Say I had a PDE:
$$u_t + u_{xxxx} + u = f(x)$$
with IC $u(t=0)=s(x)$.
How does one use Fourier transforms to solve this? If it were second order ($u_{xx}+u_{yy} = 0$), one could use a sine/cosine transform e.g.:
$$U(k,y) = \int_{-\infty}^\infty u(x,y)$$
would be the Fourier transform so that
$$-k^2~u(k,y)+\frac{\partial }{\partial y^2}u(k,y)=0$$
Where do I start to even solve a pde of this nature?