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.

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The paper proposes a modified form of differential equations for describing physical processes in applied mathematics and mechanics. It is noted that classical equations may exhibit discontinuities of the first and second kind at singular points, which are not physically meaningful and not observed in experiments. The new equations consider finite dimensions instead of infinitely small elements and include nonlocal functions averaged over the element volume. The Helmholtz equations are used to relate these nonlocal functions to actual physical variables without singular points. The paper also discusses singular problems in mathematical physics and elasticity theory and compares the obtained solutions with experimental results.