Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $$u(x,0)=f(x).$$ Let $u(x,t)=f(x)\ast\left(\dfrac{1}{2\sqrt{\pi t}}e^{-\dfrac{(at+x)^2}{4t}}\right)$, and suppose that $f\in L^1$. Show that $u(x,t)$ satisfies the equation above for $t>0,x\in\mathbb{R}$.
The function $u(x,t)$ has a convolution operation. How can I take its derivative with respect to $t$ or to $x$?