Geodesic
On a discrete boundary value problem in a quarter-plane
Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 142, pp. 169-181

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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.
Mots-clés : elliptic symbol
Keywords: invertibility, digital pseudo-differential operator, discrete equation, periodic wave factorization. 
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     title = {On a discrete boundary value problem in a quarter-plane},
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     url = {http://geodesic.mathdoc.fr/item/VTAMU_2023_28_142_a5/}
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V. B. Vasilyev; A. A. Mashinets. On a discrete boundary value problem in a quarter-plane. Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 142, pp. 169-181. http://geodesic.mathdoc.fr/item/VTAMU_2023_28_142_a5/