Diffusion equation, heat equation in one dimension
Setting all constants equal to the diffusion constant \( D \), i.e.,
$$
\begin{equation*}
D=\frac{C\rho}{\kappa},
\end{equation*}
$$
we arrive at
$$
\begin{equation*}
\nabla^2 T(\mathbf{x},t) =
D\frac{\partial T(\mathbf{x},t)}{\partial t}.
\end{equation*}
$$
Specializing to the \( 1+1 \)-dimensional case we have
$$
\begin{equation*}
\frac{\partial^2 T(x,t)}{\partial x^2}=D\frac{\partial T(x,t)}{\partial t}.
\end{equation*}
$$