Fractional-order Integral Operator Research Articles

An Efficient and Robust Numerical Solver for Impulsive Control of Fractional Chaotic Systems

This paper derives a computationally efficient and fast-running solver for the approximate solution of fractional differential equations with impulsive effects. In this connection, for approximating the fractional-order integral operator, a B-spline version of interpolation by corresponding equal mesh points is adopted. An illustrative example illustrates the accuracy of the new solver results as compared with those of the previous study. The proposed solver’s performance is evaluated by the fractional Rössler and susceptible-exposed-infectious impulsive systems. Moreover, the effect of impulsive behaviors is shown for various values of impulsive.

Volterra-Fredholm-type integral inequalities for generalized conformable fractional integrals

In this paper, fractional order integral operators are applied to the study of inequalities, which is also a research hotspot of fractional order integral operators in recent years. After introducing some basic concepts, we establish some new Volterra-Fredholm-type fractional integral inequalities. They can be used as an effective tool to study the properties of solutions of fractional differential equations and fractional integral equations. In addition, several examples are given to illustrate the effectiveness of the research results.

Numerical simulation for generalized space-time fractional Klein–Gordon equations via Gegenbauer wavelet

Abstract This paper investigates numerical solution of generalized space-time fractional Klein–Gordon equations (GSTFKGE) by using Gegenbauer wavelet method (GWM). The developed method makes use of fractional order integral operator (FOIO) for Gegenbauer wavelet, which is constructed by employing the definition of Riemann–Liouville fractional integral (RLFI) operator and Laplace transformation. The present algorithm is based on Gegenbauer wavelet jointly with FOIO to convert a GSTFKGE into a system of equations which is solved by using Newton’s technique. Additionally, the upper bound of error norm of the proposed method is calculated to validate the theoretical authenticity of the developed method. The comparison of numerical outcomes with the existing results in the literature and graphical illustrations show the accuracy and reliability of our method.

On some generalized Raina-type fractional-order integral operators and related Chebyshev inequalities

<abstract><p>In this work, we introduce generalized Raina fractional integral operators and derive Chebyshev-type inequalities involving these operators. In a first stage, we obtain Chebyshev-type inequalities for one product of functions. Then we extend those results to account for arbitrary products. Also, we establish some inequalities of the Chebyshev type for functions whose derivatives are bounded. In addition, we derive an estimate for the Chebyshev functional by applying the generalized Raina fractional integral operators. As corollaries of this study, some known results are recaptured from our general Chebyshev inequalities. The results of this work may prove useful in the theoretical analysis of numerical models to solve generalized Raina-type fractional-order integro-differential equations.</p></abstract>

An experimental analog circuit realization of Matsuda’s approximate fractional-order integral operators for industrial electronics

Analog circuit realization of fractional order (FO) elements is a significant step for the industrialization of FO control systems because of enabling a low-cost, electric circuit realization by means of standard industrial electronics components. This study demonstrates an effective operational amplifier-based analog circuit realization of approximate FO integral elements for industrial electronics. To this end, approximate transfer function models of FO integral elements, which are calculated by using Matsuda’s approximation method, are decomposed into the sum of low-pass filter forms according to the partial fraction expansion. Each partial fraction term is implemented by using low-pass filters and amplifier circuits, and these circuits are combined with a summing amplifier to compose the approximate FO integral circuits. Widely used low-cost industrial electronics components, which are LF347N opamps, resistor and capacitor components, are used to achieve a discrete, easy-to-build analog realization of the approximate FO integral elements. The performance of designed circuit is compared with performance of Krishna’s FO circuit design and performance improvements are shown. The study presents design, performance validation and experimental verification of this straightforward approximate FO integral realization method.

An improved method for image denoising based on fractional-order integration

Given that the existing image denoising methods damage the texture details of an image, a new method based on fractional integration is proposed. First, the fractional-order integral formula is deduced by generalizing the Cauchy integral, and then the approximate value of the fractional-order integral operator is estimated by a numerical method. Finally, a fractional-order integral mask operator of any order is constructed in eight pixel directions of the image. Simulation results show that the proposed image denoising method can protect the edge texture information of the image while removing the noise. Moreover, this method can obtain higher image feature values and better image vision after denoising than the existing denoising methods, because a texture protection mechanism is adopted during the iterative processing.

