Differintegral Operator Research Articles

Optimal Digital Implementation of Fractional-Order Models in a Microcontroller.

The growing number of operations in implementations of the non-local fractional differentiation operator is cumbersome for real applications with strict performance and memory storage requirements. This demands use of one of the available approximation methods. In this paper, the analysis of the classic integer- (IO) and fractional-order (FO) models of the brushless DC (BLDC) micromotor mounted on a steel rotating arms, and next, the discretization and efficient implementation of the models in a microcontroller (MCU) is performed. Two different methods for the FO model are examined, including the approximation of the fractional-order operator () using the Oustaloup Recursive filter and the numerical evaluation of the fractional differintegral operator based on the Grünwald–Letnikov definition and Short Memory Principle. The models are verified against the results of several experiments conducted on an ARM Cortex-M7-based STM32F746ZG unit. Additionally, some software optimization techniques for the Cortex-M microcontroller family are discussed. The described steps are universal and can also be easily adapted to any other microcontroller. The values for integral absolute error (IAE) and integral square error (ISE) performance indices, calculated on the basis of simulations performed in MATLAB, are used to evaluate accuracy.

Balance Adjustment of Power-Line Inspection Robot Using General Type-2 Fractional Order Fuzzy PID Controller

In this study, a general type-2 fractional order fuzzy PID (GT2FO-FPID) controller is proposed to fulfil the balance adjustment of the Power-line Inspection (PLI) robot system. It is a combination of Mamdani general type-2 fuzzy logic controller (GT2-FLC) and fractional PID controller. Since the PLI robot system is an under-actuated system, it’s necessary to get complete information of the system. However, when all state variables are treated as input to the controller, there is a problem with the rule explosion. Because of this, the information fusion method is adopt to solve the problem and simplify the controller design. At the same time, fractional-order integral-differential operators and input-output scaling factors, which are taken as design variables and optimized by genetic algorithm (GA). To assess the performance of proposed controller based on symmetry criterion, we compared it against existing controllers, i.e., interval type-2 fractional order fuzzy PID (IT2FO-FPID), type-1 fractional order fuzzy PID (T1FO-FPID), and conventional fractional order (FOPID) controllers. Furthermore, to show the anti-inference ability of the proposed controller, three common perturbed process are tested. Finally, simulation results show that the GT2FO-FPID controller outperforms other controllers in the presence of external perturbations on the PLI robot system.

Sufficient conditions for univalence obtained by using Briot-Bouquet differential subordination

In this paper, we define the operator Im : differential-integral operator, where Sm is S˘al˘agean differential operator and Lm is Libera integral operator. By using the operator Im the class of univalent functions denoted by is defined and several differential subordinations are studied. Even if the use of linear operators and introduction of new classes of functions where subordinations are studied is a well-known process, the results are new and could be of interest for young researchers because of the new approach derived from mixing a differential operator and an integral one. By using this differential–integral operator, we have obtained new sufficient conditions for the functions from some classes to be univalent. For the newly introduced class of functions, we show that is it a class of convex functions and we prove some inclusion relations depending on the parameters of the class. Also, we show that this class has as subclass the class of functions with bounded rotation, a class studied earlier by many authors cited in the paper. Using the method of the subordination chains, some differential subordinations in their special Briot-Bouquet form are obtained regarding the differential–integral operator introduced in the paper. The best dominant of the Briot-Bouquet differential subordination is also given. As a consequence, sufficient conditions for univalence are stated in two criteria. An example is also illustrated, showing how the operator is used in obtaining Briot–Bouquet differential subordinations and the best dominant.

Novel numerical solution to the fractional neutron point kinetic equation in nuclear reactor dynamics

In this work, a novel numerical solution to modified Fractional Neutron Point Kinetic (FNPK) equations is presented. The method is based on a numerical solution to linear multi-term fractional differential equations taking from scientific literature. Differential-integral operators of fractional order are numerically solved with the novel method. The impact of the order of the operators has been assessed during the process of order reduction of the fractional differential-integral equation. The numerical solution is applied to case with sinusoidal reactivity, and different values of the anomalous diffusion order are used to study the effect on the neutron density. The results of the neutron density behavior obtained with this proposed numerical novel solution were compared against the classical neutron point kinetics equations and with other results from scientific literature. The comparison showed a clear improvement of the numerical results when using a fractional differential-integral operator instead of an only fractional differential operator.

