Differential equations and the problem of singularity of solutions in applied mechanics and mathematics

A modified form of differential equations is proposed that describes physical processes studied in applied mathematics and mechanics. It is noted that the solutions of classical equations at singular points may experience discontinuities of the first and second kind, which have no physical nature and are not observed experimentally. When deriving new equations describing physical fields and processes, we consider not infinitely small elements of the medium, but elements with finite dimensions. As a result, the classical equations include non-local functions averaged over the volume of the element and are supplemented by the Helmholtz equations establishing the relationship between non-local and actual physical variables, which are smooth functions without singular points. Singular problems of the theory of mathematical physics and the theory of elasticity are considered. The obtained solutions are compared with the experimental results.