Composite Fractional Trapezoidal Rule with Romberg Integration

The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator.This would be achieved by first establishing a fractional-order version of the 2-point Trapezoidal rule and then by proposing another fractional-order version of the(n+1)-composite Trapezoidal rule.In particular,the so-called divided-difference formula is typically employed to derive the 2-point Trapezoidal rule,which has accordingly been used to derive a more accurate fractional-order formula called the(n+1)-composite Trapezoidal rule.Additionally,in order to increase the accuracy of the proposed approximations by reducing the true errors,we incorporate the so-called Romberg integration,which is an extrapolation formula of the Trapezoidal rule for integration,into our proposed approaches.Several numerical examples are provided and compared with a modern definition of the Riemann-Liouville fractional integral operator to illustrate the efficacy of our scheme.

AN INTEGRATION BY PARTS FORMULA FOR STOCHASTIC HEAT EQUATIONS WITH FRACTIONAL NOISE

In this paper,we establish the integration by parts formula for the solution of fractional noise driven stochastic heat equations using the method of coupling.As an application,we also obtain the shift Harnack inequalities.

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.

Chromatin condensation but not DNA integrity of pig sperm is greater in the sperm-rich fraction

Background Protamination and condensation of sperm chromatin as well as DNA integrity play an essential role during fertilization and embryo development.In some mammals,like pigs,ejaculates are emitted in three separate fractions:pre-sperm,sperm-rich(SRF)and post sperm-rich(PSRF).These fractions are known to vary in volume,sperm concentration and quality,as well as in the origin and composition of seminal plasma(SP),with differences being also observed within the SRF one.Yet,whether disparities in the DNA integrity and chromatin condensation and pro-tamination of their sperm exist has not been interrogated.Results This study determined chromatin protamination(Chromomycin A3 test,CMA_(3)),condensation(Dibromobi-mane test,DBB),and DNA integrity(Comet assay)in the pig sperm contained in the first 10 m L of the SRF(SRF-P1),the remaining portion of the sperm-rich fraction(SRF-P2),and the post sperm-rich fraction(PSRF).While chromatin protamination was found to be similar between the different ejaculate fractions(P>0.05),chromatin condensation was seen to be greater in SRF-P1 and SRF-P2 than in the PSRF(P=0.018 and P=0.004,respectively).Regarding DNA integrity,no differences between fractions were observed(P>0.05).As the SRF-P1 has the highest sperm concentra-tion and ejaculate fractions are known to differ in antioxidant composition,the oxidative stress index(OSi)in SP,calcu-lated as total oxidant activity divided by total antioxidant capacity,was tested and confirmed to be higher in the SRF-P1 than in SRF-P2 and PSRF(0.42±0.06 vs.0.23±0.09 and 0.08±0.00,respectively;P<0.01);this index,in addition,was observed to be correlated to the sperm concentration of each fraction(Rs=0.973;P<0.001).Conclusion While sperm DNA integrity was not found to differ between ejaculate fractions,SRF-P1 and SRF-P2 were observed to exhibit greater chromatin condensation than the PSRF.This could be related to the OSi of each fraction.

Time series analysis-based seasonal autoregressive fractionally integrated moving average to estimate hepatitis B and C epidemics in China

BACKGROUND Hepatitis B(HB)and hepatitis C(HC)place the largest burden in China,and a goal of eliminating them as a major public health threat by 2030 has been set.Making more informed and accurate forecasts of their spread is essential for developing effective strategies,heightening the requirement for early warning to deal with such a major public health threat.AIM To monitor HB and HC epidemics by the design of a paradigmatic seasonal autoregressive fractionally integrated moving average(SARFIMA)for projections into 2030,and to compare the effectiveness with the seasonal autoregressive integrated moving average(SARIMA).METHODS Monthly HB and HC incidence cases in China were obtained from January 2004 to June 2023.Descriptive analysis and the Hodrick-Prescott method were employed to identify trends and seasonality.Two periods(from January 2004 to June 2022 and from January 2004 to December 2015,respectively)were used as the training sets to develop both models,while the remaining periods served as the test sets to evaluate the forecasting accuracy.RESULTS There were incidents of 23400874 HB cases and 3590867 HC cases from January 2004 to June 2023.Overall,HB remained steady[average annual percentage change(AAPC)=0.44,95%confidence interval(95%CI):-0.94-1.84]while HC was increasing(AAPC=8.91,95%CI:6.98-10.88),and both had a peak in March and a trough in February.In the 12-step-ahead HB forecast,the mean absolute deviation(15211.94),root mean square error(18762.94),mean absolute percentage error(0.17),mean error rate(0.15),and root mean square percentage error(0.25)under the best SARFIMA(3,0,0)(0,0.449,2)12 were smaller than those under the best SARIMA(3,0,0)(0,1,2)12(16867.71,20775.12,0.19,0.17,and 0.27,respectively).Similar results were also observed for the 90-step-ahead HB,12-step-ahead HC,and 90-step-ahead HC forecasts.The predicted HB incidents totaled 9865400(95%CI:7508093-12222709)cases and HC totaled 1659485(95%CI:856681-2462290)cases during 2023-2030.CONCLUSION Under current interventions,China faces enormous challenges to eliminate HB and HC epidemics by 2030,and effective strategies must be reinforced.The integration of SARFIMA into public health for the management of HB and HC epidemics can potentially result in more informed and efficient interventions,surpassing the capabilities of SARIMA.

