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@jemoka / Jemoka Knowledge Base / wiki/concepts/solving_pdes_via_fourier_transform.md
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--- 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.