In one dimension,
$$\dfrac{\partial^2{u}}{\partial{x}^2} = C \dfrac{\partial^2{u}}{\partial{t}^2}$$
in three dimensions,
$$\sigma^2 \Biggl( \dfrac{\partial^2{u}}{\partial{x_1}^2} + \dfrac{\partial^2{u}}{\partial{x_2}^2} + \dfrac{\partial^2{u}}{\partial{x_3}^2} \Biggr) = C \dfrac{\partial^2{u}}{\partial{t}^2}$$
For the one dimensional case, the temperature at any time $i+1$
\begin{align}\underbrace{C(U_{j+1}^{i} – 2U_j^{i}+ U_{j-1}^{i})}_{\text{second order central difference for $x$}} = \underbrace{U_{j+1}^i -2U_j^i + U_{j-1}^i}_{\text{second order central difference for $t$}}
\end{align}
using explicit Euler.