For a one$-$dimensional, steady$-$state, variable thermal conductivity heat conduction with uniform internal heat generation, develop a generalized finite difference formulation for the interior nodes, with the left surface boundary node exposed to constant heat flux and the right surface boundary node exposed to a convective environment. The variable conductivity is modeled such that the thermal conductivity varies linearly with the temperature as $k(T)=k_0(1+T)$ where $T$ is the average temperature between the two nodes.