What is a fractional derivative?

This paper discusses the concepts underlying the formulation of operators capable of being interpreted as fractional derivatives or fractional integrals. Two criteria for required by a fractional operator are formulated. The Grünwald–Letnikov, Riemann–Liouville and Caputo fractional derivatives and the Riesz potential are accessed in the light of the proposed criteria. A Leibniz rule is also obtained for the Riesz potential.

Efficient Legendre spectral tau algorithm for solving two-sided space-time Caputo fractional advection-dispersion equation

In this paper we present the operational matrices of the left Caputo fractional derivative, right Caputo fractional derivative and Riemann–Liouville fractional integral for shifted Legendre polynomials. We develop an accurate numerical algorithm to solve the two-sided space–time fractional advection–dispersion equation (FADE) based on a spectral shifted Legendre tau (SLT) method in combination with the derived shifted Legendre operational matrices. The fractional derivatives are described in the Caputo sense. We propose a spectral SLT method, both in temporal and spatial discretizations for the two-sided space–time FADE. This technique reduces the two-sided space–time FADE to a system of algebraic equations that simplifies the problem. Numerical results carried out to confirm the spectral accuracy and efficiency of the proposed algorithm. By selecting relatively few Legendre polynomial degrees, we are able to get very accurate approximations, demonstrating the utility of the new approach over other numerical methods.

Modeling Vegetable Fractals by means of Fractional-order Equations

This paper characterizes four ‘fractal vegetables’: (i) cauliflower (brassica oleracea var. Botrytis); (ii) broccoli (brassica oleracea var. italica); (iii) round cabbage (brassica oleracea var. capitata) and (iv) Brussels sprout (brassica oleracea var. gemmifera), by means of electrical impedance spectroscopy and fractional calculus tools. Experimental data is approximated using fractional-order models and the corresponding parameters are determined with a genetic algorithm. The Havriliak-Negami five-parameter model fits well into the data, demonstrating that classical formulae can constitute simple and reliable models to characterize biological structures.

Nonlinear Dynamics for Local Fractional Burgers Equation Arising in Fractal Flow

The local fractional Burgers’ equation (LFBE) is investigated from the point of view of local fractional conservation laws envisaging a nonlinear local fractional transport equation with a linear non-differentiable diffusion term. The local fractional derivative transformations and the LFBE conversion to a linear local fractional diffusion equation are analyzed.

A Review of Operational Matrices and Spectral Techniques for Fractional Calculus

Recently, operational matrices were adapted for solving several kinds of fractional differential equations (FDEs). The use of numerical techniques in conjunction with operational matrices of some orthogonal polynomials, for the solution of FDEs on finite and infinite intervals, produced highly accurate solutions for such equations. This article discusses spectral techniques based on operational matrices of fractional derivatives and integrals for solving several kinds of linear and nonlinear FDEs. More precisely, we present the operational matrices of fractional derivatives and integrals, for several polynomials on bounded domains, such as the Legendre, Chebyshev, Jacobi and Bernstein polynomials, and we use them with different spectral techniques for solving the aforementioned equations on bounded domains. The operational matrices of fractional derivatives and integrals are also presented for orthogonal Laguerre and modified generalized Laguerre polynomials, and their use with numerical techniques for solving FDEs on a semi-infinite interval is discussed. Several examples are presented to illustrate the numerical and theoretical properties of various spectral techniques for solving FDEs on finite and semi-infinite intervals.

An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index

The shifted Legendre orthogonal polynomials are used for the numerical solution of a new formulation for the multi-dimensional fractional optimal control problem (M-DFOCP) with a quadratic performance index. The fractional derivatives are described in the Caputo sense. The Lagrange multiplier method for the constrained extremum and the operational matrix of fractional integrals are used together with the help of the properties of the shifted Legendre orthonormal polynomials. The method reduces the M-DFOCP to a simpler problem that consists of solving a system of algebraic equations. For confirming the efficiency and accuracy of the proposed scheme, some test problems are implemented with their approximate solutions.

Integer and Fractional Order Entropy Analysis of Earthquake Data-series

This paper studies the statistical distributions of worldwide earthquakes from year 1963 up to year 2012. A Cartesian grid, dividing Earth into geographic regions, is considered. Entropy and the Jensen–Shannon divergence are used to analyze and compare real-world data. Hierarchical clustering and multi-dimensional scaling techniques are adopted for data visualization. Entropy-based indices have the advantage of leading to a single parameter expressing the relationships between the seismic data. Classical and generalized (fractional) entropy and Jensen–Shannon divergence are tested. The generalized measures lead to a clear identification of patterns embedded in the data and contribute to better understand earthquake distributions.

Fractional Dynamics in the Rayleigh's Piston

This paper studies the dynamics of the Rayleigh piston using the modeling tools of Fractional Calculus. Several numerical experiments examine the effect of distinct values of the parameters. The time responses are transformed into the Fourier domain and approximated by means of power law approximations. The description reveals characteristics usual in Fractional Brownian phenomena.

