An introduction to the fractional continuous- time linear systems: the 21st century systems

IEEE CIRCUITS AND SYSTEMS MAGAZINE, Third Quarter

Delay approximation of fractional integrals

This paper explores the calculation of fractional integrals by means of the time delay operator. The study starts by reviewing the memory properties of fractional operators and their relationship with time delay. Based on the time response of the Mittag-Leffler function an approximation of fractional integrals consisting of time delayed samples is proposed. The tuning of the approximation is optimized by means of a genetic algorithm. The results demonstrate the feasibility of the new perspective and the limits of their application.

Integer/fractional decomposition of the impulse response of fractional linear systems

The decomposition of a fractional linear system is discussed in this paper. It is shown that it can be decomposed into an integer order part, corresponding to possible existing poles, and a fractional part. The first and second parts are responsible for the short and long memory behaviors of the system, respectively, known as characteristic of fractional systems.

Dielectric relaxation in biological materials

The study of dielectric properties concerns storage and dissipation of electric and magnetic energy in materials. Dielectrics are important in order to explain various phenomena in Solid-State Physics and in Physics of Biological Materials. Indeed, during the last two centuries, many scientists have tried to explain and model the dielectric relaxation. Starting from the Kohlrausch model and passing through the ideal Debye one, they arrived at more com- plex models that try to explain the experimentally observed distributions of relaxation times, including the classical (Cole-Cole, Davidson-Cole and Havriliak-Negami) and the more recent ones (Hilfer, Jonscher, Weron, etc.). The purpose of this thesis is to discuss a variety of models carrying out the analysis both in the frequency and in the time domain. Particular attention is devoted to the three classical models, that are studied using a transcendental function known as Mittag-Leffler function. We highlight that one of the most important properties of this function, its complete monotonicity, is an essential property for the physical acceptability and realizability of the models. Lo studio delle proprietà dielettriche riguarda l’immagazzinamento e la dissipazione di energia elettrica e magnetica nei materiali. I dielettrici sono importanti al fine di spiegare vari fenomeni nell’ambito della Fisica dello Stato Solido e della Fisica dei Materiali Biologici. Infatti, durante i due secoli passati, molti scienziati hanno tentato di spiegare e modellizzare il rilassamento dielettrico. A partire dal modello di Kohlrausch e passando attraverso quello ideale di Debye, sono giunti a modelli più complessi che tentano di spiegare la distribuzione osservata sperimentalmente di tempi di rilassamento, tra i quali modelli abbiamo quelli classici (Cole-Cole, Davidson-Cole e Havriliak-Negami) e quelli più recenti (Hilfer, Jonscher, Weron, etc.). L’obiettivo di questa tesi è discutere vari modelli, conducendo l’analisi sia nel dominio delle frequenze sia in quello dei tempi. Particolare attenzione è rivolta ai tre modelli classici, i quali sono studiati utilizzando una funzione trascendente nota come funzione di Mittag-Leffler. Evidenziamo come una delle più importanti proprietà di questa funzione, la sua completa monotonia, è una proprietà essenziale per l’accettabilità fisica e la realizzabilità dei modelli.

Well-Posedness of the Cauchy Problem for Inhomogeneous Time-Fractional Pseudo-Differential Equations

Mathematics Subject Classification: 26A33, 45K05, 35A05, 35S10, 35S15, 33E12

Time-Fractional Derivatives in Relaxation Processes: A Tutorial Survey

2000 Mathematics Subject Classification: 26A33, 33E12, 33C60, 44A10, 45K05, 74D05,

Generalized Fractional Evolution Equation

2000 Mathematics Subject Classification: Primary 46F25, 26A33; Secondary: 46G20

Fractional Integration and Fractional Differentiation of the M-Series

Mathematics Subject Classification: 26A33, 33C60, 44A15

Solution of Space-Time Fractional Schrödinger Equation Occurring in Quantum Mechanics

Dedicated to Professor A.M. Mathai on the occasion of his 75-th birthday. Mathematics Subject Classi¯cation 2010: 26A33, 44A10, 33C60, 35J10.

