Operational Calculus Research Articles

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>

Notes on various operators of fractional calculus and some of their implications for certain analytic functions

"The main purpose of this note is firstly to present certain information in relation with some elementary operators created by the well-known fractional calculus, also to determine a number of applications of them for certain complex function analytic in the open unit disc, and then to reveal (or point out) some implications of the fundamental results of this research. Keywords: Complex domains, open unit disc, normalized analytic function, fractional calculus, Operators of fractional calculus, Series expansions, inequalities in the complex plane. "

Solution and dynamics analysis of fractal-fractional multi-scroll Chen chaotic system based on Adomain decomposition method

In recent years, newly proposed fractal-fractional calculus operators have been used to predict the chaotic behavior of some attractors, and the new operators can capture self-similarity in the studied chaotic attractors. The results of varying with fractal order and fractional order are interesting, however, the dynamics are barely analyzed in these studies, and the law of fractal order and fractional order on the system is not clear. In this paper, the fractal-fractional differential operators with inverse operator property are redefined in the Caputo sense, and a new numerical scheme is proposed based on the sixth-order Adomain decomposition method. Then the dynamics of the new multi-scroll Chen chaotic system and its three degradation modes are analyzed completely in the fractal-fractional sense. The influence of the change of fractal order and fractional order on the chaotic system is revealed. Finally, a new physical phenomenon of the fractal-fractional chaotic system is captured by observing the coexistence of multi-scroll and two-scroll chaotic attractors of the new system through a wood-grain-like attractor basin, as well as transient chaos, chaotic jumps, and complex state transition behaviors. It is shown that fractal-fractional multi-scroll Chen system has complex and specific dynamical behaviors.

Sharp Estimates Involving a Generalized Symmetric Sălăgean q-Differential Operator for Harmonic Functions via Quantum Calculus

In this study, we apply q-symmetric calculus operator theory and investigate a generalized symmetric Sălăgean q-differential operator for harmonic functions in an open unit disk. We consider a newly defined operator and establish new subclasses of harmonic functions in complex order. We determine the sharp results, such as the sufficient necessary coefficient bounds, the extreme of closed convex hulls, and the distortion theorems for a new family of harmonic functions. Further, we illustrate how we connect the findings of previous studies and the results of this article.

General Fractional Calculus Operators of Distributed Order

In this paper, two types of general fractional derivatives of distributed order and a corresponding fractional integral of distributed type are defined, and their basic properties are investigated. The general fractional derivatives of distributed order are constructed for a special class of one-parametric Sonin kernels with power law singularities at the origin. The conventional fractional derivatives of distributed order based on the Riemann–Liouville and Caputo fractional derivatives are particular cases of the general fractional derivatives of distributed order introduced in this paper.

New Analytical Solutions of Mathematical Models of Heat Shock of Local Non-Equilibrium Heat Transfer

The article is devoted to practically new model representations of locally nonequilibrium heat transfer in terms of nonstationary heat conduction for hyperbolic type equations (wave equations), as well as dynamic models of heat shock based on wave equations. The results presented in the article practically open up an independent scientific direction in analytical thermal physics and in the theory of thermal shock, namely: the study of the thermal response of solids of a canonical form of finite sizes to intense heating and cooling under conditions of a locally nonequilibrium heat transfer process. This direction required the development of a special apparatus of operational calculus due to the appearance in analytical solutions of model problems in the image space according to Laplace of non-standard operational images, the originals of which are unknown and are not available in reference books on operational calculus. The presented images are typical for operational solutions of a wide class of generalized boundary value problems for equations of hyperbolic type in the theory of heat conduction, diffusion, hydrodynamics, vibrations, propagation of electricity, thermomechanics and other areas of science and technology. Illustrative examples of analytical solutions of specific model problems of locally nonequilibrium heat transfer and the theory of thermal shock for a finite region are given in both classical and generalized formulations (the latter taking into account the finite rate of heat propagation). The characteristic features of functional structures as analytical solutions of the considered mathematical models are revealed.

