This will have no explicit boundary conditions in x! Assume |U(t,x)| decays quickly as |x| \to \infty. Apply Fourier Transform Step one is to apply the Fourier Transform on our PDE
\begin{equation} \hat{U}(t, \lambda) = \int_{R} U(t,x) e^{-i\lambda x} \dd{x} \end{equation}
Leveraging the fact that Derivative of Fourier Transform is a multiplication, we can simply our Fourier transform in terms of one expression in x. Apply a Fourier Transform on f(x) This allows you to plug the initial conditions into your transformed expression above. Solve for \hat{U}(t,\lambda), and then convert back This uses the inverse Fourier transform.