Fractional Integral Operators Research Articles

The Collocation Method Based on the New Chebyshev Cardinal Functions for Solving Fractional Delay Differential Equations

The Chebyshev cardinal functions based on the Lobatto grid are introduced and used for the first time to solve the fractional delay differential equations. The presented algorithm is based on the collocation method, which is applied to solve the corresponding Volterra integral equation of the given equation. In the employed method, the derivative and fractional integral operators are expressed in the Chebyshev cardinal functions, which reduce the computational load. The method is characterized by its simplicity, adherence to boundary conditions, and high accuracy. An exact analysis has been provided to demonstrate the convergence of the scheme, and illustrative examples validate our investigation.

Analysis of (P,m)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(P,\\mathrm{m})$\\end{document}-superquadratic function and related fractional integral inequalities with applications

In the present work we establish for the first time a class of (P,m)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(P,\\mathrm{m})$\\end{document}-superquadratic functions and look into its features. Using them, we come up with the Jensen and Hermite–Hadamard inequalities, as well as the fractional versions of Hermite–Hadamard inequalities with respect to Riemann–Liouville fractional integral operators. The findings are confirmed by certain numerical calculations and graphical depictions that take a few appropriate examples into account. The study is enhanced by the addition of applications of special means, moments of random variables, and modified Bessel functions of the first kind. This is achieved by considering new functions pertaining to the uniform probability density function and taking the modified Bessel functions of the first kind into consideration. The new results clearly provide extensions and improvements of the work given in the literature.

Existence and uniqueness of the solution to initial and inverse problems for integro-differential heat equations with fractional load

This work is devoted to the unique solvability of the direct and inverse problems for a multidimensional heat equation with a fractional load in Holder spaces. In the problem under consideration, the loaded term is in the form of a fractional integral operator for the time variable. We prove the existence and uniqueness of the solution to these problems by the contraction mapping theorem and the theory of integral equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/64/abstr.html

A Novel and Accurate Algorithm for Solving Fractional Diffusion-Wave Equations

The main objective of this work is to apply a novel and accurate algorithm for solving the second-order and fourth-order fractional diffusion-wave equations (FDWEs). First, the desired equation is reduced to the corresponding Volterra integral equation (VIE). Then, the collocation method is applied, for which the Chebyshev cardinal functions (CCFs) have been considered as the bases. In this paper, the CCFs based on a Lobatto grid are introduced and used for the first time to solve these kinds of equations. To this end, the derivative and fractional integral operators are represented in CCFs. The main features of the method are simplicity, compliance with boundary conditions, and good accuracy. An exact analysis to show the convergence of the scheme is presented, and illustrative examples confirm our investigation.

Fractional integral related to Schrödinger operator on vanishing generalized mixed Morrey spaces

