Fractional Calculus Operators Research Articles

Analysis of shear creep damage model of slope rock mass under the influence of coupled damage effects (DBSM)

Abstract This article reveals the evolution law of shear creep damage of carbonaceous shale (weak interlayer of slope) under dynamic load from a mechanical perspective by combining indoor experiments and theoretical research. Firstly, a coupled damage variable D BSM was established for the initial damage D 0 and dynamic disturbance shear creep damage D BS of rock mass based on the theory of damage mechanics. Secondly, according to the fractional calculus operator theory and considering the influence of coupled damage variable D BSM on the viscosity coefficient of rock mass in the viscoelasticity and viscoplasticity creep stages, a shear creep damage model of rock mass was established. At the same time, combined with the shear creep test data under the influence of dynamic loads, the damage evolution law of shear creep in carbonaceous shale (weak interlayer of slope) was revealed, and the accuracy of the established shear creep damage model was verified. Finally, the damage evolution law of carbonaceous shale was quantitatively analyzed. Results show that: The shear creep damage model established in this article has unique advantages. The greater the initial damage, the more likely shear creep failure is to occur under the influence of dynamic disturbance and shear creep loads. Dynamic disturbance accelerates the damage of weak interlayers of slope. The cumulative coupling damage of carbonaceous shale (D ma ≤0.18) under dynamic disturbance and multi-stage shear creep loads exhibits an S-like evolution pattern. The shear creep damage mechanism of carbonaceous shale is characterized by obvious initial damage effect, dynamic disturbance damage effect, and stress response characteristics.

Some operators of fractional calculus and their applications regarding various complex functions analytic in certain domains

Abstract. In this academic research note, some familiar operators prearranged by fractional-order calculus will first be introduced and various characteristic properties of those operators will next be propounded. Through the instrumentality of various earlier results associating with both those operators and some complex- exponential forms, and also in the light of certain special information in [1], [20], [17] and [38], an extensive result together with a variety of its implications consisting of several exponential type inequalities will then be determined. A number of its possible implications will extra be pointed out. Mathematics Subject Classification (2010): 26A33, 30A10, 34A40, 35A30, 41A58, 30C45, 30C55, 30C80, 33D15, 26E05, 33E20. Keywords: Complex plane, domains, regular functions, complex exponential, series expansions, fractional calculus, operators of fractional calculus, exponential type inequalities, differential inequalities.

On the matrix versions of the k analog of ℑ‐incomplete Gauss hypergeometric functions and associated fractional calculus

In this paper, our aim is to introduce a new definition of ( ‐incomplete Wright hypergeometric matrix functions ( ‐IWHMFs) using the ‐incomplete Pochhammer matrix symbol. First, we define the ‐incomplete gamma matrix function and introduce the ‐incomplete Pochhammer matrix symbols. Furthermore, we present differential formulas and integral representation related to these ‐IWHMFs. We have also obtained some results regarding the ‐fractional calculus operators of these ‐IWHMFs. Finally, we investigate the solutions of fractional kinetic equations (FKEs) involving the ‐IWHMFs.

Data-driven discrete fractional chaotic systems, new numerical schemes and deep learning.

Parameter estimation is important in data-driven fractional chaotic systems. Less work has been reported due to challenges in discretization of fractional calculus operators. In this paper, several numerical schemes are newly derived for delay fractional difference equations of Caputo and Riemann-Liouville types. Then, loss functions are constructed and unknown parameters of the discrete fractional chaotic system are estimated by a neural network method. Parameter estimation results demonstrate high accuracy compared with real values. Robust analysis is provided under different noise levels. It can be concluded that this paper provides an efficient deep learning method based on fractional discrete-time systems.

Robustness and exploration between the interplay of the nonlinear co-dynamics HIV/AIDS and pneumonia model via fractional differential operators and a probabilistic approach