On Fractional Helmholtz Equations

MSC 2010: 26A33, 33E12, 33C60, 35R11

Matrix-Variate Statistical Distributions and Fractional Calculus

MSC 2010: 15A15, 15A52, 33C60, 33E12, 44A20, 62E15 Dedicated to Professor R. Gorenflo on the occasion of his 80th birthday

Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case

In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator $\Delta_+^{(\alpha,\beta,\gamma)}:= D_{x_0^+}^{1+\alpha} +D_{y_0^+}^{1+\beta} +D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$, and the fractional derivatives $D_{x_0^+}^{1+\alpha}$, $D_{y_0^+}^{1+\beta}$, $D_{z_0^+}^{1+\gamma}$ are in the Riemann-Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator $\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. Making use of the Mittag-Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions.

Eigenfunctions and fundamental solutions of the Caputo fractional Laplace and Dirac operators

In this paper, by using the method of separation of variables, we obtain eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator defined via fractional Caputo derivatives. The solutions are expressed using the Mittag-Leffler function and we show some graphical representations for some parameters. A family of fundamental solutions of the corresponding fractional Dirac operator is also obtained. Particular cases are considered in both cases.

Observability of nonlinear fractional dynamical systems

We study the observability of linear and nonlinear fractional differential systems of order 0 < α < 1 by using the Mittag-Leffler matrix function and the application of Banach’s contraction mapping theorem. Several examples illustrate the concepts.

Effects of bisoprolol fumarate on left ventricular size, function, and exercise capacity in patients with heart failure: analysis with magnetic resonance myocardial tagging.

BACKGROUND: Recent data suggest that beta-blockers can be beneficial in subgroups of patients with chronic heart failure (CHF). For metoprolol and carvedilol, an increase in ejection fraction has been shown and favorable effects on the myocardial remodeling process have been reported in some studies. We examined the effects of bisoprolol fumarate on exercise capacity and left ventricular volume with magnetic resonance imaging (MRI) and applied a novel high-resolution MRI tagging technique to determine myocardial rotation and relaxation velocity. METHODS: Twenty-eight patients (mean age, 57 +/- 11 years; mean ejection fraction, 26 +/- 6%) were randomized to bisoprolol fumarate (n = 13) or to placebo therapy (n = 15). The dosage of the drugs was titrated to match that of the the Cardiac Insufficiency Bisoprolol Study protocol. Hemodynamic and gas exchange responses to exercise, MRI measurements of left ventricular end-systolic and end-diastolic volumes and ejection fraction, and left ventricular rotation and relaxation velocities were measured before the administration of the drug and 6 and 12 months later. RESULTS: After 1 year, heart rate was reduced in the bisoprolol fumarate group both at rest (81 +/- 12 before therapy versus 61 +/- 11 after therapy; P <.01) and peak exercise (144 +/- 20 before therapy versus 127 +/- 17 after therapy; P <.01), which indicated a reduction in sympathetic drive. No differences were observed in heart rate responses in the placebo group. No differences were observed within or between groups in peak oxygen uptake, although work rate achieved was higher (117.9 +/- 36 watts versus 146.1 +/- 33 watts; P <.05) and exercise time tended to be higher (9.1 +/- 1.7 minutes versus 11.4 +/- 2.8 minutes; P =.06) in the bisoprolol fumarate group. A trend for a reduction in left ventricular end-diastolic volume (-54 mL) and left ventricular end-systolic volume (-62 mL) in the bisoprolol fumarate group occurred after 1 year. Ejection fraction was higher in the bisoprolol fumarate group (25.0 +/- 7 versus 36.2 +/- 9%; P <.05), and the placebo group remained unchanged. Most changes in volume and ejection fraction occurred during the latter 6 months of treatment. With myocardial tagging, insignificant reductions in left ventricular rotation velocity were observed in both groups, whereas relaxation velocity was reduced only after bisoprolol fumarate therapy (by 39%; P <.05). CONCLUSION: One year of bisoprolol fumarate therapy resulted in an improvement in exercise capacity, showed trends for reductions in end-diastolic and end-systolic volumes, increased ejection fraction, and significantly reduced relaxation velocity. Although these results generally confirm the beneficial effects of beta-blockade in patients with chronic heart failure, they show differential effects on systolic and diastolic function.