---
title: "Solving PDEs via Fourier Transform"
type: concept
related: [Fourier Transform]
source: https://www.jemoka.com/posts/kbhsolving_pdes_via_fourier_transform/
confidence: high
status: active
---

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.
