To the Problem of Discontinuous Solutions in Applied Mathematics

This paper addresses discontinuities in the solutions of mathematical physics that describe actual processes and are not observed in experiments. The appearance of discontinuities is associated in this paper with the classical differential calculus based on the analysis of infinitesimal quantities. Nonlocal functions and nonlocal derivatives, which are not specified, in contrast to the traditional approach to a point, but are the results of averaging over small but finite intervals of the independent variable are introduced. Classical equations of mathematical physics preserve the traditional form but include nonlocal functions. These equations are supplemented with additional equations that link nonlocal and traditional functions. The proposed approach results in continuous solutions of the classical singular problems of mathematical physics. The problems of a string and a circular membrane loaded with concentrated forces are used to demonstrate the procedure. Analytical results are supported with experimental data.

Automated summary

This paper addresses the issue of discontinuities in mathematical physics solutions which describe actual processes but are not observed in experiments. The author suggests that the presence of discontinuities is linked to classical differential calculus that analyzes infinitesimal quantities. To overcome this, the paper introduces nonlocal functions and nonlocal derivatives, which are obtained by averaging over small finite intervals of the independent variable instead of using the traditional point approach. By incorporating these nonlocal functions into classical equations and introducing additional equations to connect them with traditional functions, continuous solutions to classical singular problems in mathematical physics are obtained. The approach is demonstrated and supported by experimental data using the problems of a loaded string and circular membrane.