In this article, we considered a nonlinear compartmental mathematical model that assesses the effect of treatment on the dynamics of HIV/AIDS and pneumonia (H/A-P) co-infection in a human population at different infection stages. Understanding the complexities of co-dynamics is now critically necessary as a consequence. The aim of this research is to construct a co-infection model of H/A-P in the context of fractional calculus operators, white noise and probability density functions, employing a rigorous biological investigation. By exhibiting that the system possesses non-negative and bounded global outcomes, it is shown that the approach is both mathematically and biologically practicable. The required conditions are derived, guaranteeing the eradication of the infection. Furthermore, adequate prerequisites are established, and the configuration is tested for the existence of an ergodic stationary distribution. For discovering the system’s long-term behavior, a deterministic-probabilistic technique for modeling is designed and operated in MATLAB. By employing an extensive review, we hope that the previously mentioned approach improves and leads to mitigating the two diseases and their co-infections by examining a variety of behavioral trends, such as transitions to unpredictable procedures. In addition, the piecewise differential strategies are being outlined as having promising potential for scholars in a range of contexts because they empower them to include particular characteristics across multiple time frame phases. Such formulas can be strengthened via classical techniques, power law, exponential decay, generalized Mittag-Leffler kernels, probability density functions and random procedures. Furthermore, we get an accurate description of the probability density function encircling a quasi-equilibrium point if the effect of H/A-P minimizes the propagation of the co-dynamics. Consequently, scholars can obtain better outcomes when analyzing facts using random perturbations by implementing these strategies for challenging issues. Random perturbations in H/A-P co-infection are crucial in controlling the spread of an epidemic whenever the suggested circulation is steady and the amount of infection eliminated is closely correlated with the random perturbation level.

On the approximation to fractional calculus operators with multivariate Mittag-Leffler function in the kernel

Several numerical techniques have been developed to approximate Riemann–Liouville (R - L) and Caputo fractional calculus operators. Recently linear positive operators have been started to use to approximate fractional calculus operators such as R - L, Caputo, Prabhakar and operators containing bivariate Mittag-Leffler functions. In the present paper, we first define and investigate the fractional calculus properties of Caputo derivative operator containing the multivariate Mittag-Leffler function in the kernel. Then we introduce approximating operators by using the modified Kantorovich operators for the approximation to fractional integral and Caputo derivative operators with multivariate Mittag-Leffler function in the kernel. We study the convergence properties of the operators and compute the degree of approximation by means of modulus of continuity and Hölder continuous functions. The obtained results corresponds to a large family of fractional calculus operators including R- L, Caputo and Prabhakar models.

The Generalized Fox–Wright Function: The Laplace Transform, the Erdélyi–Kober Fractional Integral and Its Role in Fractional Calculus

In this paper, we consider and study in detail the generalized Fox–Wright function Ψ˜qp introduced in our recent work as an extension of the Fox–Wright function Ψqp. This special function can be seen as an important case of the so-called I-functions of Rathie and H¯-functions of Inayat-Hussain, that in turn extend the Fox H-functions and appear to include some Feynman integrals in statistical physics, in polylogarithms, in Riemann Zeta-type functions and in other important mathematical functions. Depending on the parameters, Ψ˜qp is an entire function or is analytic in an open disc with a final radius. We derive its basic properties, such as its order and type, and its images under the Laplace transform and under classical fractional-order integrals. Particular cases of Ψ˜qp are specified, including the Mittag-Leffler and Le Roy-type functions and their multi-index analogues and many other special functions of Fractional Calculus. The corresponding results are illustrated. Finally, we emphasize the role of these new generalized hypergeometric functions as eigenfunctions of operators of new Fractional Calculus with specific I-functions as singular kernels. This paper can be considered as a natural supplement to our previous surveys “Going Next after ‘A Guide to Special Functions in Fractional Calculus’: A Discussion Survey”, and “A Guide to Special Functions of Fractional Calculus”, published recently in this journal.

A gazelle optimization expedition for key term separated fractional nonlinear systems with application to electrically stimulated muscle modeling

Various real-time processes are effectively represented with a fractional nonlinear autoregressive exogenous (F-NARX) model where estimation of model parameters is considered as an essential task. In this study, a population-based gazelle optimization algorithm (GOA) inspired by the evolutionary characteristics of gazelle's is exploited for the parameter estimation of the key term-separated F-NARX system. The Grünwald-Letnikov derivative, a fractional order calculus operator is integrated to develop the F-NARX system from a conventional non-linear autoregressive system. The mean square error-based merit function is developed and the effectiveness of the GOA for the F-NARX system is analyzed in terms of speedy convergence, estimation accuracy, complexity and robustness for different noise scenarios. The extendibility of the GOA is assessed through the estimation of stiff parameters of the electrically stimulated muscle model (ESMM) required for rehabilitation of paralyzed muscles. The efficacy of the GOA is endorsed through Wilcoxon signed rank statistical test in comparison with recent counterparts of Runge Kutta optimization algorithm, Whale optimization algorithm, and Harris Hawks optimization algorithm.

