I´m having trouble finding the solution of this PDE $$\frac{\partial^2 u}{\partial x\, \partial t}=\frac{\partial^2 u}{\partial x^2}$$ on $-\infty < x < \infty$, $t>0$, and initial condition $u(x,0)=\sqrt{\frac{\pi}{2}}e^{-|x|}.$
I'm trying to use Fourier Transform and getting to: $$\tilde{u}(k,t)=\tilde{u}(x,0)e^{ikt}$$
But I'm having trouble to use the inverse Fourier Transform. I tried making the Fourier Transform $\tilde{u}(k,0)=\sqrt{\frac{\pi}{2}}\int_{-\infty}^{\infty}u(x,0)e^{-ikx}dx$, but could not get to anywhere.