---
title: "damped heat equation"
type: concept
source: https://www.jemoka.com/posts/kbhdamped_heat_equation/
confidence: high
status: active
---

we can also damp the heat equation:
\begin{equation} \pdv{u}{t} + ku = \pdv[2]{u}{x} \end{equation}
we note that substituting \(u(t,x) = e^{-kt}w(t,x)\) into the expression, we yield:
\begin{equation} \pdv{w}{t} = \pdv[2]{w}{t} \end{equation}
therefore, we simply have to solve the system normally on \(w\), then multiply the solution by \(e^{-kt}\) to obtain our solution for the damped equation.
