Reverse Minkowski Inequalities Pertaining to New Weighted Generalized Fractional Integral Operators

We investigate fractional operators with homogeneous kernel in Morrey spaces. In particular, we prove that fractional integral operators and fractional maximal operators with homogeneous kernel are bounded from the Calderón product of Morrey spaces to certain Morrey spaces. Our results can be seen as a generalization of a recent result on the relation between the boundedness of (classical) fractional operators and interpolation of Morrey spaces. What is new about this paper is not only the passage from the classical fractional integral operators to the rough integral operators. Even the case of fractional integral operators, handled in earlier papers, is significantly simplified.

A new fractional derivative operator with generalized cardinal sine kernel: Numerical simulation

The present work investigates the applicability and effectiveness of generalized proportional fractional integral (mathcal{GPFI}) operator in another sense. We aim to derive novel weighted generalizations involving a family of positive functions n (nin mathbb{N}) for this recently proposed operator. As applications of this operator, we can generate notable outcomes for Riemann–Liouville (mathcal{RL}) fractional, generalized mathcal{RL}-fractional operator, conformable fractional operator, Katugampola fractional integral operator, and Hadamard fractional integral operator by changing the domain. The proposed strategy is vivid, explicit, and it can be used to derive new solutions for various fractional differential equations applied in mathematical physics. Certain remarkable consequences of the main theorems are also figured.

The multivariate Mittag–Leffler function is introduced and used to establish fractional calculus operators. It is shown that the fractional derivative and integral operators are bounded. Some fundamental characteristics of the new fractional operators, such as the semi-group and inverse characteristics, are studied. As special cases of these novel fractional operators, several fractional operators that are already well known in the literature are acquired. The generalized Laplace transform of these operators is evaluated. By involving the explored fractional operators, a kinetic differintegral equation is introduced, and its solution is obtained by using the Laplace transform. As a real-life problem, a growth model is developed and its graph is sketched.

In the present case, we propose the novel generalized fractional integral operator describing Mittag–Leffler function in their kernel with respect to another function Ф. The proposed technique is to use graceful amalgamations of the Riemann–Liouville (RL) fractional integral operator and several other fractional operators. Meanwhile, several generalizations are considered in order to demonstrate the novel variants involving a family of positive functions n (n ∈ N) for the proposed fractional operator. In order to confirm and demonstrate the proficiency of the characterized strategy, we analyze existing fractional integral operators in terms of classical fractional order. Meanwhile, some special cases are apprehended and the new outcomes are also illustrated. The obtained consequences illuminate that future research is easy to implement, profoundly efficient, viable, and exceptionally precise in its investigation of the behavior of non-linear differential equations of fractional order that emerge in the associated areas of science and engineering.

The study discussed in this article is driven by the realization that many physical processes may be understood by using applications of fractional operators and special functions. In this study, we present and examine a fractional integral operator with an I-function in its kernel. This operator is used to solve several fractional differential equations (FDEs). FDE has a set of particular cases whose solutions represent different physical phenomena. Many mathematical physics, biology, engineering, and chemistry problems are identified and solved using FDE. Specifically, a few exciting relations involving the new fractional operator with incomplete I-function (IIF) in its kernel and classical Riemann Liouville fractional integral and derivative operators, the Hilfer fractional derivative operator, and the generalized composite fractional derivate (GCFD) operator are established. The discovery and investigation of several important exceptional cases follow this.

The main goal of this paper is to describe the new version of extended Bessel–Maitland function and discuss its special cases. Then, using the aforementioned function as their kernels, we develop the generalized fractional integral and differential operators. The convergence and boundedness of the newly operators and compare them with the existing operators such as the Saigo and Riemann–Liouville fractional operators are explored. The integral transforms of newly defined and generalized fractional operators in terms of the generalized Fox–Wright function are presented. Additionally, we discuss a few exceptional cases of the main result.

The motivation of this research is to introduce some new fractional operators called “the improved fractional (IF) operators”. The originality of these fractional operators comes from the fact that they repeat the method on general forms of conformable integration and differentiation rather than on the traditional ones. Hence the convolution kernels correlating with the IF operators are served in conformable abstract forms. This extends the scientific application scope of their fractional calculus. Also, some results are acquired to guarantee that the IF operators have advantages analogous to the familiar fractional integral and differential operators. To unveil the inverse and composition properties of the IF operators, certain function spaces with their characterizations are presented and analyzed. Moreover, it is remarkable that the IF operators generalize some fractional and conformable operators proposed in abundant preceding works. As scientific applications, the resistor–capacitor electrical circuits are analyzed under some IF operators. In the case of constant and periodic sources, this results in novel voltage forms. In addition, the overall influence of the IF operators on voltage behavior is graphically simulated for certain selected fractional and conformable parameter values. From the standpoint of computation, the usage of new IF operators is not limited to electrical circuits; they could also be useful in solving scientific or engineering problems.

In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous fluid, fracture mechanics, continuous population, equilibrium, visco-elastic deformation, heat conduction problems, and others. In this manuscript, we introduce some novel notions of generalized preinvexity, namely the (m,tgs)-type s-preinvex function, Godunova–Levin (s,m)-preinvex of the 1st and 2nd kind, and a prequasi m-invex. Furthermore, we explore a new variant of the Hermite–Hadamard (H–H), Fejér, and Pachpatte-type inequality via a generalized fractional integral operator, namely, a non-conformable fractional integral operator (NCFIO). In addition, we explore new equalities. With the help of these equalities, we examine and present several extensions of H–H and Fejér-type inequalities involving a newly introduced concept via NCFIO. Finally, we explore some special means as applications in the aspects of NCFIO. The results and the unique situations offered by this research are novel and significant improvements over previously published findings.