With b belonging to a new BMOθ(ρ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$BMO_{\ heta}(\\rho )$\\end{document} space, L=−△+V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L=-\ riangle +V$\\end{document} is a Schrödinger operator on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathbb{R}^{n}}$\\end{document} with nonnegative potential V belonging to the reverse Hölder class RHn/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$RH_{n/2}$\\end{document}. The fractional integral operator associated with L is denoted by IβL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathcal{I}}_{\\beta}^{L}$\\end{document}. We investigate the boundedness of IβL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathcal{I}}_{\\beta}^{L}$\\end{document} and [b,IβL]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$[b,{\\mathcal{I}}_{\\beta}^{L}]$\\end{document}, which are its commutators with bθ(ρ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$b_{\ heta}(\\rho )$\\end{document} on vanishing generalized mixed Morrey spaces VMp→,φα,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$VM_{\\vec{p},\\varphi}^{\\alpha ,V}$\\end{document} related to Schrödinger operation and generalized mixed Morrey spaces Mp→,φα,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$M_{\\vec{p},\\varphi}^{\\alpha ,V}$\\end{document}. The boundedness of the operator IβL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathcal{I}}_{\\beta}^{L}$\\end{document} is ensured by finding sufficient conditions on the pair (φ1,φ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\varphi _{1},\\varphi _{2})$\\end{document}, which goes from Mp→,φ1α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$M_{\\vec{p},\\varphi _{1}}^{\\alpha ,V}$\\end{document} to Mq→,φ2α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$M_{\\vec{q},\\varphi _{2}}^{\\alpha ,V}$\\end{document}, and from VMp→,φ1α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$VM_{\\vec{p},\\varphi _{1}}^{\\alpha ,V}$\\end{document} to VMq→,φ2α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$VM_{\\vec{q},\\varphi _{2}}^{\\alpha ,V}$\\end{document}, ∑i=1n1pi−∑i=1n1qi=β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\sum \\limits _{i=1}^{n}\\frac{1}{p_{i}}-\\sum \\limits _{i=1}^{n}\\frac{1}{q_{i}}=\\beta $\\end{document}. When b belongs to BMOθ(ρ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$BMO_{\ heta}(\\rho )$\\end{document} and (φ1,φ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\varphi _{1},\\varphi _{2})$\\end{document} satisfies some conditions, we also show that the commutator operator [b,IβL]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$[b,{\\mathcal{I}}_{\\beta}^{L}]$\\end{document} is bounded from Mp→,φ1α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$M_{\\vec{p},\\varphi _{1}}^{\\alpha ,V}$\\end{document} to Mq→,φ2α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$M_{\\vec{q},\\varphi _{2}}^{\\alpha ,V}$\\end{document} and from VMp→,φ1α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$VM_{\\vec{p},\\varphi _{1}}^{\\alpha ,V}$\\end{document} to VMq→,φ2α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$VM_{\\vec{q},\\varphi _{2}}^{\\alpha ,V}$\\end{document}.

Endpoint estimates for commutators with respect to the fractional integral operators on Orlicz–Morrey spaces

Endpoint estimates for commutators with respect to the fractional integral operators on Orlicz–Morrey spaces

Mild Solutions of the (ρ1,ρ2,k1,k2,φ)-Proportional Hilfer–Cauchy Problem

Inspired by prior research on fractional calculus, we introduce new fractional integral and derivative operators: the (ρ1,ρ2,k1,k2,φ)-proportional integral and the (ρ1,ρ2,k1,k2,φ)-proportional Hilfer fractional derivative. Numerous previous studied fractional integrals and derivatives can be considered as particular instances of the novel operators introduced above. Some properties of the (ρ1,ρ2,k1,k2,φ)-proportional integral are discussed, including mapping properties, the generalized Laplace transform of the (ρ1,ρ2,k1,k2,φ)-proportional integral and (ρ1,ρ2,k1,k2,φ)-proportional Hilfer fractional derivative. The results obtained suggest that the most comprehensive formulation of this fractional calculus has been achieved. Under the guidance of the findings from earlier sections, we investigate the existence of mild solutions for the (ρ1,ρ2,k1,k2,φ)-proportional Hilfer fractional Cauchy problem. An illustrative example is provided to demonstrate the main results.

Exploring new fuzzy fractional integral operators with applications over fuzzy number convex and harmonic convex mappings

Exploring new fuzzy fractional integral operators with applications over fuzzy number convex and harmonic convex mappings

(L p,λ,μ,L q,λ,μ) boundedness of the G-fractional integral operator on G-Morrey spaces

Abstract In this paper we find conditions for the boundedness of the strong and weak fractional integral and fractional maximal operator generated by Gegenbauer differential operator on G-Morrey spaces.