Some fractional calculus findings associated with the product of incomplete ℵ-function and Srivastava polynomials

Abstract The generalized fractional calculus operators introduced by Saigo and Maeda in 1996 will be examined and further explored in this paper. By combining an incomplete ℵ-function with a broad category of polynomials, we create generalized fractional calculus formulations. The findings are presented in a concise manner that are helpful in creating certain lists of fractional calculus operators. The derived outcomes of a generic nature may yield results in the form of various special functions and in the form of different polynomials as special instances of the primary findings.

Approximation with continuous functions preserving fractal dimensions of the Riemann-Liouville operators of fractional calculus

Approximation with continuous functions preserving fractal dimensions of the Riemann-Liouville operators of fractional calculus

Properties for a Certain Subclass of Analytic Functions Associated with the Salagean q-Differential Operator

Using the Salagean q-differential operator, we investigate a novel subclass of analytic functions in the open unit disc, and we use the Hadamard product to provide some inclusion relations. Furthermore, the coefficient conditions, convolution properties, and applications of the q-fractional calculus operators are investigated for this class of functions. In addition, we extend the Miller and Mocanu inequality to the q-theory of analytic functions.

New Subclass of Close-to-Convex Functions Defined by Quantum Difference Operator and Related to Generalized Janowski Function

This work begins with a discussion of the quantum calculus operator theory and proceeds to develop and investigate a new family of close-to-convex functions in an open unit disk. Considering the quantum difference operator, we define and study a new subclass of close-to-convex functions connected with generalized Janowski functions. We prove the necessary and sufficient conditions for functions that belong to newly defined classes, including the inclusion relations and estimations of the coefficients. The Fekete–Szegő problem for a more general class is also discussed. The results of this investigation expand upon those of the previous study.

Radiological image analysis of double J tube in ureteral calculus operation for clinical evaluation and nursing procedures

ObjectiveTo explore the effect of radiological imaging and continuous care in patients with double-J ureteral calculi (URL) after indwelling double-J tube. MethodsUreteral calculus is a common clinical stone disease, often cause obvious abdominal pain, and can radiate to the groin and medial thigh. After operation for ureteral calculi, some patients will be discharged with double J tube. 68 patients with double J tubes of ureteral calculi treated in our hospital were selected as research objects, and were divided into control group and observation group by lottery. The control group received routine nursing intervention, and the observation group received continuous nursing based on routine nursing intervention. The incidence of circuit-related complications, self-care ability, and quality of life were compared between the two groups. ResultsThe patients in the observation group had significantly lower occurrences of pipeline shedding, hematuria, and bladder irritation than those in the control group (P < 0.05). The self-care ability scores of the observation group at discharge were higher than those of the control group at admission, the comparison was statistically significant (P < 0.05). The quality-of-life scores of patients in the observation group were higher than those in the control group (P < 0.05) in terms of mental health, physical function, and general health. ConclusionMost of the positive ureteral stones can be diagnosed by X-ray plain film. CT was superior to conventional X-ray and ultrasound in showing clearly the anatomy of genitourinary system and other abdomen. After the operation of ureteral calculi, the continuous nursing to the patients who were discharged with double J tube after the operation can effectively reduce the incidence of postoperative complications and improve their quality of life.

Incorporating fractional operators into interaction dynamics of a chaotic biological model

In the exploration of mathematical models in biology, the traditional use of integer-order calculus may fall short in adequately capturing elements such as randomness and memory. As a viable alternative, fractional calculus has been embraced by mathematicians, extending derivatives and integrals to non-integer orders to provide more appropriate tools for managing complexity. In this research, we investigate the effectiveness of two well-known fractional operators, namely the Caputo–Fabrizio and Atangana–Baleanu derivatives, in studying the dynamics of a nonlinear food chain model consisting of three populations. This specific biological system exhibits highly chaotic behavior driven by its sensitivity to various parameters. To approximate solutions, we employ two implicit numerical approaches based on Lagrange interpolation and product integration rules. Furthermore, we analyze the equilibrium points of the system and evaluate their stability characteristics, offering insights into the long-term behavior of the model. To verify the presence of chaotic patterns across different parameter values, we employ two chaos detection techniques—the smaller alignment index and the 0-1 test. By incorporating fractional calculus operators into the dynamic equations, we are able to obtain a more precise representation of such biological systems. Future research endeavors should focus on leveraging additional fractional operators to further enhance the accuracy of modeling nonlinear systems in the field of biology.

