Analysis of a Normalized Structure of a Complex Fractal–Fractional Integral Transform Using Special Functions

<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>

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.

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.

Fractional differentiation and integration have witness a continuous revolution in the last decades as they have been found to capture behaviors that resemble properties of some mathematical functions for example power law, exponential decay and crossover from exponential decay to power law. We have presented different differential operators with power law, exponential decay and the generalized Mittag-Leffler kernels. Using the fundamental theorem of calculus, integral operators associated to these differential operators were presented. Different properties of these nonlocal operators have been presented in details. Using some approximation techniques, both fractional differential and integral operators were discretized.KeywordsFractional integralsFractional derivativesInequalityLipschitz conditionsNumerical approximation

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,

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

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.

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.

In this work, we address the existence and uniqueness of a fractional system involving nonlinear Schrödinger equations. This system of fractional partial differential equations arises in quantum mechanics. It describes how the quantum state of some physical system changes with time. We show that the system under consideration admits a global solution in appropriate functional spaces. The solution is shown to be unique. The technique is established as an analytic technique of the fixed point theorem. We suppose that all the functions are analytic in the open unit disk. The fractional differential operator is considered from the point of view of the Riemann-Liouville differential operator. Moreover, a numerical scheme for this solution is calculated.

We characterize continuity and compactness of the Volterra integral operator $T_g$ with the non-constant analytic symbol $g$ between certain weighted Fréchet or (LB)-spaces of analytic functions on the open unit disc, which arise as projective (resp. inductive) limits of intersections (resp. unions) of Bergman spaces of order $1<p<\infty$ induced by the standard radial weight $(1-|z|^2)^\alpha$ for $0<\alpha<\infty$. Motivated from the earlier results obtained for weighted Bergman spaces of standard weight, we also establish several results concerning the spectrum of the Volterra operators acting on the weighted Bergman Fréchet space $A^p_{\alpha+}$, and acting on the weighted Bergman (LB)-space $A^p_{\alpha-}$.

A useful family of fractional derivative and integral operators plays a crucial role on the study of mathematics and applied science. In this paper, we introduce an operator defined on the family of analytic functions in the open unit disk by using the generalized fractional derivative and integral operator with convolution. For this operator, we study the subordination-preserving properties and their dual problems. Differential sandwich-type results for this operator are also investigated.

Groundwater flow in media with dual porosity has long been an active area of research given the complexities that arise during such modelling. In this pursuit, various approaches have been utilized to capture some variational behaviours – methods of systems of differential equations as well as the application of some fractional differential operators. However, these models fail to accurately capture the crossover behaviour and the heterogeneity of the geological formations. In the present study, the authors make use of the piecewise modelling in dual media to capture the crossover behaviour. To achieve this, they consider fractional differential and integral operators with different kernels including the power law, exponential decay, and the generalized Mittag–Leffler functions. Furthermore, randomness is incorporated in the modified model to obtain a piecewise fractional stochastic system of partial differential equations. The numerical method based on the Newton polynomial interpolation is applied to solve the obtained systems of partial differential equations numerically. MATLAB is then used to generate simulations. For a fractional order starting from 0.95 to 0.7, the results display high flow velocity within fractures; this behaviour is known as long range dependency. Also within the matrix soil, the authors observe a high flow, meaning in this case that the fractional derivative is expressing the flow within a matrix rock with high transmissivity. For a fractional order ranging from 0.7 to 0.5, the results show normal flow behaviours; water flows with average velocity within the matrix rock and moderate to high velocity within fractures. From 0.5 to 0.2, the authors observe slow flow behaviour, the mathematical equations exhibiting flow with low velocities, expressing a flow within shale with limited or no transmissivity. Keywords: dual porosity non-local operators piecewise modelling groundwater modelling transmissivity

Automorphisms and generalized projections on spaces of analytic functions

The spectrum of the Cesaro operator $$\mathsf {C}$$ , which is always continuous (but never compact) when acting on the classical Korenblum space and other related weighted Frechet or (LB) spaces of analytic functions on the open unit disc, is completely determined. It turns out that such spaces are always Schwartz but, with the exception of the Korenblum space, never nuclear. Some consequences concerning the mean ergodicity of $$\mathsf {C}$$ are deduced.