Solving of boundary tasks by using $S$-spline

This article is dedicated to use of $S$-spline theory for solving equations in partial derivatives. For example, we consider solution of the Poisson equation. $S$-spline — is a piecewise-polynomial function. Its coefficients are defined by two states. The first part of coefficients are defined by smoothness of the spline. The second coefficients are determined by least-squares method. According to order of considered polynomial and number of conditions of first and second type we get $S$-splines with different properties. At this moment we have investigated order 3 $S$-splines of class $C^1$ and order 5 $S$-splines of class $C^2$ (they meet conditions of smoothness of order 1 and 2 respectively). We will consider how the order 3 $S$-splines of class $C^1$ can be applied for solving equation of Poisson on circle and other areas.