Abstract
We consider a weighted difference scheme approximating the heat equation with the nonlocal boundary conditions
. 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.
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Original Russian Text © A. V. Gulin, N. S. Udovichenko, 2008, published in Differentsial’nye Uravneniya, 2008, Vol. 44, No. 7, pp. 963–969.
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Gulin, A.V., Udovichenko, N.S. Difference scheme for the Samarskii-Ionkin problem with a parameter. Diff Equat 44, 991–998 (2008). https://doi.org/10.1134/S0012266108070112
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DOI: https://doi.org/10.1134/S0012266108070112