Operational and non-operational methods for obtaining fundamental solutions of~partial differential equations for nanoplates. Part III

Authors

UDC

531.39

EDN

EZIXCE

DOI:

10.31429/vestnik-23-1-8-21

Abstract

Currently, many numerical methods are known, but, unfortunately, the boundary element method (BEM) and its modifications receive undeservedly little attention, although it, like the finite element method and the finite difference method, is also one of the most successful modern numerical methods with high accuracy of the results obtained. In this regard, it seems relevant to further develop the BEM to solve problems based on the application of precomputed exact fundamental solutions. In this article (Part III), which is a logical continuation of previously published articles (Parts I and II), using operational (the method of complex integral Fourier transform) and non-operational (the method of functional analysis) methods, it was possible to solve the problem of finding a fundamental solution to a linear partial differential equation with constant coefficients using the example of the bending problem of a thin a homogeneous isotropic nanoplate, the differential equation for which is obtained within the framework of the nonlocal theory of microstructural deformation. It is shown that the method of functional analysis made it possible to significantly simplify the methodology for calculating fundamental solutions without the need for a preliminary in-depth study of the mathematical theory of generalized functions and without involving the apparatus of operational calculus. Unfortunately, the mentioned theory and apparatus are still often perceived by researchers as difficult to understand, which sometimes limits the scope of the BEM.

Keywords:

nanoplates, fundamental solutions, generalized functions, functional analysis method, isotropic plates

Funding information

The study did not have sponsorship.

Author info

  • Peter G. Velikanov

    канд. физ.-мат. наук, доцент кафедры реактивных двигателей и энергетических установок и кафедры машиноведения и инженерной графики Казанского национального исследовательского технического университета им. А.Н. Туполева–КАИ

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Issue

Vol. 23 (2026) No. 1

Pages

8-21

Section

Mechanics

Dates

Submitted

November 16, 2025

Accepted

March 2, 2026

Published

March 24, 2026

How to Cite

[1]
Velikanov, P.G., Operational and non-operational methods for obtaining fundamental solutions of~partial differential equations for nanoplates. Part III. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2026, т. 23, № 1, pp. 8–21. DOI: 10.31429/vestnik-23-1-8-21

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