Houjun Liu

Solving PDEs via Fourier Transform

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.