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 solutions to classical equations at singular points may undergo discontinuities of the first and second kind, which have no physical nature and are not experimentally observed. When new equations describing physical fields and processes are derived, elements with finite dimensions are considered instead of infinitely small elements of the medium. Thus, classical equations include nonlocal functions averaged over the element volume and are supplemented by the Helmholtz equations establishing a relationship between nonlocal and actual physical variables, which are smooth functions having no 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.