Fractional Differintegration Research Articles

On the use of fractional calculus to improve the pulse arrival time (PAT) detection when using photoplethysmography (PPG) and electrocardiography (ECG) signals.

The pulse arrival time (PAT) has been considered a surrogate measure for pulse wave velocity (PWV), although some studies have noted that this parameter is not accurate enough. Moreover, the inter-beat interval (IBI) time series obtained from successive pulse wave arrivals can be employed as a surrogate measure of the RR time series avoiding the use of electrocardiogram (ECG) signals. Pulse arrival detection is a procedure needed for both PAT and IBI measurements and depends on the proper fiducial points chosen. In this paper, a new set of fiducial points that can be tailored using several optimization criteria is proposed to improve the detection of successive pulse arrivals. This set is based on the location of local maxima and minima in the systolic rise of the pulse wave after fractional differintegration of the signal. Several optimization criteria have been proposed and applied to high-quality recordings of a database with subjects who were breathing at different rates while sitting or standing. When a proper fractional differintegration order is selected by using the RR time series as a reference, the agreement between the obtained IBI and RR is better than that for other state-of-the-art fiducial points. This work tested seven different traditional fiducial points. For the agreement analysis, the median standard deviation of the difference between the IBI and RR time series is 5.72 ms for the proposed fiducial point versus 6.20 ms for the best-performing traditional fiducial point, although it can reach as high as 9.93 ms for another traditional fiducial point. Other optimization criteria aim to reduce the standard deviation of the PAT (7.21 ms using the proposed fiducial point versus 8.22 ms to 15.4 ms for the best- and worst-performing traditional fiducial points) or to minimize the standard deviation of the PAT attributable to breathing (3.44 ms using the proposed fiducial point versus 4.40 ms to 5.12 ms for best- and worst-performing traditional fiducial points). The use of these fiducial points may help to better quantify the beat-to-beat PAT variability and IBI time series.

Consistent Approximation of Fractional Order Operators

Abstract Fractional order controllers become increasingly popular due to their versatility and superiority in various performances. However, the bottleneck in deploying these tools in practice is related to their analog or numerical implementation. Numerical approximations are usually employed in which the approximation of fractional differintegrator is the foundation. Generally, the following three identical equations always hold, i.e., 1sα1s1−α=1s, sα1sα=1, and sαs1−α=s. However, for the approximate models of fractional differintegrator sα, α∈(−1,0)∪(0,1), there usually exist some conflicts on the mentioned equations, which might enlarge the approximation error or even cause fallacies in multiple orders occasion. To overcome the conflicts, this brief develops a piecewise approximate model and provides two procedures for designing the model parameters. The comparison with several existing methods shows that the proposed methods do not only satisfy the equalities but also achieve high approximation accuracy. From this, it is believed that this work can serve for simulation and realization of fractional order controllers more friendly.

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.

A complex analysis approach to Atangana–Baleanu fractional calculus

The standard definition for the Atangana–Baleanu fractional derivative involves an integral transform with a Mittag‐Leffler function in the kernel. We show that this integral can be rewritten as a complex contour integral which can be used to provide an analytic continuation of the definition to complex orders of differentiation. We discuss the implications and consequences of this extension, including a more natural formula for the Atangana–Baleanu fractional integral and for iterated Atangana–Baleanu fractional differintegrals.

Numerical and Analytical Method for Solution of Fractional Differential Equations

Fractional differ-integrals plays very important role in the description of the dynamics of many complex systems. Many methods were proposed to explicit solution in the last two decades but rigorous convergence analysis is often not available. Thus a user who needs a systematic approach for a particular problem often has difficulties in finding suitable method. The development of suitable and well-founded new analytical and approximate methods to solve complicated Fractional ordinary and partial differential equations has become the major attraction amongst researchers. In the present paper the fractional reduced differential transform method (FRDTM) is implemented to solve linear and nonlinear initial and boundary value problems for time fractional differential equations. The relation of solution with time at given x=1 and the phase plot of solution behavior for different derivative orders like 0.5, 0.75 and 1 are depicted through graphs.

Fractional operators with generalized Mittag-Leffler kernels and their iterated differintegrals.

Depending on a previous work about fractional operators of Riemann type (ABR) and Caputo type (ABC) with kernels of Mittag-Leffler in three parameters [Eα,μ γ(λ,t-s)], we derive the corresponding fractional integrals with arbitrary order by using the infinite binomial theorem, and study their semi-group properties and their action on the ABC type fractional derivatives to prove the existence and uniqueness theorem for the ABC-fractional initial value problems. In fact, as advantages to the obtained extension, we find that for μ≠1, we obtain a nontrivial solution for the linear ABC-type initial value problem with constant coefficient and prove a certain semigroup property in the parameters μ and γ simultaneously. Iterated type fractional differ-integrals are constructed by iterating fractional integrals of order (α,μ,1) to add a fourth parameter, and a semigroup property is derived under the existence of the fourth parameter. The Laplace transforms for the Atangana-Baleanu (AB) fractional integrals and the AB iterated fractional differ-integrals are calculated. An alternative representation of the ABR-derivatives is given and is compared, in the case γ=1, with the iterated AB differ-integrals with negative order (α,μ,1),-1. An example and several remarks are given to illustrate part of the proven results and to point out some particular cases. The obtained results generalized and improved some recent results.

