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On a discrete boundary value problem in a quarter-plane


We study the solvability of a discrete analogue of a model pseudo-differential equation in a quarter-plane in discrete Sobolev–Slobodetskii spaces. Using a concept of periodic wave factorization for elliptic periodic symbol, we describe solvability conditions for the equation and for a certain boundary value problem related to this equation. In particular, for certain values of the index of periodic wave factorization, a formula for a general solution of the model discrete pseudo-differential equation is obtained, there are some arbitrary functions in the formula. For their unique determination, we introduce certain additional conditions such as a discrete analogues of integral conditions on angle sides. The existence and uniqueness theorem for the stated boundary value problem is proved and a priori estimates for the solution are obtained. A comparison between discrete and continuous solutions for a special choice of discrete objects is also given.


elliptic symbol, invertibility, digital pseudo-differential operator, discrete equation, periodic wave factorization

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