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@jemoka / Jemoka Knowledge Base / raw/concept/kbhdamped_heat_equation.md
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--- title: "damped heat equation" source: https://www.jemoka.com/posts/kbhdamped_heat_equation/ --- 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.