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

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.

The paper is devoted to the study of the function E γ ρ,μ(z) defined for complex ρ, μ, γ (Re(ρ) > 0) by which is a generalization of the classical Mittag-Leffler function E ρ,μ(z) and the Kummer confluent hypergeometric function Φ(γ, μ; z). The properties of E γ ρ,μ(z) including usual differentiation and integration, and fractional ones are proved. Further the integral operator with such a function kernel is studied in the space L(a, b). Compositions of the Riemann–Liouville fractional integration and differentiation operators with E γ ρ,μ,ω;a+ are established. An analogy of the semigroup property for the composition of two such operators with different indices is proved, and the results obtained are applied to construct the left inversion operator to the operator E γ ρ,μ,ω;a+. Since, for γ = 0, E 0 ρ,μ,ω;a+ coincides with the Riemann–Liouville fractional integral of order μ, the above operator and its inversion can be considered as generalized fractional calculus operators involving the generalized Mittag-Leffler function E γ ρ,μ(z) in the kernels. Similar assertions are presented for the integral operators containing the Mittag-Leffler and Kummer functions, E ρ,μ(z) and Φ(γ, μ; z), in the kernels, and applications are given to obtain solutions in closed form of the integral equations of the first kind.

Fractional calculus, particularly fractional integral operators, has gained significant attention in recent years due to its wide-ranging applications in various scientific disciplines.This paper provides a comprehensive review of the latest advancements in fractional integral operators, covering theoretical developments, practical applications, and computational methods.We explore the mathematical foundations of fractional integrals, discuss their applications in diverse fields such as signal processing, mathematical physics, and image processing, and examine state-of-the-art computational techniques for their efficient computation.Additionally, we highlight current research trends and future directions in the field of fractional integral operators. I. INTRODUCTIONFractional calculus, a branch of mathematics that generalizes the concepts of differentiation and integration to non-integer orders, has garnered increasing attention in recent years due to its versatile applications across diverse scientific disciplines.Central to the theory of fractional calculus are fractional integral operators, which play a pivotal role in describing non-local and memory-dependent phenomena in complex systems.In this paper, we present a comprehensive review of the latest advancements in fractional integral operators, encompassing theoretical developments, practical applications, and computational methods.Theoretical Foundations of Fractional Integrals, Fractional integral operators, such as the Riemann-Liouville and Caputo integrals, are fundamental tools in fractional calculus.We begin by examining the mathematical foundations of fractional integrals, including their definitions, properties, and relationships with fractional derivatives [1,2].Additionally, we explore important theoretical results related to fractional integrals, such as their behaviour in function spaces and their connection to fractional differential equations [3,4].Applications of Fractional Integrals, Fractional integral operators find applications across a wide range of scientific disciplines, including signal processing, mathematical physics, and image processing.Recent research has demonstrated their utility in modelling anomalous diffusion phenomena [5], describing fractal behaviour in complex systems [6], and analysing non-local effects in viscoelastic materials [7].Moreover, fractional integrals have been applied to noise reduction, feature extraction, and data compression in signal and image processing applications [8,9].Computational Methods for Fractional Integrals, Efficient computation of fractional integrals is essential for their practical use in scientific research and engineering applications.We survey state-of-the-art computational methods for approximating fractional integrals, including numerical integration techniques, spectral methods, and fractional calculus software packages [10,11].Additionally, we discuss challenges and considerations in implementing these methods and provide insights into their computational complexity and accuracy.Current Research Trends and Future Directions, despite significant advancements in the theory and applications of fractional integral operators, several research avenues remain unexplored.Current research trends in the field include the application of fractional calculus to machine learning and data science [12], as well as the development of novel computational methods for solving fractional differential equations [13].Chinta Mani Tiwari (2006).Given A note on Dirac delta function [14], again Chinta Mani Tiwari.(2007).Worked on the Neutrix product of three distributions [15], recently Chinta Mani Tiwari (2023).explained Generalized function and distribution [19].Future research directions may focus on exploring emerging applications and addressing theoretical challenges in the analysis of complex systems using fractional integral operators.In summary, fractional integral operators represent a powerful mathematical tool with widespread applications in science and engineering.By providing a comprehensive overview of theoretical foundations,

The main aim of this article is to design a novel framework to study a generalized fractional integral operator that unifies two existing fractional integral operators. To ensure the suitable selection of the operator and with the discussion of special cases, it is shown that our considered fractional integral generalizes the well-known Atangana–Baleanu fractional integral (AB-fractional integral) and the ABK-fractional integral. Conditions are stated for the generalized AB-fractional integral operator (GAB-fractional integral operator) to be bounded in the space Xcp (γ1,γ2). We also provide a fractional product-integration formula for this operator. Furthermore, we generalize the reverse Minkowski’s inequality and the reverse Hölder-type inequality by utilizing the GAB-fractional integral operator. Additionally, some other types of integral inequalities are established, and several special cases are noted. The concepts in this article may influence further research in fractional calculus.

