Singular Solutions in the Problems of Mechanics and Mathematical Physics

Abstract

A problem of the solutions singularity for applied problems is discussed. It is proposed to qualify such solutions as formal mathematical results that arise from the discrepancy between the mathematical and physical models of the phenomenon or object being studied. As examples, we consider the singular solution of the Schwarzschild problem in the general theory of relativity (serving as the mathematical basis for the existence of objects called the Black Holes), the solution of the mathematical physics problem for a circular membrane loaded in the center by a concentrated force, and the solution for the problems of the theory of elasticity about a cylindrical punch and an expandable plate with a crack. A generalization of the classical definition for a function and its derivative is proposed. This generalization makes it possible to obtain regular solutions of traditional singular problems.