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。

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 FRACTIONAL TYPE MARCINKIEWICZ INTEGRALS AND COMMUTATORS ON WEIGHTED HARDY SPACES

This paper is devoted to studying the behaviors of the fractional type Marcinkiewicz integralsμΩ,βand the commutatorsμΩ,βb generated byμΩ,βwith b b∈Lloc(Rn)on weighted Hardy spaces.Under the assumption of that the homogeneous kernelΩsatisfies certain regularities,the authors obtain the boundedness ofμΩ,βfrom the weighted Hardy spaces Hωpp(Rn)to the weighted Lebesgue spaces Lωqq(Rn)for n/(n+β)≤<p≤1 with 1/q=1/p-β/n,as well as the same(Hωpp,Lωqq)-boudedness ofμΩ,βb when b belongs to BMOωp,p(Rn),which is a non-trivial subspace of BMO(Rn).

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。

Fractional differential equations of motion in terms of combined Riemann-Liouville derivatives

In this paper, we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system. A combined Riemann-Liouville fractional derivative operator is defined, and a fractional Hamilton principle under this definition is established. The fractional Lagrange equations and the fractional Hamilton canonical equations are derived from the fractional Hamilton principle. A number of special cases are given, showing the universality of our conclusions. At the end of the paper, an example is given to illustrate the application of the results.

Study for System of Nonlinear Differential Equations with Riemann-Liouville Fractional Derivative

In this work, we study existence theorem of the initial value problem for the system of fractional differential equations where Dα denotes standard Riemann-Liouville fractional derivative, 0 and A ?is a square matrix. At the same time, power-type estimate for them has been given.

An inverse problem to estimate an unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid

In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.

Nonnegative Solutions for a Riemann-Liouville Fractional Boundary Value Problem

We investigate the existence of nonnegative solutions for a Riemann-Liouville fractional differential equation with integral terms, subject to boundary conditions which contain fractional derivatives and Riemann-Stieltjes integrals. In the proof of the main results, we use the Banach contraction mapping principle and the Krasnosel’skii fixed point theorem for the sum of two operators.

Existence of Solutions for Fractional Differential Equations Involving Two Riemann-Liouville Fractional Orders

In this work, we study existence and uniqueness of solutions for multi-point boundary value problem of nonlinear fractional differential equations with two fractional derivatives. By using the variety of fixed point theorems, such as Banach’s fixed point theorem, Leray-Schauder’s nonlinear alternative and Leray-Schauder’s degree theory, the existence of solutions is obtained. At the end, some illustrative examples are discussed.

THE NONEMPTINESS AND COMPACTNESS OF MILD SOLUTION SETS FOR RIEMANN-LIOUVILLE FRACTIONAL DELAY DIFFERENTIAL VARIATIONAL INEQUALITIES

This paper investigates the nonemptiness and compactness of the mild solution set for a class of Riemann-Liouville fractional delay differential variational inequalities,which are formulated by a Riemann-Liouville fractional delay evolution equation and a variational inequality.Our approach is based on the resolvent technique and a generalization of strongly continuous semigroups combined with Schauder's fixed point theorem.

THE HADAMARD INEQUALITY FOR CONVEX FUNCTION VIA FRACTIONAL INTEGRALS

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

Quantum entangled fractional Fourier transform based on the IWOP technique

In our previous papers,the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics,and the calculation produced quantum mechanical operators corresponding to the generation of fractional Fourier transform.The core function of the coordinate-momentum exchange operators in the addition law of fractional Fourier transform was analyzed too.In this paper,the bivariate operator Hermite polynomial theory and the technique of integration within an ordered product of operators(IWOP)are used to establish the entanglement fractional Fourier transform theory to the extent of quantum.A new function generating formula and an operator for generating quantum entangled fractional Fourier transform are obtained using the fractional Fourier transform relationship in a pair of conjugated entangled state representations.

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.

FRACTIONAL ORDER BOUNDARY VALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS INVOLVING PETTIS INTEGRAL

In this article, we investigate the existence of Pseudo solutions for some fractional order boundary value problem with integral boundary conditions in the Banach space of continuous function equipped with its weak topology. The class of such problems constitute a very interesting and important class of problems. They include two, three, multi-point and nonlocal boundary-value problems as special cases. In our investigation, the right hand side of the above problem is assumed to be Pettis integrable function. To encompass the full scope of this article, we give an example illustrating the main result.

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-SEPARATED TYPE INTEGRAL BOUNDARY CONDITIONS

In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.