Calculation of fractional integrals using partial sums of Fourier series for structural mechanics problems

Anis Galimyanov,Tatyana Gorskaya,E Vdovin

doi:10.1051/e3sconf/202127403011

Abstract

The goal of this study is to develop and apply an approximate method for calculating integrals that are part of models using Riemann-Liouville integrals, and to create a software product that allows such calculations for given functions. The main results of the study consist in the construction of a quadrature formula for an integral, and the cases where the density of the integral is a function from the spaces of continuous functions with generalized derivatives with weight and the Helder classes of functions with weight were considered. For the proposed quadrature formula we further investigated the error of its approximation in the spaces of continuous functions and quadratic-summing functions with weight. As a result of the study, effective error estimates of the approximating apparatus in the proposed classes of functions have been established. In addition, the approximated method has been implemented on the computer in the form of a program in the C language. The significance of the obtained results for the construction industry consists in the fact that when solving problems, including problems on finding the shapes of structures, taking into account the properties of materials, environmental changes, in the models of which the Riemann-Liouville integrals are used, it will be possible to apply an approximate approach, the quadrature formula proposed in the article.

Highlights

There has been increasing interest in the study of fractional order differential equations in which the unknown function is contained in the fractional order derivative, as well as in fractional order integral equations
Jrbashyan's research related to the modern theory of fractional calculus. Extensive applications of this mathematical apparatus in various fields of science and industry [4], especially in fields related to nanotechnology, diffusion problems, as well as in creating
In particular, using fractional order integrals in solving problems in continuum mechanics related to studies of elasticity theory, the authors [5] propose a generalized theory capable of capturing both stiffness and softening effects, under selected external loads and boundary conditions

Summary

Introduction

There has been increasing interest in the study of fractional order differential equations in which the unknown function is contained in the fractional order derivative, as well as in fractional order integral equations. This is due to a number of reasons. In particular, using fractional order integrals in solving problems in continuum mechanics related to studies of elasticity theory, the authors [5] propose a generalized theory capable of capturing both stiffness and softening effects, under selected external loads and boundary conditions. In a number of technical problems (in diffusion problems, for example), economic (for example, in sustainable development problems) unstable systems are used, description of which is connected with fractional integrals, on the basis of which models of processes under study are created [6], and for regulation of such processes PID regulators are created, the basis of which uses fractional integration, the developers [7, 8] suggest ways to improve them

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