New energy effect in non-cylindrical domains with a thermally insulated moving boundary

Objectives. To develop mathematical model representations of the energy effect in non-cylindrical domains having a thermally insulated moving boundary; to introduce a new boundary condition for thermal insulation of a moving boundary both for locally equilibrium heat transfer processes in the framework of classical Fourier phenomenology, as well as for more complex locally non-equilibrium processes in the framework of Maxwell–Cattaneo–Lykov–Vernott phenomenology, taking into account the finite rate of heat propagation into analytical thermophysics and applied thermomechanics; to consider an applied problem of analytical thermophysics according to the theory of thermal shock for a domain with a moving thermally insulated boundary free from external and internal influences; to obtain an exact analytical solution of the formulated mathematical models for hyperbolic type equations; to investigate the solutions obtained using a computational experiment at various values of the parameters included in it; to describe the wave nature of the kinetics of the processes under consideration.Methods. Methods and theorems of operational calculus, Riemann–Mellin contour integrals are used in calculating the originals of complex images with two branch points. A new mathematical apparatus for the equivalence of functional constructions for the originals of the obtained operational solutions, which considers the computational difficulties in finding analytical solutions to boundary value problems for equations of hyperbolic type in the domain with a moving boundary, is developed.Results. Developed mathematical models of locally nonequilibrium heat transfer and the theory of thermal shock for equations of hyperbolic type in a domain with a moving thermally insulated boundary are presented. It is shown that, despite the absence of external and internal sources of heat, the presence of a thermally insulated moving boundary leads to the appearance of a temperature gradient in the domain and, consequently, to the appearance of a temperature field and corresponding thermoelastic stresses in the domain, which have a wave character. A stochastic analysis of this energy effect forms the basis for a proposed transition of the kinetic energy of a moving thermally insulated boundary into the thermal energy of the domain. The presented model representations of the indicated effect confirmed the stated assumption.Conclusions. Mathematical models for locally nonequilibrium heat transfer processes and the theory of thermal stresses are developed and investigated on the basis of constitutive relations of the theory of thermal shock for equations of hyperbolic type in a domain with a thermally isolated moving boundary. A numerical experiment is presented to demonstrate the possibility of transiting from one form of analytical solution of a thermophysical problem to another equivalent form of a new type. The described energy effect manifests itself both for parabolic type equations based on the classical Fourier phenomenology, as well as for hyperbolic type equations based on the generalized Maxwell–Cattaneo–Lykov–Vernott phenomenology.

METHOD OF SOLVING THE CAUCHY PROBLEM FOR THE WAVE EQUATION IN ONE SPACE DIMENSION BY REDUCING THE ORDER OF DIFFERENTIATIONS

In this article, we present method of solving the Cauchy problem for the wave equation in one space dimension by reducing the order of differentiations. To obtain its solution, we use the solution of the Cauchy problem for a linear partial differential equation of the first order with constant coefficients. These methods for solving hyperbolic partial differential equations can be computationally cumbersome and involve much more elementary transformations, but with their help we can quite simply find the class of functions for which we obtain a given solution. In this way, it is possible to transform many types of partial differential equations with constant coefficients, in which differentiation of some given order is carried out in both variables, to an equation whose partial derivative order is one less. The main point of the method for obtaining a solution is the choice of the correct sequence of application of elementary transformations and the simplest operations of differential and integral calculus on the initial given equation.

