Results 221 to 230 of about 10,319 (253)

The evaluation of a class of fractional part integrals

Integral Transforms and Special Functions, 2015
The paper is about calculating in closed form the following multiple fractional part integral Ik=∫01⋯∫01x1x2⋯xn−1xnkdx1dx2⋯dxn, where n≥3, k≥1 are integers and {x} denotes the fractional part of x. We prove that the integral Ik can be expresses in terms of a difference of two rational numbers and a series involving Riemann zeta function values and some
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On Finite Part Integrals and Hadamard-Type Fractional Derivatives

Journal of Computational and Nonlinear Dynamics, 2018
This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist.
Li Ma, Changpin Li
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Fractional Part Integrals

2013
This chapter introduces the reader to a collection of problems that are rarely seen: the evaluation of exotic integrals involving a fractional part term, called fractional part integrals. The problems were motivated by the interesting formula \(\int _{0}^{1}\left \{1/x\right \}\mathrm{d}x = 1-\gamma ,\) which connects an exotic integral to the Euler ...
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Some results on generalized multiple fractional part integrals

Integral Transforms and Special Functions, 2015
In this paper, the following multiple fractional part integrals In,mp1,p2,…,pn=∫[0,1]n∏j=1nxjpj{Sn−1}mdx1⋯dxn and Jn,mp=∫[0,1]nSnp{Sn−1}mdx1⋯dxn are calculated for non-negative integers p1,p2,…,pn,p,m and positive integer n, where {u} denotes the fractional part of u and Sn=x1+x2+⋯+xn.
Aijuan Li, Zhongfeng Sun, Huizeng Qin
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Representation of a class of multiple fractional part integrals and their closed form

Integral Transforms and Special Functions, 2016
ABSTRACTIn this paper, the following generalized multiple fractional part integrals: In,μα1,α2,…,αn(a)=∫01∫01⋯∫01x1α1x2α2⋯xnαn1ax1x2⋯xnμ×dx1dx2⋯dxn, and k1,k2,…,knIn,μα1,α2,…,αn(a)=∫01∫01⋯∫011ax1x2⋯xnμ×∏i=1nxiαilnki⁡1xidx1dx2⋯dxn, are considered for complex numbers μ,αi (i=1,2,3,…,n), positive integers ki (i=1,2,…,n) and positive number a, where {u ...
Aijuan Li, Zhongfeng Sun, Huizeng Qin
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A class of fractional part integrals and zeta function values

Integral Transforms and Special Functions, 2013
The paper is about calculating a special class of fractional part integrals of the form where n is a positive integer and x denotes the fractional part of x. We show that these integrals can be expressed as rational linear combination of Riemann zeta function values and a series involving the difference of two zeta function values of consecutive ...
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The memory-dependent FPK equation for fractional Gaussian noise

Probabilistic Engineering Mechanics
This paper aims to explore non-Markovian dynamics of nonlinear dynamical systems subjected to fractional Gaussian noise (FGN) and Gaussian white noise (GWN). A novel memory-dependent Fokker-Planck-Kolmogorov (memFPK) equation is developed to characterize
Lifang Feng, B. Pei, Yong Xu
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Optimal Feedback Control of Nonlinear Fractional-Order Systems

2025 7th International Conference on Electronic Engineering and Informatics (EEI)
This paper presents an optimal feedback control design framework for a class of nonlinear fractional-order dynamical systems arising in batch fermentation processes.
Qiong Li, Jinxu Cui
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On distortion of the real part in a class of analytic functions related to fractional integration

Complex Variables, Theory and Application: An International Journal, 1986
In close relation to fractional integration a one-parameter family of operators acting on a class of analytic functions regular in the unit disk is observed. In considering functionals referring to the real part, the author derives distortion inequalities showing how their bounds change after application of operators.
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Parameter estimation of fractional chaotic systems based on stepwise integration and response sensitivity analysis

Nonlinear dynamics, 2023
Tao Zhang, Zhong-rong Lu, Jike Liu, Guang Liu   +3 more
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