Mathematical modeling of bending of a circular plate with the use of $S$-splines

The present paper is concerned with the application of newly developed high-order semi-local smoothing splines (or $S$-splines) in solving problems in elasticity. We will consider seventh-degree $S$-splines, which preserve the four continuous derivatives ($C^4$-smooth splines) and remain stable. The problem in question can be reduced to solving an inhomogenous biharmonic equation by the Galerkin method, where as a system of basis functions we take the $C^4$-smooth fundamental $S$-splines. Such an approach is capable of not only delivering high accuracy of the resulting numerical solution under fairly small number of basis function, but may also easily deliver the sought-for loads. In finding the loads, as is known, one has to twice numerically differentiate the resulting bipotential, which is the solution of the biharmonic equation. This results in roundoff propagation.