Operational calculus for Hilfer-Prabhakar operator Applications to inverse problems

Mikusiński's operational calculus has proven to be a powerful tool for tackling various fractional order differential equations. Notably, it has been extended to encompass the Hilfer-Prabhakar fractional order operator. By employing this operational calculus, we have been able to recover and connect previously derived results pertaining to the Hilfer, Riemann-Liouville, and Caputo operators. Furthermore, we address two inverse problems that involve determining space and time dependent source terms and diffusion concentration. Using our operational calculus results, we obtain series solutions for these inverse problems. It is essential to note that the inverse problems belong to the class of ill-posed problems according to Hadamard's definition. To establish the validity and uniqueness of the solutions to both inverse problems, we rigorously prove the existence and uniqueness results. Additionally, we provide specific examples of inverse problems, thereby illustrating the practical applications and versatility of the operational calculus approach.

General fractional integrals and derivatives and their applications

In this survey paper, some recent results obtained for the general fractional integrals and derivatives and for the regularized general fractional derivatives are discussed. We start with a short historical survey of the general Fractional Calculus operators with the Sonin kernels and continue with a presentation of the recent developments regarding the general fractional derivatives and their applications. In particular, we introduce the first and the second 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 consider some Cauchy problems for the fractional differential equations with the general fractional derivatives.

An event-triggered sliding mode control mechanism to exponential consensus of fractional-order descriptor nonlinear multi-agent systems

The problem of leader-following exponential consensus is addressed for fractional-order descriptor nonlinear multi-agent systems by designing an event-triggered sliding mode control strategy. Such an exponential consensus is achieved by defining a suitable sliding surface and a controller under the Mittag–Leffler stability of fractional-order systems. Due to the memory nature of the fractional-order calculus operator, a new condition is established to avoid Zeno behaviors happening, which is different from existing event-triggered schemes. Finally, to illustrate the effectiveness and feasibility of the proposed method, two examples are provided.

Approximating fractional calculus operators with general analytic kernel by Stancu variant of modified Bernstein–Kantorovich operators

The main aim of this paper is to approximate the fractional calculus (FC) operator with general analytic kernel by using auxiliary newly defined linear positive operators. For this purpose, we introduce the Stancu variant of modified Bernstein–Kantorovich operators and investigate their simultaneous approximation properties. Then we construct new operators by means of these auxiliary operators, and based on the obtained results, we prove the main theorems on the approximation of the general FC operators. We also obtain some quantitative estimates for this approximation in terms of modulus of continuity and Lipschitz class functions. Additionally, we exhibit our approximation results for the well‐known FC operators such as Riemann–Liouville integral, Caputo derivative, Prabhakar integral, and Caputo–Prabhakar derivative.

Mild solutions and controllability of fractional evolution inclusions of Clarke's subdifferential type with nonlocal conditions in Hilbert spaces

In this work, our aim is to ensure the existence of mild solutions and study the controllability of fractional evolution inclusions of Clarke's subdifferential type of order q∈(1,2) with nonlocal conditions in the setting of Hilbert spaces. Using fixed point techniques, fractional calculus, multivalued maps, cosine and sine function operators, we discuss the existence of mild solutions for the considered system. Moreover, under suitable conditions, we investigate the controllability of the proposed system. Finally, we present an illustrated example.

Leader–follower sliding mode formation control of fractional-order multi-agent systems: A dynamic event-triggered mechanism

This paper examines the concept of the leader–follower formation for fractional-order multi-agent systems (FO-MASs) by utilizing dynamic event-triggered sliding mode control (SMC) method. Firstly, a dynamic event-triggered mechanism (DETM) is designed relying on the combined measurement vector, in which an auxiliary variable is introduced to dynamically adjust the threshold for each fractional-order agent system. Unlike most existing event-triggered communication mechanisms, whose threshold parameters are always fixed, the threshold parameters in our designed event-triggered conditions can be dynamically adjusted according to the fractional-order dynamic rule. Numerical results show that the proposed DETM can achieve a better system performance in reducing the sampling frequency and the expected formation performance. Secondly, the leader–follower formation control problem is transformed into checking the asymptotic stability problem of the closed-loop system. In addition, due to the memorability of fractional calculus operators, a distinctive condition is established to avoid the occurrence of Zeno behaviors reported in existing dynamic event-triggered schemes. Finally, a simulation example is presented to illustrate the effectiveness and feasibility of the dynamic event-triggered SMC method proposed in this paper.