In this paper we investigate several interesting properties of the fractional calculus operators, the Carlson-Shaffer linear operator L(a,c) and a certain integral operator , associated with various subclasses of analytic and univalent functions. We also investigate the relationships of some of these classes with the Hardy space H ∞ (of bounded analytic functions in the open unit disk μ and with some families of multiplier transformations.

This paper deals with extended M-series, which is extension of the generalized M-series [12]. Mittag-Leffler function, $\omega$ - hypergeometric function, generalized $\omega-$ Gauss hypergeometric function, $\omega-$ confluent hypergeometric function, Bessel-Maitland function can be deduced as special cases of our finding. Moreover, we obtain some theorem for extended M-series by using fractional calculus operators and many results associated with Riemann-Liouville, Weyl and ErdelyiKober operators. We begin our study from the following definitions.

The article proposes a generalized polynomial projection method for solving linear integral equations with a fractional Riemann - Liouville integral in the main part. Fractional-integral equations have numerous applications in various applied problems, in particular, in problems of plasma diagnostics, optical and X-ray diffraction, in ultrasonic measurements in moving media, in metallurgy, biology, microscopy, seismology, in astrophysics and other fields of science.The equation we are considering is an equation of the first kind and therefore, generally speaking, refers to incorrectly set by Hadamard equations. The latter circumstance is connected with the fact that in known function spaces the fractional integral Riemann-Liouville operator is completely continuous. All this imposes its own characteristics on the construction and study of approximate methods for solving integral equations with a fractional integral operator in the main part. It should be noted that the operator of the projection method is not necessarily projective, which allows the construction of computational schemes using the methods of summation of Fourier series and interpolation polynomials. A theoretical and functional substantiation of the proposed projection method was carried out, based on the correct formulation of the equation in a pair of specially selected different Holder spaces. In particular, it proved the unique solvability of the system of linear algebraic equations of the method and the convergence of the constructed approximations to the exact solution in the norm of the space of Holder functions and, as a consequence, in the uniform metric. This also implies the substantiation of specific well-known projection methods such as the Galerkin methods, colocations, and subdomains.

We consider non-standard generalized Holder spaces of functions defined on a segment of the real axis, whose local continuity modulus has a majorant varying from point to point. We establish some properties of fractional integration operators of variable order acting from variable generalized Holder spaces to those with a “better” majorant, as well as properties of fractional differentiation operators of variable order acting from the same spaces to those with a “worse” majorant.

<abstract><p>For generalizations of concepts of different fields fractional derivative operators as well as fractional integral operators are useful notions. Our aim in this paper is to discuss boundedness of the integral operators which contain Mittag-Leffler function in their kernels. The results are obtained for strongly $ (\alpha, h-m) $-convex functions which hold for different kinds of convex functions at the same time. They also give improvements/refinements of many already published results.</p></abstract>

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.

<abstract><p>In this article, a new fractional mathematical model is presented to investigate the contagion of the human immunodeficiency virus (HIV). This model is constructed via recent improved fractional differential and integral operators. Other operators like Caputo, Riemann-Liouville, Katugampola, Jarad and Hadamard are being extended and generalized by these improved fractional differential and integral operators. Banach's and Leray-Schauder nonlinear alternative fixed point theorems are utilized to examine the existence and uniqueness results of the proposed fractional HIV model. Moreover, different kinds of Ulam stability for the fractional HIV model are established. It is simple to recognize that the extracted results can be reduced to some results acquired in multiple works of literature.</p></abstract>

In this paper, a weighted generalized fractional integral operator based on the Mittag-Leffler function is established, and it exhibits symmetric characteristics concerning classical operators. We demonstrate the semigroup property as well as the boundedness of the operator in absolute continuous like spaces. In this work, some applications with graphical representation are also considered. Finally, we modify the weighted generalized Laplace transform and then applied it to the newly defined weighted fractional integral operator. The defined operator is an extension and generalization of classical Riemann–Liouville and Prabhakar integral operators.

A numerical study on fractional order financial system with chaotic and Lyapunov stability analysis

Fractional calculus operators of special functions? The result is well predictable!

Saigo and Maeda (Transform Methods and Special Functions, Varna, Bulgaria, pp. 386-400, 1996) introduced and investigated certain generalized fractional integral and derivative operators involving the Appell function . Here we aim at presenting four unified fractional integral and derivative formulas of Saigo and Maeda type, which are involved in a product of ℵ-function and a general class of multivariable polynomials. The main results, being of general nature, are shown to be some unification and extension of many known formulas given, for example, by Saigo and Maeda (Transform Methods and Special Functions, Varna, Bulgaria, pp. 386-400, 1996), Saxena et al. (Kuwait J. Sci. Eng. 35(1A):1-20, 2008), Srivastava and Garg (Rev. Roum. Phys. 32:685-692, 1987), Srivastava et al. (J. Math. Anal. Appl.193:373-389, 1995) and so on. Our main results are also shown to be further specialized to yield a large number of known and (presumably) new formulas involving, for instance, Saigo fractional calculus operators, several special functions such as H-function, I-function, Mittag-Leffler function, generalized Wright hypergeometric function, generalized Bessel-Maitland function. MSC:26A33, 33E20, 33C45, 33C60, 33C70.