Stability of the Two-parameter Set of Nonlocal Difference Schemes

The paper deals with difference schemes for the heat-conduction equation with nonlocal boundary conditions containing two real parameters, α and γ. Such schemes have been investigated for some special parameter values, but the general case was not considered previously. The eigenvalue problem arises as a result of variable division and is solved here explicitly. The so-called reality domains were selected on the (α, γ) plane for which all eigenvalues and eigenfunctions are real. It was demonstrated that the difference schemes in question are symmetrizable in reality domains, that is their transition operators are similar to self-adjoint ones. The necessary and sufficient stability conditions for difference schemes under consideration are obtained with respect to the initial data in the specially constructed norm. The equivalence of the above-mentioned norm to the grid L2-norm has been proved. 2000 Mathematics Subject Classification: 65N06, 65N12, 65N25.