The heat-flow equation

We wish to solve equation (5) by a method involving splitting. Since equation (5) is an unfamiliar one, we turn to the heat-flow equation which besides being familiar, has no complex numbers. A two-sentence derivation of the heat-flow equation follows.  (1) The heat flow H_x in the x-direction equals the negative of the gradient $- \partial / \partial x$ of temperature T times the heat conductivity $\sigma$.(2) The decrease of temperature $- \partial T / \partial t$ is proportional to the divergence of the heat flow $\partial H_x / \partial x$ divided by the heat storage capacity C of the material. Combining these, extending from one dimension to two, taking $\sigma$ constant and $C \,=\, 1$,gives the equation  

$${ \partial T \over \partial t} \eq \left( \sigma\ {\partial... ...x^2} \ +\ \sigma\ {\partial^2 \over \partial y^2} \right)\ T$$ (6)