The paper is devoted to the basic properties of fractional integrals. It is a survey of the well-known properties of fractional integrals, however, the authors tried to present the known information about fractional integrals as short and transparently as possible. We introduce fractional integrals on the compact interval and on the semi-axes, consider the famous Hardy-Littlewood theorem and other properties of integrability of fractional integrals. Among other basic properties, we consider Holder continuity and establish to what extent fractional integration increases the smoothness of the integrand. Also, we establish continuity of fractional integrals according to the index of fractional integration, both at strictly positive value and at zero. Then we consider properties of restrictions of fractional integrals from semi-axes on the compact interval. Generalized Minkowsky inequality is applied as one of the important tools. Some examples of calculating fractional integrals are provided.
In this paper, we studied the natural oscillations in the vertical and horizontal surface of viaduct of Ivanci. A number of schemes which allow determining the frequency on a more simple dependencies are given
The article, using the example of a three-flight irregular beam, studies the influence of irregularities of linear, stiffness- and mass-related parameters on the change of own frequencies. It has been shown that at certain parameters of irregularity it is possible to use in practical calculations simple solutions for regular schemes, the obtained approximated dependences, charts and nomograms.
In this paper on the example of continuous spans of bridges the asymmetry influence of the scheme on the frequency and forms of natural oscillations is investigated. It is shown that under certain span lengths of regular schemes the use of them to determine the span frequencies is possible.
The paper is devoted to the approximate solutions of the Fredholm integral equations of the second kind with the weak singular kernel that can have additional singularity in the numerator. We describe two problems that lead to such equations. They are the problem of minimization of small deviation and the entropy minimization problem. Both of them appear when considering dynamical system involving mixed fractional Brownian motion. In order to deal with the kernel with additional singularity applying well-known methods for weakly singular kernels, we prove the theorem on the approximation of solution of integral equation with the kernel containing additional singularity by the solutions of the integral equations whose kernels are weakly singular but the numerator is continuous. We demonstrate numerically how our methods work being applied to our specific integral equations.
The article is devoted to the possibility of replacement of complex inseparable irregular bridge structures by their regular counterparts, with minimal errors in definition of the own frequencies. The analysis has been performed on a real five-span bridge structure, designed under a 66 +126 +147 +115 + 76 – meter scheme.
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