Difference scheme for the Samarskii-Ionkin problem with a parameter

We consider a weighted difference scheme approximating the heat equation with the nonlocal boundary conditions

$$u(0,t) = 0, \frac{{\partial u}}{{\partial x}}(0,t) + \frac{{\partial u}}{{\partial x}}(1,t) = 0$$

. We show that in this case the system of eigenfunctions of the main difference operator is not a basis but can be supplemented with associated functions to form a basis. Using the method of expansions in the basis of eigenfunctions and associated functions, we find a necessary and sufficient condition for stability with respect to the initial data in some energy norm. We show that this stability condition cannot be weakened by choosing a different norm. The above-mentioned energy norm is shown to be equivalent to the grid L ₂-norm.