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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*}