I have the following PDE problem which I think sounds like a job for the Fourier transform:
$ u_t + 2u_x = u_{xx} \space \space \space -\infty < x < \infty \space \space \space t>0 $
$ u(x,0) = \left\{ \begin{array}{ll} 0 & \mbox{if } x > 0 \\ e^x & \mbox{if } x < 0 \end{array} \right. $
I thought about using an integral transform to solve this and since the added condition is with a specific $t $ I thought it would be natural to transform with respect to $x$ and since $x$ can be any real number , I figured the Fourier Transform is the right choice so transforming the equation gives an ODE int terms of $t$ and transforming the initial condition is feasible but the problem comes with the solution which at first is of course stated in terms of the Fourier transform with respect to $x$ the function at the end is simply awful, so awful that I cannot even write the solution in terms of a convolution integral. Here is what I got
$ U(\omega,t) = \frac{1}{i\omega + 1} e^{-t(\omega^2+2i\omega)} $
Perhaps my solution was incorrect somewhere or the Fourier transform is not the way to go but I would like to ask for help on this in terms of how to solve it properly, but of course there is the matter of different but equivalent definitions of the Fourier transform (I use the one with no normalization constant and positive complex exponent). All help appreciated thanks.