Theoretical and mathematical codynamics of nonlinear tuberculosis and COVID-19 model pertaining to fractional calculus and probabilistic approach

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a novel virus known as coronavirus 2 (SARS-CoV-2) that affects the pulmonary structure and results in the coronavirus illness 2019 (COVID-19). Tuberculosis (TB) and COVID-19 codynamics have been documented in numerous nations. Understanding the complexities of codynamics is now critically necessary as a consequence. The aim of this research is to construct a co-infection model of TB and COVID-19 in the context of fractional calculus operators, white noise and probability density functions, employing a rigorous biological investigation. By exhibiting that the system possesses non-negative and bounded global outcomes, it is shown that the approach is both mathematically and biologically practicable. The required conditions are derived, guaranteeing the eradication of the infection. Sensitivity analysis and bifurcation of the submodel are also investigated with system parameters. Furthermore, existence and uniqueness results are established, and the configuration is tested for the existence of an ergodic stationary distribution. For discovering the system’s long-term behavior, a deterministic-probabilistic technique for modeling is designed and operated in MATLAB. By employing an extensive review, we hope that the previously mentioned approach improves and leads to mitigating the two diseases and their co-infections by examining a variety of behavioral trends, such as transitions to unpredictable procedures. In addition, the piecewise differential strategies are being outlined as having promising potential for scholars in a range of contexts because they empower them to include particular characteristics across multiple time frame phases. Such formulas can be strengthened via classical technique, power-law, exponential decay, generalized Mittag–Leffler kernels, probability density functions and random procedures. Furthermore, we get an accurate description of the probability density function encircling a quasi-equilibrium point if the effect of TB and COVID-19 minimizes the propagation of the codynamics. Consequently, scholars can obtain better outcomes when analyzing facts using random perturbations by implementing these strategies for challenging issues. Random perturbations in TB and COVID-19 co-infection are crucial in controlling the spread of an epidemic whenever the suggested circulation is steady and the amount of infection eliminated is closely correlated with the random perturbation level.

Computational and stability analysis of Ebola virus epidemic model with piecewise hybrid fractional operator.

In this manuscript, we developed a nonlinear fractional order Ebola virus with a novel piecewise hybrid technique to observe the dynamical transmission having eight compartments. The existence and uniqueness of a solution of piecewise derivative is treated for a system with Arzel'a-Ascoli and Schauder conditions. We investigate the effects of classical and modified fractional calculus operators, specifically the classical Caputo piecewise operator, on the behavior of the model. A model shows that a completely continuous operator is uniformly continuous, and bounded according to the equilibrium points. The reproductive number R0 is derived for the biological feasibility of the model with sensitivity analysis with different parameters impact on the model. Sensitivity analysis is an essential tool for comprehending how various model parameters affect the spread of illness. Through a methodical manipulation of important parameters and an assessment of their impact on Ro, we are able to learn more about the resiliency and susceptibility of the model. Local stability is established with next Matignon method and global stability is conducted with the Lyapunov function for a feasible solution of the proposed model. In the end, a numerical solution is derived with Newton's polynomial technique for a piecewise Caputo operator through simulations of the compartments at various fractional orders by using real data. Our findings highlight the importance of fractional operators in enhancing the accuracy of the model in capturing the intricate dynamics of the disease. This research contributes to a deeper understanding of Ebola virus dynamics and provides valuable insights for improving disease modeling and public health strategies.

On Certain Subclass of Meromorphic Multivalent Functions Associated with Fractional Calculus Operator

In this paper, the class of meromorphic multivalent functions of the form by using fractional differ-integral operators is introduced. We get Coefficients estimates, radii of convexity and star likeness. Also closure theorems and distortion theorem for the class , is calculaed.

Design of Runge-Kutta optimization for fractional input nonlinear autoregressive exogenous system identification with key-term separation

Population-based metaheuristic algorithms have gained significant attention in research community due to its effectiveness in solving complex optimization problems in diverse fields. In this study, knacks of population-based Runge-Kutta optimizer (RUN) are exploited for the identification of fractional input non-linear exogenous auto-regressive (FINARX) system with key term separation. The fractional order calculus operator of the Grünwald-Letnikov derivative is exploited to develop FINARX from a conventional non-linear auto-regressive exogenous system. The identification scheme for FINARX model is implemented through a mean-square-error-based fitness function. RUN utilizes the slope variations calculated by the well-known Runge-Kutta method for an effective search mechanism in the exploration and exploitation phases. Moreover, an enhanced solution quality mechanism is employed for speedy convergence and keeping the movement toward the best solution by escaping the local optima. The robustness of the algorithm is analyzed by multiple variations of non-linearity as well as different noise scenarios. The performance of the RUN to identify the FINARX system is validated in terms of convergence rate, fitness value, robustness, and accuracy in weight estimation. The effectiveness of the RUN is further assessed through exhaustive simulations with their statistics as well as comparison with the standard recent counterparts, including the Whale optimization algorithm, Reptile Search algorithm, and Aquila optimizer on different performance indices for the FINARX system.

