Method of infinite systems of equations for solving an elliptic problem in a semistrip

Many problems of mechanics and physics are posed in unbounded (or infinite) domains. For solving these problems one typically limits them to bounded domains and finds ways to set appropriate conditions on artificial boundaries or use quasi-uniform grid that maps unbounded domains to bounded ones. Differently from the above methods we approach to problems in unbounded domains by infinite systems of equations. In this paper we develop this approach for an elliptic problem in an infinite semistrip. Using the idea of Polozhii in the method of summary representations we transform the infinite system of three-point vector equations to infinite systems of three-point scalar equations and obtain the approximate solution with a given accuracy. Numerical experiments for several examples show the effectiveness of the proposed method.