The method of intrinsic boundary conditions in the theory of difference boundary value problems

V. S. Ryaben'kii

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 26 (1971) no. 3, pp. 117-176

Résumé

In this article we consider general systems of difference equations with constant coefficients in arbitrary many-dimensional network domains. We define the boundary of a network domain in a certain way and give a formula that expresses the values of a solution at each point of the network domain in terms of its values at points of the boundary. We use this formula to derive necessary and sufficient conditions, which we call 'intrinsic boundary conditions', for a vector function given on the boundary of a network domain to be extendable everywhere in this domain to a solution. This formula allows us to appreciate that it is natural to consider a general boundary value problem for the systems in question. The method we suggest for investigating and calculating solutions of difference boundary value problems consists in going from the original problem to the problem on the boundary that arises when we consider a combination of given and intrinsic boundary conditions. We present results obtained by the method of intrinsic boundary conditions. In the main they relate to non-stationary problems in simple and composite domains and have various degrees of effectiveness. We refer in the paper to other methods only to appreciate the position of the new method among those already available; it interacts with them and supplements them.