Analog of the Gellerstedt problem for the mixed type equation with integral-differential operators of fractional order

Analog of the Gellerstedt problem for the mixed type equation with integral-differential operators of fractional order

Applications of differential subordinations involving a generalized fractional differintegral operator

Using the third-order differential subordination basic results, we introduce certain classes of admissible functions and investigate some applications of third-order differential subordination for p-valent functions associated with generalized fractional differintegral operator.

Comparative analysis on landsat image enhancement using fractional and integral differential operators

In this paper, Landsat image enhancement based on the fractional and integral differential methods is compared. Enhancement techniques used in remote sensing are based on the traditional integral order differential mask operators, such as Sobel, Prewitt and Laplacian operators. Other techniques involve the fractional calculus and general masks using Grunwald-Letnikov with eight directions. Notably, it is crucial to perform fractional filtering of I weight (intensity or brightness) in color space of (hue, saturation, intensity or brightness) according to the filtering rule. We study the quantitative analysis of enhancement performance via the gray-scale histogram and information entropy. Finally, we demonstrated that fractional differential operator is capable of enhanced performance superior to that of the integral differential operators. The optimal fractional order for image enhancement of Landsat Thematic Mapper is 1.15, 1.4, and 0.6 based on the (3 × 3), (5 × 5), and (7 × 7) masks, respectively.

The Principle of Differential Subordination and Its Application to Analytic and p-Valent Functions Defined by a Generalized Fractional Differintegral Operator

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.

Subordination and inclusion theorems for higher order derivatives of a generalized fractional differintegral operator

The main object of this paper is to investigate some subordination results of certain subclasses of multivalent analytic functions which are defined by a generalized fractional differintegral operator. Inclusion relations for functions in the class mathcal {S}_{p,q}^{lambda,mu,eta }left (zeta ;A,Bright) and the images of these functions by the generalized Bernardi-Libera-Livingston integral operator are also considered.

Fractional differential and integral operations via cumulative approach

In this study, a fractal operator model of cumulative processes is described. Accordingly, differential and integral operators of the fractional calculus are derived by the fractal operator model of a cumulative process. In order to exhibit the relation between our cumulative approach and fractional calculus, vertical motion of a body is handled within these frameworks. Thereby, regard to our assessments, the underlying physical mechanism of the success of the fractional differintegral operators in describing stochastic complex systems is uncovered to some extent.

Study of new class of q-fractional integral operator

In this paper, we study on the new class of q-fractional integral operator. In the aid of iterated Cauchy integral approach to fractional integral operator, we applied tp f(t) in these integrals and a new class of q-fractional integral operator with parameter p, is introduced. Recently, the q-analogue of fractional differential integral operator is studied and all of the operators defined in these studies are q-analogue of Riemann fractional differential operator. We show that our new class of operator generalize all the operators in use, and additionally, it can cover the q-analogue of Hadamard fractional differential operator, as well. Some properties of this operator are investigated.

A Computational Fractional Signal Derivative Method

We propose an efficient computational method to obtain the fractional derivative of a digital signal. The proposal consists of a new interpretation of the Grünwald–Letnikov differintegral operator where we have introduced a finite Cauchy convolution with the Grünwald–Letnikov dynamic kernel. The method can be applied to any signal without knowing its analytical form. In the experiments, we have compared the proposed Grünwald–Letnikov computational fractional derivative method with the Riemman–Louville fractional derivative approach for two well-known functions. The simulations exhibit similar results for both methods; however, the Grünwald–Letnikov method outperforms the other approach in execution time. Finally, we show an application of how our proposal can be useful to find the fractional relationship between two well-known biomedical signals.

Non-smooth convex Lyapunov functions for stability analysis of fractional-order systems

Based on proximal subdifferentials and subgradients, and instrumented with an extended Caputo differintegral operator, the stability analysis of a general class of fractional-order nonlinear systems is considered by means of non-smooth but convex Lyapunov functions. This facilitates concluding the Mittag–Leffler stability for fractional-order systems whose solutions are not necessarily differentiable in any integer-order sense. As a solution to the problem of robust command of fractional-order systems subject to unknown but Lebesgue-measurable and bounded disturbances, a unit-vector-like integral sliding mode controller is proposed. Numerical simulations are conducted to highlight the reliability of the proposed method in the analysis and design of fractional-order systems closed by non-smooth robust controllers.