Depth images are often accompanied by unavoidable and unpredictable noise. Depth image denoising algorithms mainly attempt to fill hole data and optimise edges. In this paper, we study in detail the problem of effectively filtering the data of depth images under noise interference. The classical filtering algorithm tends to blur edge and texture information, whereas the fractional integral operator can retain more edge and texture information. In this paper, the Grünwald–Letnikov-type fractional integral denoising operator is introduced into the depth image denoising process, and the convolution template of this operator is studied and improved upon to build a fractional integral denoising model and algorithm for depth image denoising. Depth images from the Redwood dataset were used to add noise, and the mask constructed by the fractional integral denoising operator was used to denoise the images by convolution. The experimental results show that the fractional integration order with the best denoising effect was −0.4 ≤ ν ≤ −0.3 and that the peak signal-to-noise ratio was improved by +3 to +6 dB. Under the same environment, median filter denoising had −15 to −30 dB distortion. The filtered depth image was converted to a point cloud image, from which the denoising effect was subjectively evaluated. Overall, the results prove that the fractional integral denoising operator can effectively handle noise in depth images while preserving their edge and texture information and thus has an excellent denoising effect.

Fractional operators are widely used in mathematical models describing abnormal and nonlocal phenomena. Although there are extensive numerical methods for solving the corresponding model problems, theoretical analysis such as the regularity result, or the relationship between the left-side and right-side fractional operators is seldom mentioned. Instead of considering the fractional derivative spaces, this paper starts from discussing the image spaces of Riemann-Liouville fractional integrals of LP(Ω) functions, since the fractional derivative operators that are often used are all pseudo-differential. Then the high regularity situation—the image spaces of Riemann-Liouville fractional integral operators on the Wm,p(Ω) space is considered. Equivalent characterizations of the defined spaces, as well as those of the intersection of the left-side and right-side spaces are given. The behavior of the functions in the defined spaces at both the nearby boundary point/points and the points in the domain is demonstrated in a clear way. Besides, tempered fractional operators are shown to be reciprocal to the corresponding Riemann-Liouville fractional operators, which is expected to contribute some theoretical support for relevant numerical methods. Last, we also provide some instructions on how to take advantage of the introduced spaces when numerically solving fractional equations.

A unified fast time-stepping method for both fractional integral and derivative operators is proposed. The fractional operator is decomposed into a local part with memory length $\Delta T$ and a history part, where the local part is approximated by the direct convolution method and the history part is approximated by a fast memory-saving method. The fast method has $O(n_0+\sum_{\ell}^L{q}_{\alpha}(N_{\ell}))$ active memory and $O(n_0n_T+ (n_T-n_0)\sum_{\ell}^L{q}_{\alpha}(N_{\ell}))$ operations, where $L=\log(n_T-n_0)$, $n_0={\Delta T}/\tau,n_T=T/\tau$, $\tau$ is the stepsize, $T$ is the final time, and ${q}_{\alpha}{(N_{\ell})}$ is the number of quadrature points used in the truncated Laguerre--Gauss (LG) quadrature. The error bound of the present fast method is analyzed. It is shown that the error from the truncated LG quadrature is independent of the stepsize, and can be made arbitrarily small by choosing suitable parameters that are given explicitly. Numerical examples are presented to verify the effectiveness of the current fast method.

Fractional differential and integral operators, Dirichlet averages, and splines of complex order are three seemingly distinct mathematical subject areas addressing different questions and employing different methodologies. It is the purpose of this paper to show that there are deep and interesting relationships between these three areas. First a brief introduction to fractional differential and integral operators defined on Lizorkin spaces is presented and some of their main properties exhibited. This particular approach has the advantage that several definitions of fractional derivatives and integrals coincide. We then introduce Dirichlet averages and extend their definition to an infinite-dimensional setting that is needed to exhibit the relationships to splines of complex order. Finally, we focus on splines of complex order and, in particular, on cardinal B-splines of complex order. The fundamental connections to fractional derivatives and integrals as well as Dirichlet averages are presented.

Motivated by the demonstrated potential for their applications in various research areas such as those in mathematical, physical, engineering, and statistical sciences, our main object in this paper is to introduce and investigate a fractional integral operator that contains a certain generalized multi‐index Mittag‐Leffler function in its kernel. In particular, we establish some interesting expressions for the composition of such well‐known fractional integral and fractional derivative operators as (for example) the Riemann‐Liouville fractional integral and fractional derivative operators, the Hilfer fractional derivative operator, and the above‐mentioned fractional integral operator with the generalized multi‐index Mittag‐Leffler function in its kernel. The main findings in this paper are shown to generalize the results that were derived earlier by Kilbas et al and Srivastava et al. Finally, in this paper, we derive integral representations for the product of 2 generalized multi‐index Mittag‐Leffler functions in terms of the familiar Fox‐Wright hypergeometric function.

Fractional integral operators connected with real-valued scalar functions of matrix argument are applied in various problems of mathematics, statistics and natural sciences. In this survey we start with considering the case of a Gauss hypergeometric function with the argument being a rectangular matrix. Subsequently, some fractional integral operators are introduced which complement theresults available on fractional operators in the matrix variate cases. Several properties and limiting forms are derived. Then the pathway idea is incorporated to move among several different functional forms. When these are used as models for problems in the natural sciences, then these can cover the ideal situations, neighborhoods, in between stages and paths leading to optimal situations.