Fractional Integral Operators Research Articles

Mittag-Leffler wavelets and their applications for solving fractional optimal control problems

Herein, we design a new scheme for finding approximate solutions to fractional optimal control problems (OCPs) with and without delay. In this strategy, we introduce Mittag-Leffler wavelet functions and develop a new Riemann–Liouville fractional integral operator for these functions utilizing the hypergeometric function. The properties of the operational matrix have reflected well in the process of the numerical method and affect the accuracy of the proposed method directly. Employing the Riemann–Liouville fractional integral operator, delay operational matrix, and Galerkin method, the considered problems lead to systems of algebraic equations. An error analysis is proposed. Finally, some illustrative numerical tests are given to show the precision and validity of the suggested technique. The proposed method is very efficient for solving the OCPs with delay and without delay, and gives very accurate results.

Refinements and Applications of Hermite–Hadamard-Type Inequalities Using Hadamard Fractional Integral Operators and GA-Convexity

In this paper, several applications of the Hermite–Hadamard inequality for fractional integrals using GA-convexity are discussed, including some new refinements and similar extensions, as well as several applications in the Gamma and incomplete Gamma functions.

New Applications of Fractional q-Calculus Operator for a New Subclass of q-Starlike Functions Related with the Cardioid Domain

Recently, a number of researchers from different fields have taken a keen interest in the domain of fractional q-calculus on the basis of fractional integrals and derivative operators. This has been used in various scientific research and technology fields, including optics, mathematical biology, plasma physics, electromagnetic theory, and many more. This article explores some mathematical applications of the fractional q-differential and integral operator in the field of geometric function theory. By using the linear multiplier fractional q-differintegral operator Dq,λmρ,σ and subordination, we define and develop a collection of q-starlike functions that are linked to the cardioid domain. This study also investigates sharp inequality problems like initial coefficient bounds, the Fekete–Szego problems, and the coefficient inequalities for a new class of q-starlike functions in the open unit disc U. Furthermore, we analyze novel findings with respect to the inverse function (μ−1) within the class of q-starlike functions in U. The findings in this paper are easy to understand and show a connection between present and past studies.

On Certain Analogues of Noor Integral Operators Associated with Fractional Integrals

In this paper, we employ a q-Noor integral operator to perform a q-analogue of certain fractional integral operator defined on an open unit disc. Then, we make use of the Hadamard convolution product to discuss several related results. Also, we derive a class of convex functions by utilizing the q-fractional integral operator and apply the inspired presented theory of the differential subordination, to geometrically explore the most popular differential subordination properties of the aforementioned operator. In addition, we discuss an exciting inclusion for the given convex class of functions. Over and above, we investigate the q-fractional integral operator and obtain some applications for the differential subordination.

Going Next after “A Guide to Special Functions in Fractional Calculus”: A Discussion Survey

In the survey Kiryakova: “A Guide to Special Functions in Fractional Calculus” (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions pΨq and a large number of their representatives. Among these, the Mittag-Leffler-type functions are the most popular and frequently used in fractional calculus. Naturally, these also include all “Classical Special Functions” of the class of the Meijer’s G- and pFq-functions, orthogonal polynomials and many elementary functions. However, it so happened that almost simultaneously with the appearance of the Mittag-Leffler function, another “fractionalized” variant of the exponential function was introduced by Le Roy, and in recent years, several authors have extended this special function and mentioned its applications. Then, we introduced a general class of so-called (multi-index) Le Roy-type functions, and observed that they fall in an “Extended Class of SF of FC”. This includes the I-functions of Rathie and, in particular, the H¯-functions of Inayat-Hussain, studied also by Buschman and Srivastava and by other authors. These functions initially arose in the theory of the Feynman integrals in statistical physics, but also include some important special functions that are well known in math, like the polylogarithms, Riemann Zeta functions, some famous polynomials and number sequences, etc. The I- and H¯-functions are introduced by Mellin–Barnes-type integral representations involving multi-valued fractional order powers of Γ-functions with a lot of singularities that are branch points. Here, we present briefly some preliminaries on the theory of these functions, and then our ideas and results as to how the considered Le Roy-type functions can be presented in their terms. Next, we also introduce Gelfond–Leontiev generalized operators of differentiation and integration for which the Le Roy-type functions are eigenfunctions. As shown, these “generalized integrations” can be extended as kinds of generalized operators of fractional integration, and are also compositions of “Le Roy type” Erdélyi–Kober integrals. A close analogy appears with the Generalized Fractional Calculus with H- and G-kernel functions, thus leading the way to its further development. Since the theory of the I- and H¯-functions still needs clarification of some details, we consider this work as a “Discussion Survey” and also provide a list of open problems.

