Approximation in the Mean of Classes of the Functions with Fractional Derivatives by their Abel-Poisson Integrals

Abstract

The constant development of the applied mathematics is due to its close connection with the fundamental directions of research in the related fields of natural sciences. One of the most important areas of modern science is the study of linear and nonlinear mathematical game models of various phenomena and processes of nature. The emergence of such models is due to the use in modern physics and techniques of influence on matter of electric fields of high intensity, beams of high-energy particles, powerful laser coherent radiation of shock waves of high intensity and powerful heat fluxes. The differential equations in partial derivatives, one of which is the equation of the elliptic type, describing the stationary processes of different physical nature, are the basis of such models. The simplest and most widespread equation of the elliptic type is the Laplace equation whose solution, under given conditions, on the boundary of the considered region, is the well-known Abel-Poisson integral. Approximate properties of the solution of an elliptic boundary value problem with the given boundary conditions at the boundary of the domain on classes of functions with fractional derivatives have been investigated. The solution to this problem finds its application in the study and further application of methods of resolving functions for game dynamics problems. Here we found the asymptotic equalities for the exact upper bounds of the deviations of classes of functions with fractional derivatives from their Abel-Poisson integrals in the integral metric. We establish the equivalence of the approximation characteristics of solutions of an elliptic boundary value problem with the given boundary conditions at the boundary of domain both in the uniform and in integral metrics for classes of functions with fractional derivatives.