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 $$ \begin{equation*} \frac{\partial^2 T(x,t)}{\partial x^2}=D\frac{\partial T(x,t)}{\partial t}. \end{equation*} $$