Computational Physics Lectures: Partial differential equations

Processing math: 100%

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

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