Generalized Functions and Generalized Regular Solutions for Traditionally Singular Problems of Mathematical Physics

Abstract

The paper is concerned with the equations of mathematical physics which describe the actual physical processes. The actual are the processes that can be, in principle, simulated experimentally. Particularly, the discussion is restricted to the singular solutions which are formally exact, but have no physical meaning at the points where the parameters of the actual processes become infinitely high. The existence of such solutions is explained in the paper by the shortcomings of the classical differential calculus based on the analysis of infinitesimal quantities. In contrast to the classical equations of mathematical physics, which are derived as the conservation equations for a media element with infinitely small dimensions, the proposed approach is based on the analysis of an element with small but finite dimensions. As a result, the obtained generalized equations retain the form of the corresponding classical equations, but include the generalized parameters studied process instead of the actual parameters. The generalized parameters are expressed of the actual parameters trough the Helmholtz operator which includes the scale parameter linked with the element dimension. For various particular problems, this parameter is found analytically or experimentally. Thus, in the proposed generalized equations of mathematical physics, the classical equations are supplemented with Helmholtz’s equations for the actual parameters of the process. The right-hand sides of these equations coincide with the corresponding classical solutions of the problem. If the expression in the right-hand side is regular, the final solution reduces to the classical solution. However, if this expression is singular, the fundamental solutions of the homogeneous Helmholtz equations allow us to eliminate the singularities of the classical solution and to arrive at the regular solution. The proposed approach is demonstrated for the classical singular problem of mathematical physics — a circular membrane loaded with a concentrated force applied at the center, the contact and the crack problems of the theory of elasticity.