Nonlocal Solutions to Singular Problems of Mathematical Physics and Mechanics

Resumo

Second-order partial differential equations are considered that describe the behavior of elastic bodies and have singular solutions. In contrast to the common differential calculus based on the analysis of the function behavior in the neighborhood of a point at infinitesimal variation of arguments, the nonlocal function and its derivative, which describe the behavior of the function in a small, but finite, interval of variation of the argument, are introduced. As a result, the order of the equations under consideration increases up to fifth, and the solution to the traditionally singular problems of mathematical physics appears regular. The solution to the nonlocal problem depends on the constant coefficient, which is suggested to be determined experimentally. Possible applications we consider include the generalized solutions to the equation of mathematical physics and mechanics in Cartesian, polar, and spherical coordinates, governing the bending of the thin membrane and the stress state of the elastic orthotropic sphere.