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.