Some New Delay Integral Inequalities Based on Modified Riemann-Liouville Fractional Derivative and Their Applications

By using the properties of modified Riemann-Liouville fractional derivative, some new delay integral inequalities have been studied. First, we offered explicit bounds for the unknown functions, then we applied the results to the research concerning the boundness, uniqueness and continuous dependence on the initial for solutions to certain fractional differential equations.

协同拟凸函数的Riemann-Liouville分数阶积分不等式

基于Riemann-Liouville分数阶积分,对协同拟凸函数的Hermite-Hadamard分数阶积分不等式进行了研究。剖析Riemann-Liouville分数阶积分的运算特点,深入探究SAR?KAYA给出的Riemann-Liouville分数阶积分恒等式。在该Riemann-Liouville分数阶积分恒等式的基础上,利用二元函数的单调性和协同拟凸性,巧妙应用三角不等式和H?lder不等式等经典不等式,建立了若干个协同拟凸函数的Hermite-Hadamard分数阶积分不等式。

基于Riemann-Liouville分数阶积分Bernstein-Kantorovich算子的逼近性质

文章构造一种基于Riemann-Liouville分数阶积分的Bernstein-Kantorovich算子。利用一阶、二阶光滑模、Peetre’-K泛函和Lipschitz型极大函数等工具研究该算子的近似性质。然后利用一阶、二阶和四阶中心矩对该算子建立Vorononskaja型渐进公式。该算子的构建使得在曲线曲面造型方面对于给定的函数将会有更小的近似误差。

A Technique for Estimation of Box Dimension about Fractional Integral

This paper discusses further the roughness of Riemann-Liouville fractional integral on an arbitrary fractal continuous functions that follows Rfs. [1]. A novel method is used to reach a similar result for an arbitrary fractal function , where is the Riemann-Liouville fractional integral. Furthermore, a general resultis arrived at for 1-dimensional fractal functions such as with unbounded variation and(or) infinite lengths, which can infer all previous studies such as [2] [3]. This paper’s estimation reveals that the fractional integral does not increase the fractal dimension of f(x), i.e. fractional integration does not increase at least the fractal roughness. And the result has partly answered the fractal calculus conjecture and completely answered this conjecture for all 1-dimensional fractal function (Xiao has not answered). It is significant with a comparison to the past researches that the box dimension connection between a fractal function and its Riemann-Liouville integral has been carried out only for Weierstrass type and Besicovitch type functions, and at most Hlder continuous. Here the proof technique for Riemann-Liouville fractional integral is possibly of methodology to other fractional integrals.

THE HADAMARD INEQUALITY FOR CONVEX FUNCTION VIA FRACTIONAL INTEGRALS

In this paper,we establish several inequalities for some diferantiable mappings that are connected with the Riemann-Liouville fractional integrals.The analysis used in the proofs is fairly elementary.

On the Connection between the Order of Riemann-Liouvile Fractional Calculus and Hausdorff Dimension of a Fractal Function

This paper investigates the fractal dimension of the fractional integrals of a fractal function. It has been proved that there exists some linear connection between the order of Riemann-Liouvile fractional integrals and the Hausdorff dimension of a fractal function.

关于k-Riemann-Liouville分数阶积分的Ostrowski型积分不等式

引入k-Riemann-Liouville分数阶积分和h-凸函数,通过建立k-Riemann-Liouville分数阶积分的Ostrowski型积分不等式,构造了一些新的Hermite-Hadamard型积分不等式.相比较于已有的一些结果,所得结果在积分形式和数据点类型两个方面有一定的优势,使得Hermite-Hadamard型积分不等式适用范围更广.最后给出Ostrowski型k-Riemann-Liouville分数阶积分不等式在概率方面的应用.

Hermite-Hadamard Type Fractional Integral Inequalities for Preinvex Functions

In this article,we extend some estimates of the right-hand side of the Hermite-Hadamard type inequality for preinvex functions with fractional integral.The notion of logarithmically s-Godunova-Levin-preinvex function in second sense is introduced and then a new Hermite-Hadamard inequality is derived for the class of logarithmically s-Godunova-Levin-preinvex function.

凸函数的关于Riemann-Liouville分式积分的Hermite-Hadamard型不等式

凸函数的Hermite-Hadamard型不等式具有重要的理论意义,并且有着广泛的应用.首先建立了一个关于Riemann-Liouville分式积分的等式,然后讨论凸函数的关于Riemann-Liouville分式积分的Hermite-Hadamard型积分不等式,得到了若干个结果.

一个含参量积分例题的研究性学习和启示——引入Riemann-Liouville分数阶积分和导数的一种方法

本文将一个关于含参量积分的例题拓展成了针对学生的研究性学习课题.通过对这一例题的研究性学习,对整数阶微积分进行了推广,给出了Riemann-Liouville分数阶积分和导数的定义,并通过例题强化了学生对这一定义的认识,扩大了学生的知识视野.最后强调了研究性学习在数学分析教学中的重要性.

