Fractional Differential Equations

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion.

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Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking.

A new difference scheme for the time fractional diffusion equation

Fractional derivative has a history as long as that of classical calculus, but it is much less popular than it should be. What is the physical meaning of fractional derivative? This is still an open problem. In modeling various memory phenomena, we observe that a memory process usually consists of two stages. One is short with permanent retention and the other is governed by a simple model of fractional derivative. With the numerical least square method, we show that the fractional model perfectly fits the test data of memory phenomena in different disciplines, not only in mechanics, but also in biology and psychology. Based on this model, we find that a physical meaning of the fractional order is an index of memory.

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Measuring memory with the order of fractional derivative

This monograph covers some selected problems of positive and fractional electrical circuits composed of resistors, coils, capacitors and voltage (current) sources. The book consists of 8 chapters, 4 appendices and a list of references. Chapter 1 is devoted to fractional standard and positive continuous-time and discrete-time linear systems without and with delays. In chapter 2 the standard and positive fractional electrical circuits are considered and the fractional electrical circuits in transient states are analyzed. Descriptor linear electrical circuits and their properties are investigated in chapter 3, while chapter 4 is devoted to the stability of fractional standard and positive linear electrical circuits. The reachability, observability and reconstructability of fractional positive electrical circuits and their decoupling zeros are analyzed in chapter 5. The fractional linear electrical circuits with feedbacks are considered in chapter 6. In chapter 7 solutions of minimum energy control for standard and fractional systems with and without bounded inputs is presented. In chapter 8 the fractional continuous-time 2D linear systems described by the Roesser type models are investigated.

Fractional Linear Systems and Electrical Circuits

A new class of fractional 2D linear discrete-time systems is introduced. The fractional difference definition is applied to each dimension of a 2D Roesser model. Solutions of these systems are derived using a 2D Z-transform. The classical Cayley-Hamilton theorem is extended to 2D fractional systems described by the Roesser model. Necessary and sufficient conditions for the positivity and stabilization by the state-feedback of fractional 2D linear systems are established. A procedure for the computation of a gain matrix is proposed and illustrated by a numerical example.

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Positivity and stabilization of fractional 2D linear systems described by the Roesser model

The paper presents an analysis of the fractional electrical circuit in the transient state described by the fractional-order state-space equations. General solutions to the fractional state-space equations containing two types of definitions of fractional derivative: Caputo fractional order derivative and the Conformable Fractional Derivative (CFD) definitions. The solutions are given in three cases: (1) nonzero initial conditions and zero input signal, (2) the input signal in the form of a step function and zero initial conditions; (3) the general case where the input and initial conditions are nonzero. The results for voltages across the capacitors for derivatives of fractional orders equal \(\alpha =0.7; 0.8; 0.9\) are shown. The comparison of those two definitions is presented and discussed.

Analysis of Fractional Electrical Circuit Using Caputo and Conformable Derivative Definitions

In recent years, there has been an increased interest in the old NACA four-digit series when designing wind turbines or small aircraft. One of the airfoils frequently used for this purpose is the NACA 0018 profile. However, since 1933, for over 70 years, almost no new experimental studies of this profile have been carried out to investigate its performance in the regime of small and medium Reynolds numbers as well as for various turbulence parameters. This paper discusses the effect of the Reynolds number and the turbulence intensity on the lift and drag coefficients of the NACA 0018 airfoil under the low Reynolds number regime. The research was carried out for the range of Reynolds numbers from 50,000 to 200,000 and for the range of turbulence intensity on the airfoil from 0.01% to 0.5%. Moreover, the tests were carried out for the range of angles of attack from 0 to 10 degrees. The uncalibrated γ−Reθ transition turbulence model was used for the analysis. Our research has shown that airfoil performance is largely dependent on the Reynolds number and less on the turbulence intensity. For this range of Reynolds numbers, the characteristic of the lift coefficient is not linear and cannot be analyzed using a single aerodynamic derivative as for large Reynolds numbers. The largest differences in both aerodynamic coefficients are observed for the Reynolds number of 50,000.

https://www.mdpi.com/2227-9717/10/5/1004/pdf?version=1653635714

Numerical Study of the Effect of the Reynolds Number and the Turbulence Intensity on the Performance of the NACA 0018 Airfoil at the Low Reynolds Number Regime

Krzysztof Rogowski

Papers

Fractional Differential Equations

Fractional Linear Systems and Electrical Circuits

Positivity and stabilization of fractional 2D linear systems described by the Roesser model

Analysis of Fractional Electrical Circuit Using Caputo and Conformable Derivative Definitions

Numerical Study of the Effect of the Reynolds Number and the Turbulence Intensity on the Performance of the NACA 0018 Airfoil at the Low Reynolds Number Regime