Analysis and implementation of new fractional-order multi-scroll hidden attractors* *Project supported by the National Natural Science Foundation of China (Grant Nos. 61561022 and 61672226).

To improve the complexity of chaotic signals, in this paper we first put forward a new three-dimensional quadratic fractional-order multi-scroll hidden chaotic system, then we use the Adomian decomposition algorithm to solve the proposed fractional-order chaotic system and obtain the chaotic phase diagrams of different orders, as well as the Lyaponov exponent spectrum, bifurcation diagram, and SE complexity of the 0.99-order system. In the process of analyzing the system, we find that the system possesses the dynamic behaviors of hidden attractors and hidden bifurcations. Next, we also propose a method of using the Lyapunov exponents to describe the basins of attraction of the chaotic system in the matlab environment for the first time, and obtain the basins of attraction under different order conditions. Finally, we construct an analog circuit system of the fractional-order chaotic system by using an equivalent circuit module of the fractional-order integral operators, thus realizing the 0.9-order multi-scroll hidden chaotic attractors.

On positive solutions of a system of equations generated by Hadamard fractional operators

This paper is devoted to studying some systems of quadratic differential and integral equations with Hadamard-type fractional order integral operators. We concentrate on general growth conditions for functions generating right-hand side of considered systems, which leads to the study of Hadamard-type fractional operators on Orlicz spaces. Thus we need to prove some properties of such type of operators. In contrast to the case of Caputo or Riemann–Liouville type of fractional operators, it is not a convolution-type operator, so we need to study some of their new properties. Some more general problems than systems of quadratic integral equations are also studied, and the results are new even in the context of a single integral equation with the Hadamard fractional operator. The paper concludes with illustrative examples.

Parameter Identification of Fractional Order Systems Using a Collocation Method Based on Hybrid Functions

Abstract In this paper, we propose a novel collocation method based on hybrid functions to identify the parameters and differential orders of fractional order systems (FOS). The hybrid functions consist of block-pulse functions and Taylor polynomials. The analytical form of Riemann–Liouville fractional order integral operator of these hybrid functions is derived using the Laplace transform. Then the integral operator is utilized, in conjunction with collocation points, to convert the FOS into an algebraic system directly. The parameters and differential orders of the FOS are estimated by minimizing the error between the output of the actual system and that of the estimated system. The effectiveness of the proposed method is verified through four examples.

Complexity Analysis and Synchronization Control of Fractional-Order Jafari-Sprott Chaotic System

Aiming at the complexity problem of fractional-order Jafari-Sprott chaotic system, in this paper, Adomian decomposition method is used to study its numerical analysis and a complexity analysis method of fractional-order Jafari-Sprott chaotic system based on fuzzy entropy algorithm, sample entropy algorithm and dispersion entropy algorithm is proposed. For the synchronization and control of fractional-order Jafari-Sprott chaotic system, sliding mode control is used to achieve synchronization of fractional-order Jafari-Sprott chaotic system and a control method of fractional-order Jafari-Sprott chaotic system is proposed based on frequency distribution model of fractional-order integral operator. The main results are as follows: (1) The complexity of the fractional-order Jafari-Sprott chaotic system is greater than the integer-order Jafari-Sprott chaotic system, and fractional-order chaotic system has better application prospects. (2) Moreover, it is concluded that the effect of the dispersion entropy algorithm on detecting complexity is the best, which provides theoretical and experimental basis for the practical engineering application of the fractional-order Jafari-Sprott chaotic system. (3) Synchronization and control of fractional-order Jafari-Sprott chaotic system is accomplished by sliding model control and frequency distribution model of fractional-order integral operator respectively. In particular, the control effect of each variable is accomplished by designing a control law based on the frequency distribution model of fractional integral operator.

