I have finished the transformation into the Heat Equation. And I am now at the point of establishing the initial conditions. The article I read said the $\max(S-K,0)$ is now the initial condition rather than the terminal condition, which I agree.
But then the article said that $u_0(x) \equiv K(e^{\max(x,0)} - 1) \theta(x)$ with $\theta(x)$ being the Heaviside Step Function. I have a vague idea of $\theta(x)$ being the derivative of max(x,0), but doesn't see how this comes together.
In solving the equation, I looked up the Green's Function for $\partial_t - k \partial^2_x$ ,which in my opinion is quite fitting, is $\theta(t)(\frac{1}{4 \pi k t})^{\frac{1}{2}}e^{\frac{-x^2}{4kt}}$. But the integral in that the article presented is $\frac{1}{\sigma \sqrt{2 \pi \tau}}\int_{\mathbb{R}}u_0(y)e^{-\frac{(x-y)^2}{2 \sigma^2 \tau}}dy$. My question is why the hell is the $y$ term in the exponential.