The solution of stochastic evolution equation with the fractional derivative

In this paper, we consider the stochastic evolution equation driven by the Gaussian noise with white time and colored space, where the noise coefficient is the Marchaud fractional derivative. The key idea is that we transform our model into a stochastic space-fractional equation by taking the Marchaud fractional derivative, and then use Chaos expansion to prove the mild solution. There are three main results in this paper. First, we apply Chaos expansion to obtain the existence, uniqueness and Lyapunove exponent of the solution of the transformed equation. Second, we prove that there exists an unique mild solution of the original equation, the approach is taking the fractional integral operator into the transformed equation. Finally, we explore Hölder continuity of the mild solution.

Research on Application of Fractional Calculus Operator in Image Underlying Processing

Fractional calculus extends traditional, integer-based calculus to include non-integer orders, offering a powerful tool for a range of engineering applications, including image processing. This work delves into the utility of fractional calculus in two crucial aspects of image processing: image enhancement and denoising. We explore the foundational theories of fractional calculus together with its amplitude–frequency characteristics. Our focus is on the effectiveness of fractional differential operators in enhancing image features and reducing noise. Experimental results reveal that fractional calculus offers unique benefits for image enhancement and denoising. Specifically, fractional-order differential operators outperform their integer-order counterparts in accentuating details such as weak edges and strong textures in images. Moreover, fractional integral operators excel in denoising images, not only improving the signal-to-noise ratio but also better preserving essential features such as edges and textures when compared to traditional denoising techniques. Our empirical results affirm the effectiveness of the fractional-order calculus-based image-processing approach in yielding optimal results for low-level image processing.

Compactness of commutators of fractional integral operators on ball Banach function spaces

<abstract><p>Let $ 0 < \alpha < n $ and $ b $ be a locally integrable function. In this paper, we obtain the characterization of compactness of the iterated commutator $ (T_{\Omega, \alpha})_{b}^{m} $ generated by the function $ b $ and the fractional integral operator with the homogeneous kernel $ T_{\Omega, \alpha} $ on ball Banach function spaces. As applications, we derive the characterization of compactness via the commutator $ (T_{\Omega, \alpha})_b^m $ on weighted Lebesgue spaces, and further obtain a necessary and sufficient condition for the compactness of the iterated commutator $ (T_{\alpha})_{b}^{m} $ generated by the function $ b $ and the fractional integral operator $ T_\alpha $ on Morrey spaces. Moreover, we also show the necessary and sufficient condition for the compactness of the commutator $ [b, T_{\alpha}] $ generated by the function $ b $ and the fractional integral operator $ T_\alpha $ on variable Lebesgue spaces and mixed Morrey spaces.</p></abstract>

Applications of $ q- $Ultraspherical polynomials to bi-univalent functions defined by $ q- $Saigo's fractional integral operators

<abstract><p>This study established upper bounds for the second and third coefficients of analytical and bi-univalent functions belonging to a family of particular classes of analytic functions utilizing $ q- $Ultraspherical polynomials under $ q- $Saigo's fractional integral operator. We also discussed the Fekete-Szegö family function problem. As a result of the specialization of the parameters used in our main results, numerous novel outcomes were demonstrated.</p></abstract>

Refinements of Various Types of Fractional Inequalities via Generalized Convexity

This paper aims to find generalizations of inequalities that hold for unified integral operators by applying strongly exponentially (α, ℏ − m) − p‐convex functions. These inequalities generate results for several fractional integral operators and simultaneously hold for convex, strongly convex, and exponentially convex functions. The particular cases of presented inequalities are directly connected with many recently published results.

Some well known inequalities on two dimensional convex mappings by means of Pseudo $ \mathcal{L-R} $ interval order relations via fractional integral operators having non-singular kernel

<abstract><p>Fractional calculus and convex inequalities combine to form a comprehensive mathematical framework for understanding and analyzing a variety of problems. This note develops Hermite-Hadamard, Fejér, and Pachpatte type integral inequalities within pseudo left-right order relations by applying fractional operators with non-singular kernels. Recently, results have been developed using classical Riemann integral operators in addition to various other partial order relations that have some defects that we explained in literature in order to demonstrate the unique characteristics of pseudo order relations. To verify the developed results, we constructed several interesting examples and provided a number of remarks that demonstrate that this type of fractional operator generalizes several previously published results when different things are set up. This work can lead to improvements in mathematical theory, computational methods, and applications across a wide range of disciplines.</p></abstract>