关于Riemann-Liouville分数积分的Hermite-Hadamard型不等式

主要建立2个关于已知函数导数的重要Hermite-Hadamard型Riemann-Liouville分数积分恒等式,进而得到关于某些特殊凸函数有意义的Riemann-Liouville分数积分的Hermite-Hadamard型不等式,如s-凸函数、m-凸函数、(s,m)-凸函数等.这些结果改进了一些文献中的有关结果,并结合几个常用的平均值给出应用.

Riemann-Liouville分数阶积分的图像去噪算法

传统图像去噪算法易丢失图像边缘和纹理细节,使图像模糊不清,为后续图像分析处理带来困难。为克服传统图像去噪算法的缺点,根据Riemann-Liouville分数阶积分,构造一种分数阶积分掩膜算子,对测试图像进行图像去噪仿真实验。同时,引入客观评价标准峰值信噪比和灰度共生矩阵,对分数阶积分掩膜算子的去噪效果进行分析。结果表明,不同于传统图像去噪算法,该分数阶积分掩膜算子可在去除图像噪声的同时,有效保留图像的边缘和纹理细节信息。

Riemann-Liouville分数阶微积分的定义及其性质

作为微积分理论的发展,分数阶微积分的概念早已提出,实际应用的介入为它注入了新的生机.分数阶微积分成为研究分数阶微分方程,分形函数的有力工具,它被应用于分形集合、分形函数、分形PDE、函数空间等领域,近年来分数阶微积分被广泛的应用到建立各种数学模型.

Fractional Versions of the Fundamental Theorem of Calculus

The concept of fractional integral in the Riemann-Liouville, Liouville, Weyl and Riesz sense is presented. Some properties involving the particular Riemann-Liouville integral are mentioned. By means of this concept we present the fractional derivatives, specifically, the Riemann-Liouville, Liouville, Caputo, Weyl and Riesz versions are discussed. The so-called fundamental theorem of fractional calculus is presented and discussed in all these different versions.

含三个Riemann-Liouville分数阶导数的脉冲Langevin型方程的边值问题的可解性

设n,l,k为正整数且α∈(n-1,n),β∈(l-1,l),γ∈(k-1,k).该文首先利用迭代方法给出具有三个分数阶导数的Langevin方程[D0+^αD0+^β?λD0+^γ]x(t)=P(t)的连续通解.然后,该文使用数学归纳法获得脉冲分数阶Langevin方程[D0+^αD0+^β?λD0+^γ]x(t)=P(t),t∈(ti,ti+1],i∈N0^m分片连续通解.接下来,该文运用获得的结果研究具有三个分数阶导数α,β∈(1,2),γ∈(0,1)的非线性脉冲Langevin方程的一类边值问题,通过将其化为积分方程,运用不动点定理建立这类边值问题解的存在性定理.最后,该文给出例子说明了主要结果的应用.

区间值Riemann-Liouville型分数阶积分等式

利用区间分析理论以及区间值Riemann-Liouville型分数阶积分证明了gH-可导区间凸函数的2个新等式,并给出了相应的例子证明结果的准确性.

Numerical Solutions of Fractional Differential Equations by Using Fractional Taylor Basis

In this paper, a new numerical method for solving fractional differential equations(FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.

Weierstrass函数的盒维数与Riemann-Liouville分数阶积分的阶之间联系更进一步的研究

本文中,我们更完整地对IE上Weierstrass函数分形维数与Riemann-Liouville分数阶微积分的阶之间进行了研究。即当α + v不再小于1时,Weierstrass函数的Riemann-Liouville分数阶积分的分形维数被证明是1。

涉及Riemann-Liouville分数阶积分的一个多参数Hermite-Hadamard型不等式

目的:研究Hermite-Hadamard不等式中间部分与右侧之差的上界估计的一类不等式的推广。方法:建立一个涉及高阶导数的关于Riemann-Liouville分数阶积分的含参数恒等式,利用该恒等式并借助s-凸函数、超几何函数的性质及分析不等式技巧,来构造和证明Hermite-Hadamard型不等式。结果:在|f^((n))|^(q)为s-凸函数的条件下,建立一个含Riemann-Liouville分数阶积分的多参数新Hermite-Hadamard型不等式。结论:所得到的多参数Hermite-Hadamard型不等式统一推广了以往文献中的一些结果,当其参数取特定值时可导出一些已有的不等式以及新的不等式。

On the Approximate Solution of Fractional Logistic Differential Equation Using Operational Matrices of Bernstein Polynomials

In this paper, operational matrices of Bernstein polynomials (BPs) are presented for solving the non-linear fractional Logistic differential equation (FLDE). The fractional derivative is described in the Riemann-Liouville sense. The operational matrices for the fractional integration in the Riemann-Liouville sense and the product are used to reduce FLDE to the solution of non-linear system of algebraic equations using Newton iteration method. Numerical results are introduced to satisfy the accuracy and the applicability of the proposed method.