Subordination Properties for Multivalent Functions Associated with a Generalized Fractional Differintegral Operator

Using of the principle of subordination, we investigate some subordination and convolution properties for classes of multivalent functions under certain assumptions on the parameters involved, which are defined by a generalized fractional differintegral operator under certain assumptions on the parameters involved.

On a New Class of Meromorphic Multivalent Functions Defined by Fractional Differ-integral Operator

In this work, we submite and study a newlclass of meromorphiccmultivalent functions defined by fractional differ-integral operator . We gain some geometric properties , such as, coefficient inequality, growth and distortion bounds, convolution properties, integral representation, radii of starlikeness and convexity, extreme points, weighted mean and arithmetic mean for functions belonging to the class

Subordinating results for p-valent functions associated with the Srivastava–Saigo–Owa fractional differintegral operator

In this paper, by making use of the principle of subordination, we investigate some subordination and convolution properties of certain subclasses of\(\ p\)-valent analytic functions which are defined by the Srivastava–Saigo–Owa fractional differintegral operator.

Inclusion relations for subclasses of multivalent functions defined by Srivastava–Saigo–Owa fractional differintegral operator

The purpose of this paper is to introduce subclasses of multivalent functions by using Srivastava–Saigo–Owa fractional differintegral operator and investigate various properties for these subclasses. Also, we investigate inclusion relations involving the operator \(F_{p,c}\).

Fractional-order algorithms for tracking Rayleigh fading channels

This paper presents the tracking behavior of fractional-order (FO) variants of the normalized least mean square (NLMS) algorithm in a nonstationary environment modeled as time-varying Rayleigh fading sequence. The celebrated recursive least squares (RLS) or its variant extended RLS (E-RLS) algorithms fail in such situations although they exhibit faster convergence but with the undesired feature of higher computational complexity. The FO algorithms are based on the Riemann–Liouville differintegral operator which is used in the gradient calculation; such schemes provide two step sizes and an FO to control the rate of convergence. In evaluation, we consider a high-speed mobile environment with a Rayleigh channel which results in different Doppler frequency shifts depending upon the transmission frequency, relative velocity of the transmitter and receiver. The proposed algorithms are compared with the NLMS, RLS and E-RLS schemes, and numerical experiments show the superiority of the FO variants over these schemes in terms of stability and model accuracy in the steady state. A hybrid scheme is also shown where the weights of an FO variant are initially trained with RLS and then performs self-adaptation; the FO scheme is confirmed to have better performance than all traditional counterparts.

Efficient control of integrated power system using self-tuned fractional-order fuzzy PID controller

The integrated power system (IPS) uses various autonomous generation and energy storage systems like aqua electrolyzer, battery, diesel engine, flywheel, fuel cell, solar photovoltaic, ultracapacitor, wind turbine, etc. These may be switched on/off and may run at higher/lower power outputs, at different times. Additionally, IPS is also subjected to parameter variations of its components and the load. As a result, the frequency of an IPS fluctuates from the nominal desired value and therefore it requires a robust controller to accomplish the above-mentioned task. In this work, a self-tuned fractional-order fuzzy PID (STFOFPID) controller, tuned using cuckoo search algorithm, is investigated for efficient control of IPS. STFOFPID is essentially a Takagi–Sugeno model-based fuzzy adaptive controller comprising of non-integer-order differ-integral operators. To assess the relative performance of STFOFPID controller, it is compared with its integer-order counterpart on the basis of their respective objective function value defined as the sum of integral of squared error and integral of squared deviation of controller output. Intensive LabVIEW-based simulation studies have indicated the robustness and hence superiority of STFOFPID controller over its integral counterpart.

Multivalent functions associated with Srivastava–Saigo–Owa fractional differintegral operator

The purpose of the present paper is to investigate some subordination and other properties of several subclasses of multivalent functions which are defined by the Srivastava–Saigo–Owa fractional differintegral operator. Inclusion relations for functions in the class $$\mathcal {S}_{p,\lambda ,\mu ,\eta }^{m,\delta ,\zeta }(\alpha; A, B)$$ and the images of these functions by the generalized Bernardi–Libera–Livingston integral operator are also considered.