A Numerical Algorithm of Discrete Fractional Calculus by using Inhomogeneous Sampling Data

Abstract

This paper presents an efficient numerical method to realize discrete models of fractional derivatives and integrals which imply derivatives and integrals of arbitrary real order. This approach is based on a class of Stieltjes integrals transferred from the Riemann-Liouville definition. It is to calculate on inhomogeneous sampling periods which are getting longer as the operation points go back toward the initial time. It leads to the effective quality which has low computational costs and enough accuracy. The calculation times and precision of the proposed procedure are compared with those of a conventional procedure for a practical numerical simulation and the effectiveness of this procedure is verified.

Peripheral arterial disease screening for hemodialysis patients using a fractional‐order integrator and transition probability decision‐making model

“…This index has been validated so as to identify patients with and without PAD [15, 17, 18]. A quantitative method based on discrete fractional‐order integrators [7, 20, 21] with finite computations, short‐memory requirements, and finite power series was designed to calculate the bilateral pulse AUSPs. This method could overcome the limitations of Fourier analysis and wavelet transformation methods, such as the large number of numerical computations, memory and sampling data requirements, and choices of specific frequency features for healthy subjects and PAD subjects [12, 22].…”

Detection of venous needle dislodgement during haemodialysis using fractional order shape index ratio and fuzzy colour relation analysis

“…Discrete FI: Traditional integer derivative or integral computations use integer-order differential operators with Tustin transforms and trapezoidal integration methods for geometrical interpretations. In contrast, fractional order calculus is with respect to time [16]. Thus, it is employed to describe dynamic phenomena both in the time domain and frequency domain behaviours.…”

“…Thus, it is employed to describe dynamic phenomena both in the time domain and frequency domain behaviours. For signal processing, the fractional order integral implies all non-integer and irrational numbers to deal with signals, and the summation is computed using the ratios of the gamma function, Γ(α), incorporating the number of sampling data points in the range [t 0 , t 1 ], and the fractional order parameters, α [16][17][18][19]. Therefore, there is a broad range to analyse time-varying signals, such as fluid flows and bio-signals.…”

“…To quantify the differences in TVPs between the normal condition and a VND event occurrence, TVP variation is primarily used to define the risk levels of blood loss, with the blood flows of 200-500 ml/min, >500 ml blood loss, and <2.5 min reaction time for an adult [11]. Discrete fractional order integrator (FI) with finite computations, short-memory requirement, and a finite power series [16][17][18][19] is employed to calculate the shape index ratio (SIR) for in-line signal processing, while fractional order SIR is validated to identify the risk level, referring to TVP variations. Then, a fuzzy CRA-based classifier [20,21] utilises a visual message manner to automatically detect VND during HD treatment.…”

Controlling and Synchronizing of Fractional- Order Chaotic Systems via Simple and Optimal Fractional-Order Feedback Controller

“…According to this princip le, the length of system memory can be substantially reduced in the numerical algorith m to get reliable results. So me authors propose other ideas to imp rove the capacity of storage and computation time of fractional-order systems [23][24][25].…”