Study on A Combined Optimal Rational Approximation Method of Fractional-order Calculus Operator and Its Application

Background: Because the fractional-order calculus operator is an irrational function of the complex variable, it is required to deal with rational approximation. Objective: It is difficult to directly implement the fractional-order calculus operator in the numerical simulation and practical application. Methods: So the minimum amplitude frequency approximation error and phase frequency approximation error of rational approximation function are comprehensively considered in the same rational function class. And the rational approximation function construction method and combined optimal rational approximation method of fractional-order calculus operator based on the function approximation theory and logarithmic frequency characteristic asymptote idea are described in detail. Then, an approximation method of fractional-order calculus operator based on the combined optimal rational approximation is proposed in this paper. The approximation effectiveness of fractional-order calculus operator is analyzed by using s-0.6 . And the combined optimal rational approximation method is applied to the design and digital realization of fractional-order controller. In order to simplify the controller design, the controller structure of parallel connection mode is used and the simplified rational approximation expression and transform of fractionalorder integral operator in the different order is obtained. Results: The analyzed results show that the combined optimal rational approximation method of fractional-order calculus operator is consistent with the ideal c in amplitude frequency domain and has smallest error for the phase frequency characteristic. And the applied results show that the proposed combined optimal rational approximation method can conveniently control and simulate the practical computer control system by analyzing the simplified rational approximation expression and z transform of fractional-order integral operator. Conclusion: So the combined optimal rational approximation method can provide better engineering applications of fractional-order controller in the practical control system. Keywords: Fractional-order calculus, rational approximation of operator, combined optimal rational approximation, discretization, fractional-order PIα Dβ controller, parallel connection structure.

Comments on time-varying fractional order

The calculus of arbitrary order, known as the fractional calculus, allows for real- and complex-valued orders. Interest in time-varying order is growing in order to study transient behavior under varying environmental conditions. Popular forms currently in use suffer from fundamental flaws of dimensional inconsistency and predict physically implausible behaviors such as violation of causality and/or violation of conservation of energy. This article reviews some of the motivation behind the study of time-varying fractional order and makes suggestions as to how to overcome the flaws in the forms in current usage. A new fractional-order integral operator is proposed that may allow for modeling time-varying fractional-order systems in a dimensionally consistent and physically plausible manner. Possible experimental tests of the revised fractional-order model are proposed.

Tail asymptotics under beta random scaling

Let X , Y , B be three independent random variables such that X has the same distribution function as YB. Assume that B is a beta random variable with positive parameters α , β and Y has distribution function H with H ( 0 ) = 0 . In this paper we derive a recursive formula for calculation of H, if the distribution function H α , β of X is known. Furthermore, we investigate the relation between the tail asymptotic behaviour of X and Y, which is closely related to asymptotics of Weyl fractional-order integral operators. We present three applications of our asymptotic results concerning the extremes of two random samples with underlying distribution functions H and H α , β , respectively, and the conditional limiting distribution of bivariate elliptical distributions.

Fractional Dual Solutions and Corresponding Sources - Abstract

Fractional dual solutions to the Maxwell's equations for a TEM plane wave propagating in an arbitrary direction are derived. The results obtained are utilized to calculate fractional dual solutions for spectrum of plane waves radiated by a line source. It is shown that fractional dual field for a plate source and for a line source, in the far-zone along the axis of symmetry, are related via fractional order integral operator. Field that can be regarded as intermediate step between the fractional dual field for a plate source and for a line source is determined. Source distributions corresponding to the fractional dual solutions and intermediate fractional dual solutions are also derived.

Fractional Solutions for the Helmholtz's Equation in a Multilayered Geometry

A generalized fractional order integral operator is derived that relates the one and two dimensional Green's function for a geometry with parallel plane interfaces. It is found that order of the operator is independent of the geometry while lower limit and operational variable are functions of the geometry. Utilizing the general fractional order operator, it is possible to calculate the fractional solutions for Helmholtz's equation in a geometry with parallel plane interfaces. A generalized expression is proposed for far-zone fractional solution in a geometry with parallel plane interfaces.

Fractional Solutions for the Helmholtz's Equation in a Multilayered Geometry - Abstract *

A generalized fractional order integral operator is derived that relates the one and two dimensional Green's function for a geometry with parallel plane interfaces. It is found that order of the operator is independent of the geometry while lower limit and operational variable are functions of the geometry. Utilizing the general fractional order operator, it is possible to calculate the fractional solutions for Helmholtz's equation in a geometry with parallel plane interfaces. A generalized expression is proposed for far-zone fractional solution in a geometry with parallel plane interfaces.