En una dimensión $$ \begin{equation*} k \frac{\partial^{2} T}{\partial x^{2}} = \frac{\partial T}{\partial t} \end{equation*} $$
aproximando \( T_{xx} \) $$ \begin{equation*} \frac{\partial^{2} T}{\partial x^{2}} = \frac{T^{l}_{i-1} - 2 T^{l}_{i} + T^{l}_{i+1}}{\Delta x^{2}} \end{equation*} $$
aproximando \( T_{t} \) $$ \begin{equation*} \frac{\partial T}{\partial t} = \frac{T^{l+1}_{i} - T^{l}_{i}}{\Delta t} \end{equation*} $$
reemplazando $$ \begin{equation*} k \frac{T^{l}_{i-1} - 2 T^{l}_{i} + T^{l}_{i+1}}{\Delta x^{2}} = \frac{T^{l+1}_{i} - T^{l}_{i}}{\Delta t} \end{equation*} $$
despejando \( T^{l+1}_{i} \) $$ \begin{equation*} T^{l+1}_{i} = T^{l}_{i} + k \frac{\Delta t}{\Delta x^{2}} (T^{l}_{i-1} - 2 T^{l}_{i} + T^{l}_{i+1}) \end{equation*} $$