Bilinear Θ-type generalized fractional integral and its commutator on product of generalized fractional weighted Morrey spaces

The aim of this paper is to establish the boundedness of a bilinear θ-type generalized fractional integral operator T˜α and its commutator T˜α,b1,b2 on generalized fractional weighted Morrey spaces Lωp,η,φ(Rn). Under assumption that the function φ satisfies doubling conditions, the authors prove that the T˜α is bounded from product of generalized fractional weighted Morrey spaces Lω1p1p1,η1,φ(Rn)×Lω2p2p2,η2,φ(Rn) into generalized fractional weighted Morrey spaces Lνω→qq,η−α,φ(Rn), and it is also bounded from product of spaces Lω1p1p1,η1,φ(Rn)×Lω2p2p2,η2,φ(Rn) into generalized fractional weighted weak Morrey spaces WLνω→qq,η−α,φ(Rn), where η=η1+η2, ω→=(ω1,ω2)∈A(P→,q), P→=(p1, p2), 1q=1p−αn, 1p=1p1+1p2 for 1<p<nα and 0<α<η. Furthermore, via the known result of fractional integral operators, the boundedness of the commutator T˜α,b1,b2 which is generated by b1,b2∈BMO(Rn) and the T˜α on spaces Lωp,η,φ(Rn) and on spaces WLωp,η,φ(Rn) is also obtained.

Generalized Mittag-Leffler-confluent hypergeometric functions in fractional calculus integral operator with numerical solutions

The Mittag-Leffler and confluent hypergeometric functions were originally developed to extend the exponential function and its area of applications. This study aims to examine some operators involving generalized Mittag-Leffler-type functions in the kernels, employing the generalized Fox-Wright function in specific circumstances. Furthermore, we investigate some of the commonly utilized generalized fractional integral operators in fractional calculus. Moreover, a numerical technique is developed to solve fractional differential equations of both kinds, linear and nonlinear. The graphic results of the examples show how effective this method is at solving fractional differential equations. Lastly, various effects and implications of these results are thoroughly examined.

Fractal-fractional estimations of Bullen-type inequalities with applications

The study of inequalities inside fractal domains has been stimulated by the growing interest in fractional calculus for the applied and mathematical sciences. This work uses extended fractional integrals in fractal domains to prove new Bullen-type inequalities for differentiable convex functions. By showing how these operators and inequalities can convert classical inequalities into fractal sets, it fills a vacuum in the literature on fractal-fractional integral inequalities. Using extended fractional integral operators, we derive new fractal-fractional estimates and Bullen-type inequalities, accompanied by thorough mathematical derivations and graphical validations. We show optimal results derived from new Bullen-type inequalities for fractal domains. We confirm our results with mathematical examples and graphs, improving models for complicated problems in fractal contexts. In particular, our findings have important ramifications for special means on fractal sets, probability density functions, and quadrature formulas, as well as for future study and applications.

An accurate wavelets‐collocation technique for neutral delay distributed‐order fractional optimal control problems

AbstractIn this study, a new class of optimal control problems called neutral delay distributed‐order fractional optimal control problems is introduced, this problem is solved based on an efficient computational scheme. To solve the problem, we derive an exact formula for the Riemann–Liouville fractional integral operator of Genocchi wavelets based on beta functions for the first time. By taking into account this operator, collocation method, and Gauss–Legendre integration formula, the solution of fractional optimal control problems (FOCPs) under consideration is converted to a nonlinear programming one to which existing well‐developed algorithms may be applied. The mentioned scheme is applied to both FOCPs with or without delay. Error analysis associated with the proposed idea is also investigated under several mild conditions. The effectiveness of the strategy is showed by several illustrative examples, furthermore, a comparison with the previous methods highlights the preference of this scheme.

On a Generalized Class of Non-Singular Kernel Operators and Their Singular Kernel Extensions: Useful Modeling Insights

Abstract Some possible definitions of fractional derivative operators with non-singular analytic kernels have been introduced. In this paper, we propose a new generalized class of fractional derivative operators of Caputo-type with non-singular analytic kernels which includes some known operators as special cases. We demonstrate a relationship between the fractional derivative operators of the proposed generalized class and the Riemann-Liouville fractional integral operator. We also, using this relationship, introduce the corresponding fractional integral operators. Then, mainly, we provide extensions to the fractional derivative operators of the proposed generalized class that display integrable singular kernels. The extended fractional derivative operators provide useful insights regarding the modeling issue so that the initialization problem can be overcome. Finally, we discuss some basic properties of the proposed operators that are expected to be widely used in fractional calculus.