Fractional State Space Analysis of Economy Systems

This paper examines modern economic growth according to the multidimensional scaling (MDS) method and state space portrait (SSP) analysis. Electing GDP per capita as the main indicator for economic growth and prosperity, the long-run perspective from 1870 to 2010 identifies the main similarities among 34 world partners’ modern economic growth and exemplifies the historical waving mechanics of the largest world economy, the USA. MDS reveals two main clusters among the European countries and their old offshore territories, and SSP identifies the Great Depression as a mild challenge to the American global performance, when compared to the Second World War and the 2008 crisis.

Utilidade do Doppler Tecidular na Predição de Eventos Arrítmicos em Adultos com Tetralogia de Fallot Corrigida

INTRODUCTION: Adults with repaired tetralogy of Fallot (TOF) may be at risk for progressive right ventricular (RV) dilatation and dysfunction, which is commonly associated with arrhythmic events. In frequently volume-overloaded patients with congenital heart disease, tissue Doppler imaging (TDI) is particularly useful for assessing RV function. However, it is not known whether RV TDI can predict outcome in this population. OBJECTIVE: To evaluate whether RV TDI parameters are associated with supraventricular arrhythmic events in adults with repaired TOF. METHODS: We studied 40 consecutive patients with repaired TOF (mean age 35 +/- 11 years, 62% male) referred for routine echocardiographic exam between 2007 and 2008. The following echocardiographic measurements were obtained: left ventricular (LV) ejection fraction, LV end-systolic volume, LV end-diastolic volume, RV fractional area change, RV end-systolic area, RV end-diastolic area, left and right atrial volumes, mitral E and A velocities, RV myocardial performance index (Tei index), tricuspid annular plane systolic excursion (TAPSE), myocardial isovolumic acceleration (IVA), pulmonary regurgitation color flow area, TDI basal lateral, septal and RV lateral peak diastolic and systolic annular velocities (E' 1, A' 1, S' 1, E' s, A' s, S' s, E' rv, A' rv, S' rv), strain, strain rate and tissue tracking of the same segments. QRS duration on resting ECG, total duration of Bruce treadmill exercise stress test and presence of exercise-induced arrhythmias were also analyzed. The patients were subsequently divided into two groups: Group 1--12 patients with previous documented supraventricular arrhythmias (atrial tachycardia, fibrillation or flutter) and Group 2 (control group)--28 patients with no previous arrhythmic events. Univariate and multivariate analysis was used to assess the statistical association between the studied parameters and arrhythmic events. RESULTS: Patients with previous events were older (41 +/- 14 vs. 31 +/- 6 years, p = 0.005), had wider QRS (173 +/- 20 vs. 140 +/- 32 ms, p = 0.01) and lower maximum heart rate on treadmill stress testing (69 +/- 35 vs. 92 +/- 9%, p = 0.03). All patients were in NYHA class I or II. Clinical characteristics including age at corrective surgery, previous palliative surgery and residual defects did not differ significantly between the two groups. Left and right cardiac chamber dimensions and ventricular and valvular function as evaluated by conventional Doppler parameters were also not significantly different. Right ventricular strain and strain rate were similar between the groups. However, right ventricular myocardial TDI systolic (Sa: 5.4+2 vs. 8.5 +/- 3, p = 0.004) and diastolic indices and velocities (Ea, Aa, septal E/Ea, and RV free wall tissue tracking) were significantly reduced in patients with arrhythmias compared to the control group. Multivariate linear regression analysis identified RV early diastolic velocity as the sole variable independently associated with arrhythmic history (RV Ea: 4.5 +/- 1 vs. 6.7 +/- 2 cm/s, p = 0.01). A cut-off for RV Ea of < 6.1 cm/s identified patients in the arrhythmic group with 86% sensitivity and 59% specificity (AUC = 0.8). CONCLUSIONS: Our results suggest that TDI may detect RV dysfunction in patients with apparently normal function as assessed by conventional echocardiographic parameters. Reduction in RV early diastolic velocity appears to be an early abnormality and is associated with occurrence of arrhythmic events. TDI may be useful in risk stratification of patients with repaired tetralogy of Fallot.

Fractional Analysis of the DNA Information

Proceedings of the 12th Conference on 'Dynamical Systems -Theory and Applications'

Fractional Calculus: Application in modeling and control

This contribution introduces the fractional calculus (FC) fundamental mathematical aspects and discuses some of their consequences. Based on the FC concepts, the chapter reviews the main approaches for implementing fractional operators and discusses the adoption of FC in control systems. Finally are presented some applications in the areas of modeling and control, namely fractional PID, heat diffusion systems, electromagnetism, fractional electrical impedances, evolutionary algorithms, robotics, and nonlinear system control.

Bipedal Locomotion: A Fractional CPG for Generating Patterns

Proceedings of the 10th Conference on Dynamical Systems Theory and Applications

Fractional Dynamics Representation in the Pseudo Phase Plane

Proceedings of the 10th Conference on Dynamical Systems Theory and Applications

Control and Dynamics of Fractional Order Systems

Fractional Calculus (FC) goes back to the beginning of the theory of differential calculus. Nevertheless, the application of FC just emerged in the last two decades due to the progress in the area of nonlinear dynamics. This article discusses several applications of fractional calculus in science and engineering, namely: the control of heat systems, the tuning of PID controllers based on fractional calculus concepts and the dynamics in hexapod locomotion.