Cauchy problems for Laplace equation on compact sets

摘要

In this paper a Cauchy problem for a two-dimensional Laplace equation under the condition that an exact solution belongs to a compact set is considered. We solve this problem as an operator equation. The errors of the operator and the right-hand side are found under a condition that the solution belongs to sets of monotonic, convex functions or functions with a Lipschitz constant. We also consider piecewise functions on the considered segment, i.e., the segment is divided on several segments, where a function belongs to one of the simple compact sets mentioned above. The method to cut convex polyhedrons is used to construct areas, which approximate solutions of the considered problems to which the given errors belong.