Coefficient bounds and second Hankel determinant for a subclass of symmetric bi-starlike functions involving Euler polynomials

Various operators of fractional calculus, as well as their quantum (or q-) extensions have been used widely and successfully in the study of the Taylor-Maclaurin coefficient estimation problems for many different families of normalized analytic, univalent and bi-univalent functions in Geometric Function Theory of Complex Analysis. On the other hand, Numerous writers have extensively employed orthogonal polynomials in the framework of Geometric Function Theory in Complex analysis. The recent and ongoing studies on this topic is primarily what drives us. In this paper, we solve the Fekete-Szegö problem for the symmetric function classes of analytical and bi-univalent functions involving the Euler polynomials. We also give estimates for the coefficients and an upper bound for the second Hankel determinant.

On the analysis of fractional calculus operators with bivariate Mittag Leffler function in the kernel

Bivariate Mittag-Leffler (ML) functions are a substantial generalization of the univariate ML functions, which are widely recognized for their significance in fractional calculus. In the present paper, our initial focus is to investigate the fractional calculus properties of the integral and derivative operators with kernels including the Bivariate ML functions. Further, certain fractional Cauchy-type problems including these operators are considered. Also the numerical approximations of the Caputo type derivative operator are investigated. The theoretical results are justified by applications on examples. Furthermore, the theory of applying the same operators with respect to arbitrary monotonic functions is analyzed in this research.

An analysis on the approximate controllability outcomes for fractional stochastic Sobolev‐type hemivariational inequalities of order 1<r<2$$ 1<r<2 $$ using sectorial operators

In this paper, we deal with the approximate controllability of fractional stochastic Sobolev‐type hemivariational differential systems of order with sectorial operators. Firstly, by using stochastic analysis, fractional calculus, sine and cosine family operators, sectorial operators, and the fixed‐point theorems of multivalued maps, we show the existence of mild solutions for the fractional stochastic evolution systems. Then, we provide a sufficient condition to guarantee the approximate controllability of the stochastic evolution systems. Next, our results cover problems involving nonlocal conditions. Finally, we present theoretical and practical applications to support the validity of the study.

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.

Mikusiński’s Operational Calculus for Fractional Operators with Different Kernels

Mikusiński’s operational calculus provides a way of applying the machinery of abstract algebra to the spaces and operators of calculus, thus allowing integro-differential equations to be solved by reducing them to algebraic equations. We summarise the application of this method to several operators of fractional calculus, defined by various convolution kernel functions at different levels of generality, and how the corresponding fractional differential equations can be solved.

The Multi-Index Mittag-Leffler-Le Roy Functions as I- and H¯-Functions and New Fractional Calculus Operators

In the last decade, the interest in a special function named after the French mathematician Le Roy has provoked several authors to introduce and study its various extensions. Historically, the Le Roy function appeared almost in same time and in similar way of goals like the Mittag-Leffler function that has important role in Fractional Calculus. The Le Roy type functions are also “fractional exponentials” but the fractionalizing parameters appear as power indices of the involved Gamma functions. In our recent works, we have introduced rather general multi-index Le Roy type functions, studied their analytical properties and proposed their association to the class of Special Functions of Fractional Calculus.A newly developed idea is to show that the Le Roy type functions can be represented in terms of the I-functions of Rathie and H-functions of Inayat-Hussain that are further extensions of the Fox H¯-functions and Fox-Wright pΨq-functions. It happens that other important mathematical functions as the polylogarithms, Riemann Zeta function and its extensions, also belong to this more general class of special functions. We have solved, at least in some simpler case, the problem to find integral and differential-like operators for which the Le Roy type functions are eigenfunctions. This lead us to a new class of operators with I-functions kernels that can be considered as further extensions of the operators of generalized fractional calculus.

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>

General Fractional Calculus Operators with the Sonin kernels and Some of Their Applications

In this survey paper, the general fractional integrals and derivatives with the Sonin kernels, the regularized general fractional derivatives, the sequential generalized fractional derivatives as well as some of their applications and the fractional differential equations with these derivatives are discussed. We start with a short summary of different kinds of the general Fractional Calculus operators with the Sonin kernels and then proceed with a presentation of some recent results including the fundamental theorems of Fractional Calculus for the general fractional derivatives of arbitrary order, for the regularized general fractional derivatives of arbitrary order, and for the sequential general fractional derivatives. As an application of these results, the generalized convolution Taylor formulas and the generalized convolution Taylor series with the coefficients and remainders in terms of the general fractional derivatives and the regularized general fractional derivatives are presented. Finally, we discuss a method for construction of solutions to the fractional differential equations with the general fractional derivatives in terms of the so called convolution series.