How to inverse transform this function, when solving PDE

Question

I'm trying to solve this following PDE problem $$ \begin{align}\begin{cases} u_{t} = u_{xx} + au_{x}, \;- \infty < x < \infty , t > 0 \\ \\ u(x,0) = \frac{1}{x^{2}+1} \end{cases} \end{align} $$ by using Fourier transform. I'm not experienced with this method, but after some steps, what I get is the solution to the ODE, which is $$U(\omega,t) = \sqrt{\frac{\pi}{2}}e^{((ai\omega-\omega^2)t-|\omega|)}.$$ Now I want to inverse transform this function to get the solution of the original problem $u(x,t)$ but I don't know how to follow. Any help would be appreciated.