Optimization of parameters for image denoising algorithm pertaining to generalized Caputo-Fabrizio fractional operator

The aim of the present paper is to optimize the values of different parameters related to the image denoising algorithm involving Caputo Fabrizio fractional integral operator of non-singular type with the Mittag-Leffler function in generalized form. The algorithm aims to find the coefficients of a kernel to remove the noise from images. The optimization of kernel coefficients are done on the basis of different numerical parameters like Mean Square Error (MSE), Peak Signal-to-Noise Ratio (PSNR), Structure Similarity Index measure (SSIM) and Image Enhancement Factor (IEF). The performance of the proposed algorithm is investigated through above-mentioned numeric parameters and visual perception with the other prevailed algorithms. Experimental results demonstrate that the proposed optimized kernel based on generalized fractional operator performs favorably compared to state of the art methods. The uniqueness of the paper is to highlight the optimized values of performance parameters for different values of fractional order. The novelty of the presented work lies in the development of a kernel utilizing coefficients from a fractional integral operator, specifically involving the Mittag-Leffler function in a more generalized form.

Fractional Calculus for Non-Discrete Signed Measures

In this paper, we suggest a first-ever construction of fractional integral and differential operators based on signed measures including a vector-valued case. The study focuses on constructing the fractional power of the Riemann–Stieltjes integral with a signed measure, using semigroup theory. The main result is a theorem that provides the exact form of a semigroup for the Riemann–Stieltjes integral with a measure having a countable number of extrema. This article provides examples of semigroups based on integral operators with signed measures and discusses the fractional powers of differential operators with partial derivatives.

Advancements in Bullen-type inequalities via fractional integral operators and their applications

In this paper, we investigate Bullen-type inequalities applicable to functions that are twice-differentiable. To explore these advanced inequalities, we utilize generalized convexity and Riemann-type fractional integrals. A comparative analysis is provided to highlight the more refined inequalities from among the explored results. By exploring the limiting cases, a relation with existing literature is established. Several examples are also presented to illustrate the outcomes and their accuracy is validated through graphical analysis. Additionally, applications in generalized means are also discussed.

Weighted Hardy factorizations in terms of Multilinear Calderón-Zygmund and fractional integral operators

The foundational paper of Coifman-Rochberg-Weiss provided a constructive proof of the weak factorizations of the classical Hardy space in terms of Riesz transforms. In this paper, we establish the factorization theorems for the weighted Hardy spaces. The results are also extended to the multilinear setting. Furthermore, we show that the weighted BMO space and weighted Lipschitz spaces are necessary for the weighted boundedness of commutators of the multilinear Calderón-Zygmund and fractional integral operators, respectively.

Integral inequalities of h‐superquadratic functions and their fractional perspective with applications

The purpose of this article is to provide a number of Hermite–Hadamard and Fejér type integral inequalities for a class of ‐superquadratic functions. We then develop the fractional perspective of inequalities of Hermite–Hadamard and Fejér types by use of the Riemann–Liouville fractional integral operators and bring up with few particular cases. Numerical estimations based on specific relevant cases and graphical representations validate the results. Another motivating component of the study is that it is enriched with applications of modified Bessel function of first type, special means, and moment of random variables by defining some new functions in terms of modified Bessel function and considering uniform probability density function. The results in this paper have not been initiated before in the frame of ‐superquadraticity. We are optimistic that this effort will greatly stimulate and encourage additional research.

An extension of Schweitzer's inequality to Riemann-Liouville fractional integral

Abstract This note focuses on establishing a fractional version akin to the Schweitzer inequality, specifically tailored to accommodate the left-sided Riemann-Liouville fractional integral operator. The Schweitzer inequality is a fundamental mathematical expression, and extending it to the fractional realm holds significance in advancing our understanding and applications of fractional calculus.