On a new definition of fractional differintegrals with Mittag-Leffer kernel

We introduce and study the properties of a new family of fractional differential and integral operators which are based directly on an iteration process and therefore satisfy a semigroup property. We also solve some ODEs in this new model and discuss applications of our results.

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 the distributed order PID controller

This paper proposes a generalized fractional controller that covers the complete range of differintegrators of order −1 to order 1. This controller is named distributed order PID controller (DOPID) due to the fact that it broadens the structure of the conventional PID controller. Furthermore, a new method for implementation of distributed order PID controller (DOPID), as well as analysis of its properties in both time and frequency domain are proposed. As DOPID controller represents generalized classical PID controller, it is approximated by a compound fractional controller with multiple fractional differintegrators, with multiple gains, connected in parallel. Orders of these differintegrators have been equally spaced, with the first one being the first order classical integrator, and the last one being the first order classical differentiator. Noise cancellation filter is considered explicitly in the controller’s structure.

Design of new practical phase shaping circuit using optimal pole–zero interlacing algorithm for fractional order PID controller

This paper presents the implementation of fractional order PID (FO-PID) controller using hardwired modules of constant phase element (CPE). A new approach of phase shaping by slope cancellation of asymptotic phase plots for zeros and poles within the given bandwidth is realized. Analog circuits, which exhibit analog fractional-order integrator and fractional-order differentiator, are used for building the FO-PID controller. The design procedure is developed to obtain the optimal pole---zero pairs and respective "Fractance" components to realize for any value of fractional differ-integrator. These CPE elements give minimum error tolerance over the set phase value by using commercially available (R---C) components and Op-Amps. The pole---zero location in the root locus plot with constant asymptotic angle under various feed-forward gains is achieved with these analog integrodifferential circuits of the FO-PID. The iso-damping feature of the controller is practically demonstrated. A comparative performance is demonstrated under various settings of feed forward gains, which indicate the constant overshoot with FO-PID against the conventional PID. These circuits are developed and implemented with a DC motor emulator to confirm the designed performance of the controller.

Some Characterizations of Integral Operators Associated with Certain Classes of p-Valent Functions Defined by the Srivastava–Saigo–Owa Fractional Differintegral Operator

The purpose of this paper is to introduce new integral operators associated with Srivastava–Saigo–Owa fractional differintegral operator. We investigate some properties for the integral operators \({\mathcal {F}}_{p,\eta ,\mu }^{\lambda ,\delta }(z)\) and \({\mathcal {G}}_{p,\eta ,\mu }^{\lambda ,\delta }(z)\) to be in the classes \({\mathcal {R}}_{k}^{\zeta }\left( p,\rho \right) \) and \({\mathcal {V}}_{k}^{\zeta }\left( p,\rho \right) \).

Some Subclasses of p-Valent Functions Defined by Generalised Fractional Differintegral Operator - I

In this paper, by applying a generalized extended fractional differintegral operator S λ;μ;η 0;z (z 2 4 ; p 2 N; μ;η 2 R; μ < p+ 1;−∞ < λ < η + p+ 1) we define a new class convex functions C V λ;μ;η p (α; A;B) and several sharp inclusion relationships and other interesting properties were discussed by using the te chniques of differential subordination.

Approximations of higher-order fractional differentiators and integrators using indirect discretization

This paper describes new approximations of fractional order integrators (FOIs) and fractional order differentiators (FODs) by using a continued fraction expansion-based indirect discretization scheme. Different tenth-order fractional blocks have been derived by applying three different s-to-z transforms described earlier by Al-Alaoui, namely new two-segment, four-segment, and new optimized four-segment operators. A new addition has been done in the new optimized four-segment operator by modifying it by the zero reflection method. All proposed half (s$^{\pm 1/2})$ and one-fourth (s$^{\pm 1/4})$ differentiator and integrator models fulfill the stability criterion. The tenth-order fractional differ-integrators (s$^{\pm \alpha })$ based on the modified new optimized four-segment rule show tremendously improved results with relative magnitude errors (dB) of $\le $ -15 dB for $\alpha $ = 1/2 and $\le $ -20 dB for $\alpha $ = 1/4 in the full range of Nyquist frequency so these have been further analyzed. The main contribution of this paper lies in the reduction of these tenth-order blocks into four new fifth-order blocks of half and one-fourth order models of FODs and FOIs. The analyses of magnitude and phase responses show that the proposed new fifth-order half and one-fourth differ-integrators closely approximate their ideal counterparts and outperform the existing ones.

