Fractional Calculus Operators Research Articles

Transformation formulas of incomplete hypergeometric functions via fractional calculus operators

The desire for present article is to derive from the application of fractional calculus operators a transformation that expresses a potentially useful incomplete hypergeometric function in various forms of a countable sum of lesser-order functions. Often listed are numerous (known or new) specific cases and implications of the findings described herein

Certain <i>k</i>-fractional calculus operators and image formulas of <i>k</i>-Struve function

In this article, the Saigo's k-fractional order integral and derivative operators involving k-hypergeometric function in the kernel are applied to the k-Struve function; outcome are expressed in the term of k-Wright function, which are used to present image formulas of integral transforms including beta transform. Also special cases related to fractional calculus operators and Struve functions are considered.

Fractional-Order Modeling and Analysis of a Three-Phase Voltage Source PWM Rectifier

In this paper, the fractional-order modeling and analysis of a three-phase voltage source PWM rectifier (VSR) are researched considering the fractional-order characteristics of actual inductors and capacitors. While the Caputo fractional calculus operator is used to describe the fractional-order characteristics of an inductor and a capacitor, the fractional-order model of a three-phase VSR is established in the three-phase static coordinate system. With the coordinate transformation, the fractional-order model is expressed in the two-phase static coordinate system and in the synchronous rotation coordinate system, respectively. Simulation results verify the effectiveness of the established models, and indicate that the fractional-order models can describe the operating characteristics of a three-phase VSR more accurately. Besides, compared with the traditional integer-order three-phase VSRs, the three-phase VSRs which contain the fractional-order inductors and the fractional-order capacitor can present better static and dynamic performance indexes (such as smaller overshoot and shorter regulation time) by appropriately selecting the orders of the inductors and the capacitor.

On an integral and consequent fractional integral operators via generalized convexity

Fractional calculus operators are very useful in basic sciences and engineering. In this paper we study an integral operator which is directly related with many known fractional integral operators. A new generalized convexity namely exponentially (α, h?m)-convexity is defined which has been applied to obtain the bounds of unified integral operators. A generalized Hadamard inequality is established for the generalized convex functions. The established theorems reproduce several known results.

On A Certain Class of Meromorphic Multivalent Functions Defined by Fractional Calculus Operator

In this work, we study a new class of meromorphicmultivalent functions, defined by fractional differ-integral operator.We obtain some geometricproperties, such ascoefficient inequality, growth and distortion bounds, convolution properties, integral representation, radii of starlikeness, convexity, extreme pointsproperties, weighted mean and arithmetic meanproperties.

A Fractional Quadratic autocatalysis associated with chemical clock reactions involving linear inhibition

Abstract Our main aim in this article is to introduce and investigate a new model of fractional-order quadratic autocatalysis with linear inhibition. In particular, we evaluate the approximate solutions of this model by means of the power law, the exponential law and the Mittag-Leffler kernel. The approximate solutions are based upon the fundamental theorem of fractional calculus as well as the Lagrange polynomial interpolation. We compare the approximate solutions with that derived by using the finite-difference method, thereby showing excellent agreement which we have found by applying the power law and the Mittag-Leffler kernel. We study the effect of the variation the fractional-order on the behavior of the solutions due to the presence of definitions of new fractional-calculus operators. We observe the chaotic fractional behavior and illustrate the chaotic fractional-order quadratic autocatalysis with linear inhibition system by plotting the solutions in the plane.

Implementation of Simplified Fractional-Order PID Controller Based on Modified Oustaloup’s Recursive Filter

The aim of this paper is to simplify the design of fractional-order PID controllers. Because the analytical expressions and operations of fractional-order systems are complex, numerical approximation tool is needed for the simulation analysis and engineering practice of fractional-order control systems. The key to numerical approximation tool is the exact approximation of the fractional calculus operator. A commonly used method is to approximate the fractional calculus operator with an improved Oustaloup’s recursive filter. Based on the modified Oustaloup’s recursive filter, a mathematical simplification method is proposed in this paper, and a simplified fractional-order PID controller (SFOC) is designed. The controller parameters are tuned by using genetic algorithm (GA). Effectiveness of the proposed control scheme is verified by simulation. The performance of SFOC has been compared with that of the integer-order PID controller and conventional fractional-order PID controller (CFOC). It is observed that SFOC requires smaller effort as compared with its integer and conventional fractional counterpart to achieve the same system performance.

On a Class of Meromorphic Multiivalent Functiions Convoluted withi Higher Derivatiives of Fractionali Calculus Operator

The main goal of this paper is to study and discuss a new class of meromorphici "functions[ which are multivalent defined by [fractional calculus operators. Coefficients iestimates , radiisi of satarlikeness , convexityi and closed-to-iconvexity are studied. Also distortion iand closure theorems for the classi" , are considered.

A New Extension of the τ-Gauss Hypergeometric Function and Its Associated Properties

In this article, we define an extended version of the Pochhammer symbol and then introduce the corresponding extension of the τ-Gauss hypergeometric function. The basic properties of the extended τ-Gauss hypergeometric function, including integral and derivative formulas involving the Mellin transform and the operators of fractional calculus, are derived. We also consider some new and known results as consequences of our proposed extension of the τ-Gauss hypergeometric function.