Certain geometric properties of the fractional integral of the Bessel function of the first kind

<abstract><p>This paper revealed new fractional calculus applications of special functions in the geometric function theory. The aim of the study presented here was to introduce and begin the investigations on a new fractional calculus integral operator defined as the fractional integral of order $ \lambda $ for the Bessel function of the first kind. The focus of this research was on obtaining certain geometric properties that give necessary and sufficient univalence conditions for the new fractional calculus operator using the methods associated to differential subordination theory, also referred to as admissible functions theory, developed by Sanford S. Miller and Petru T. Mocanu. The paper discussed, in the proved theorems and corollaries, conditions that the fractional integral of the Bessel function of the first kind must comply in order to be a part of the sets of starlike functions, positive and negative order starlike functions, convex functions, positive and negative order convex functions, and close-to-convex functions, respectively. The geometric properties proved for the fractional integral of the Bessel function of the first kind recommend this function as a useful tool for future developments, both in geometric function theory in general, as well as in differential subordination and superordination theories in particular.</p></abstract>

Some extensions of classes involving pair of weights related to the boundedness of multilinear commutators associated to generalized fractional integral operators

Some extensions of classes involving pair of weights related to the boundedness of multilinear commutators associated to generalized fractional integral operators

Milne-Type inequalities via expanded fractional operators: A comparative study with different types of functions

<abstract><p>This study focused on deriving Milne-type inequalities using expanded fractional integral operators. We began by establishing a key equality associated with these operators. Using this equality, we explored Milne-type inequalities for functions with convex derivatives, supported by an illustrative example for clarity. Additionally, we investigated Milne-type inequalities for bounded and Lipschitzian functions utilizing fractional expanded integrals. Finally, we extended our exploration to Milne-type inequalities involving functions of bounded variation.</p></abstract>

Multilinear Commutators of Fractional Integral Operator on Central Morrey Spaces with Variable Exponent

Multilinear Commutators of Fractional Integral Operator on Central Morrey Spaces with Variable Exponent

Fractional integral inequalities for preinvex functions via Atangana-Baleanu integral operators

Different types of functions have been used in many various areas of mathematics until today, and today they are used to obtain new equality and inequalities in fractional analysis, which is one of the important areas of mathematics. In this study, we obtained integral inequalities for preinvex functions with the help of Atangana Baleanu fractional integral operator.

Boundedness Criterion for Multi-sublinear Operators and Their Commutators Generated by Multilinear Fractional Integral Operator on Product Generalized Mixed Morrey Spaces

Boundedness Criterion for Multi-sublinear Operators and Their Commutators Generated by Multilinear Fractional Integral Operator on Product Generalized Mixed Morrey Spaces

Copson-type Inequalities via the k-Hadamard Operator

Abstract In this paper, we present a novel class of Copson-type inequalities involving the right-sided k-Hadamard fractional integral operator with all parameters of integrability p ≠ 0 and obtain some classical special cases of Copson inequalities. The main results will be proved by employing the Hölder inequality and the Fubini theorem.

Two-weight norm inequalities for rough fractional integral operators on Morrey spaces

We establish the two-weight norm inequalities for the rough fractional integral operators on Morrey spaces.

A constant proportional caputo operator for modeling childhood disease epidemics

We propose a constant proportional-Caputo (CPC) operator for modeling childhood disease epidemics by treating positivity, boundedness, well-posedness, and biological viability and checking the reproductive and strength numbers for the biological system. For local and global stability analysis, we used the Volterra-type Lyapunov function with the first and second derivatives, providing symmetric results for fractional-order derivatives. We used different techniques to invert the proportional-Caputo (PC) and constant proportional-Caputo operators to evaluate the fractional integral operator. We also used our suggested model’s fractional differential equations to derive the eigenfunctions of the CPC operator. Modern mathematical control with the fractional operator has many applications, including the complex and crucial study of systems with symmetry. Symmetry analysis is a powerful tool that enables the user to construct numerical solutions to a given fractional differential equation fairly systematically. Therefore, we construct the comparative results with the Caputo derivative and constant proportional-Caputo operator to understand the dynamic behavior of the disease. We also present an additional analysis of the constant proportional-Caputo and Hilfer generalized proportional operators. Using the Laplace Adomian Decomposition Method, we numerically simulate a system of fractional differential equations through tables and graphs.