Certain Families of Multivalent Analytic Functions Associated with Iterations of the Owa-Srivastava Fractional Differintegral Operator

By making use of a multivalent analogue of the Owa-Srivastava fractional differintegral operator and its iterations, certain new families of analytic functions are introduced. Several interesting properties of these function classes, such as convolution theorems, inclusion theorems, and class-preserving transforms, are studied.

Twenty‐four hour variation in heart rate variability indices derived from fractional differintegration

Assuming that RR time-series behave as a fractionally differintegrated Gaussian process, García-González etal. (2003) recently proposed new indices for quantifying variability and structure in RR data. One of these was the 'fractional noise quantifier' (fnQ), measuring the departure of an RR time-series from a monofractal structure (i.e. a measure of its multifractality). Sixty-nine participants (aged=34·5±12·4years, body mass index (BMI)=23·9±2·9kgm(-2) , maximal oxygen uptake rate (V˙O2peak )=42·4±10·9mlmin(-1) kg(-1) , 39 males) provided continuous beat-to-beat ECG recordings for a 24-h period. Fractional differintegration was used to quantify fnQ, and heart rate variability was calculated in the time domain. All variables were evaluated during consecutive 1-h periods and also during four 6-h blocks corresponding to morning, afternoon, evening and night periods. Apart from RR, circadian trends in all variables were independent of gender (P=0·11-0·59). Apart from fnQ, all variables exhibited circadian variation (0·0005<P<0·012). Although fnQ was statistically uniform during the 24-h period, it showed a trend towards elevated values during evening and night. The main finding of this study was that fnQ was elevated by around 10% during the evening and night, although this was not statistically significant. This suggests that the structure of RR time-series in healthy individuals is most strongly 'multifractal' during evening and night periods. fnQ appears to be a plausible surrogate measure of multifractality in RR time-series.

Some Inclusion Relations Associated with Generalized Fractional Differintegral Operator

A generalized fractional differintegral operator is used to define some new subclasses of analytic functions in the open unit disk . For each of these new function classes, several inclusion relationships are established.

An application of fractional differintegration to heart rate variability time series

Fractional differintegration is used as a new tool to characterize heart rate variability time series. This paper proposes and focuses in two indexes (αc and fnQ) derived from the fractional differintegration operator. Both indexes are applied to fractional Gaussian noise (fGn) and actual RR time series in order to test their behavior. In the analysis of monofractal time series, αc is linearly related with the Hurst exponent and the estimation of the exponent by the proposed index has lower variance than by using Detrended Fluctuation Analysis (DFA) or the periodogram. The other index fnQ quantifies how the time series adjust to a monofractal time series. Age, postural changes and paced breathing cause significant changes on fnQ while αc only shows significant changes due to posture. In the analyzed actual HRV time series, αc shows good correlation with the short term scaling exponent obtained by DFA, LF/HF and RMSSD while no correlations have been found for fnQ.

Coefficient bounds for some subclasses of p-valently starlike functions

For functions of the form \[f(z) = z^{p} + \sum_{n = 1}^{\infty} a_{p + n} z^{p + n}\] we obtain sharp bounds for some coefficients functionals in certain subclasses of starlike functions. Certain applications of our main results are also given. In particular, Fekete-Szego-like inequality for classes of functions defined through extended fractional differintegrals are obtained.

Numerical evaluation of fractional differ-integrals of some periodical functions via the IMT transformation

Abstract. This paper presents accuracy evaluation of the numerical calculations of the fractional differ-integrals. We focus on applying the Riemann-Liouville formula, on singularity, which appears while using classical form of this formula. To calculate it we use the Newton-Cotes’ Quadrature and additionally two Gaussian rules. Using this different approach to the IMT Transformation, transforming the “core” integrand of Riemann-Liouville formula, we point the possible way of increasing the accuracy of the calculations. We use our own tools and compare obtained results with, where possible, exact values, where not - values obtained using an excellent method of integration incorporated in Mathematica.

Thiele’s continued fractions in digital implementation of noninteger differintegrators

A rational approximation is the preliminary step of all the indirect methods for implementing digital fractional differintegrators sν, with \({\nu \in \mathbb{R}, 0<|\nu| <1 }\) , and where \({s \in \mathbb{C}}\) . This paper employs the convergents of two Thiele’s continued fractions as rational approximations of sν. In a second step, it uses known s-to-z transformation rules to obtain a rational, stable, and minimum-phase z-transfer function, with zeros interlacing poles. The paper concludes with a comparative analysis of the quality of the proposed approximations in dependence of the used s-to-z transformations and of the sampling period.