Novel technique of Atangana and Baleanu for heat dissipation in transmission line of electrical circuit

Abstract The new idea of Atangana and Baleanu introduced recently for fractional derivatives has received tremendous attention from researchers of various fields. However, this idea has not been applied for heat dissipation in transmission line of electrical circuit. Although the phenomenon of heat dissipation potentially damages the electrical devices which leads to catastrophic failure and shortens the life expectancy of costly electrical components. This manuscript signifies an electro-analog model of fractional diffusion process which enables to simulate heat dissipation in circuit board. The modeling of the problem has been established through modern operators of fractional calculus via the governing differential equation. By invoking Laplace transform on fractional governing equation of heat dissipation in transmission line of circuit, the analytic solutions are investigated through the exponential and Mittage Leffler kernel of non-integer order differentiations. The duality of analytic solutions has been compared with Caputo–Fabrizio and Atangana–Baleanu fractional differentiations graphically using embedded pertinent parameters. The comparison of both approaches of fractional differentiations has disclosed the heat transfer of circuit board and diffusion process with dissipation vividly.

Neighborhoods and Partial Sums of a New Class of Meromorphic Multivalent Functions Defined by Fractional Calculus

In this paper, we introduce and study a new class A^* (λ,μ,ν,η,p,ζ,τ) of meromorphic multivalent functions defined by fractional calculus operator of the punctured unit disc U^*. On this class we obtain several results like, coefficient inequality, modified Hadamard product, (N,δ)- neighborhood, partial sums, convex linear combination and integral operator.

Extended Mittag-Leffler function and associated fractional calculus operators

Abstract Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

Fractional Calculus of Extended Mittag-Leffler Function and Its Applications to Statistical Distribution

Several fractional calculus operators have been introduced and investigated. In this sequence, we aim to establish the Marichev-Saigo-Maeda (MSM) fractional calculus operators and Caputo-type MSM fractional differential operators of extended Mittag-Leffler function (EMLF). We also investigate the statistical distribution associated with the EMLF. Finally, we derive some of the particular cases of the main results.

Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time–space fractional Bloch–Torrey equations on irregular convex domains

Models based on partial differential equations containing time–space fractional derivatives have attracted considerable interest in the past decade because of their ability to model anomalous transport phenomena. These phenomena are strongly connected to the interactions within complex and non-homogeneous media exhibiting spatial heterogeneity. The class of equations with multi-term time–space derivatives of fractional orders has been found to be very useful in the description of such interactions. This motivates the extension of the classical Bloch–Torrey equation through the application of the operators of fractional calculus to new multi-term time–space fractional Bloch–Torrey equations with Riesz fractional operators.In this paper, we firstly propose an unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time–space fractional diffusion equation with Riesz fractional operators on irregular convex domains. Secondly, we rigorously establish the stability and convergence of the numerical scheme. Thirdly, we extend the computational model to solve a system of coupled two-dimensional multi-term time–space fractional Bloch–Torrey equations. Finally, some numerical results are given to demonstrate the versatility and application of the models.

On new extensions of the generalized Hermite matrix polynomials

Various families of generating matrix functions have been established in diverse ways. The objective of the present paper is to investigate these generalized Hermite matrix polynomials, and derive some important results for them, such as, the generating matrix functions, matrix recurrence relations, an expansion of xnI, finite summation formulas, addition theorems, integral representations, fractional calculus operators, and certain other implicit summation formulae.

On the class of multivalent analytic functions defined by differential operator for derivative of first order

In the submitted search ,by making use of Differential operator ,we drive coefficient bounds and some important properties of the subclass ( ) of analytic
 and multivalent function with negative coefficients .Distortion property for functions in the class are investigated once by using the composition involving an integral operator and certain fractional calculus operator and other once by using the composition involving an integral operator and certain fractional calculus inverse operator .

Discrete fractional solution of a nonhomogeneous non-Fuchsian differential equations

In this article, we also present new fractional solutions of the non-homogeneous and homogeneous non-Fuchsian differential equation by using nabla-discrete fractional calculus operator ??(0 < ? < 1). So, we acquire new solution of these equation in the discrete fractional form via a newly developed method.

Fractional Integration of the Product of two Multivariable Gimel-Functions and a General Class of Polynomials

A significantly large number of earlier works on the subject of fractional calculus give the interesting account of the theory and applications of fractional calculus operators in many different areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, special functions, the summation of series, etc.). The object of the present paper is to study and develop the Saigo-Maeda operators. First, we establish four results that give the images of the product of two multivariable Gimel-functions and a general class of multivariable polynomials in Saigo- Maeda operators. On account of the general nature of the Saigo-Maeda operators, multivariable Gimel-functions and a class multivariable polynomials a large number of new and known theorems involving Riemann-Liouville and Erdelyi- Kober fractional integral operators and several special functions.

Operators of Fractional Calculus and Their Applications

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Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions

We consider an integral transform introduced by Prabhakar, involving generalised multi-parameter Mittag-Leffler functions, which can be used to introduce and investigate several different models of fractional calculus. We derive a new series expression for this transform, in terms of classical Riemann–Liouville fractional integrals, and use it to obtain or verify series formulae in various specific cases corresponding to different fractional-calculus models. We demonstrate the power of our result by applying the series formula to derive analogues of the product and chain rules in more general fractional contexts. We also discuss how the Prabhakar model can be used to explore the idea of fractional iteration in connection with semigroup properties.