme on the Black scholes PDE without using any transofrmation, just substitute the approximations of the derivatives.
In that case i realized that i have no initial condition $V(S,0)$ to start solving. And one of the boundary condition works for $S \rightarrow \infty$, will it work if i restrict $S=S_{max}$.
Is it possible to solve the Black scholes PDE this way, if it is what is the initial condition and the boundary condition i should take .