A coherent approach to non-integer order derivatives

Signal Processing, vol. 86, nº 10

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.

On Multidimensional Analogue of Marchaud Formula for Fractional Riesz-Type Derivatives in Domains in R^n

2000 Mathematics Subject Classification: 26A33, 42B20

Nonexistence Results of Solutions of Semilinear Differential Inequalities with Temperal Fractional Derivative on the Heinsenberg Group

2000 Mathematics Subject Classification: 26A33, 33C60, 44A15, 35K55

Piperamides and their derivatives as potential anti-trypanosomal agents

We describe herein an evaluation of the trypanocidal effect of eight piperamides (1-8) isolated from Piper tuberculatum bearing dihydropyridone, piperidine, and isobutyl moieties against epimastigote forms of Trypanosoma cruzi, the causative agent of Chagas` disease. Based on such results, three hydrogenated and two hydrolyzed derivatives (10-14) were prepared and evaluated as well. The dihydropyridone amides (1-3) displayed higher anti-trypanosomal activity. The (Z)-piplartine (1) showed higher activity with a 50% inhibition concentration (IC(50)) value of 10.5 mu M, almost four times more potent than the positive control, benznidazole (IC(50) = 42.7 mu M), and should be further evaluated as a suitable hit for the design of new antiprotozoal agents.

Synthesis, NMR studies and conformational analysis of oxazolidine derivatives of the beta-adrenoreceptor antagonists metoprolol, atenolol and timolol

Formaldehyde-derived oxazolidine derivatives 4-7 of the beta-adrenoreceptor antagonists metoprolol 1, atenolol 2 and timolol 3 have been synthesised. Conformational analysis of 1-3 and the oxazolidine derivatives 4-7 has been performed using H-1 NMR spectroscopy and computational methods. The H-1 NMR studies show that for the aryloxypropanolamine beta-adrenoreceptor antagonists there is a predominance of the conformer in which the amine group is approximately antiperiplanar or trans to the aryloxymethylene group. Both H-1 NMR data and theoretical studies indicate that the oxazolidine derivatives 4-7 and the aryloxypropanolamine beta-adrenoreceptor antagonists 1-3 adopt similar conformations around the beta-amino alcohol moiety. Thus, oxazolidine ring formation does not dramatically alter the preferred conformation adopted by the beta-amino alcohol moiety of 1-3. Oxazolidine derivatives of aryloxypropanolamine beta-adrenoreceptor antagonists may therefore be appropriate as prodrugs, or semi-rigid analogues, when greater lipophilicity is required for drug delivery.

Effects of Caffeoylquinic Acid Derivatives and C-Flavonoid from Lychnophora ericoides on in vitro Inflammatory Mediator Production

In this study we aimed at evaluating the effect of the major polar constituents of the medicinal plant Lychnophora ericoides on the production of inflammatory mediators produced by LPS-stimulated U-937 cells. The 6,8-di-C-beta-glucosylapigenin (vicenin-2) presented no effect on tumor necrosis factor (TNF)-alpha production, but inhibited, in a dose-dependent manner, the production of prostaglandin (PG) E(2) without altering the expression of cyclooxygenase (COX) -2 protein. 3,5-Dicaffeoylquinic acid and 4,5-dicaffeoylquinic acid, at lower concentrations, had small but significant effects on reducing PG E, levels; at higher doses these compounds stimulated PGE(2) and also TNF-alpha production by the cells. All the caffeoylquinic acid derivatives, in a dose-dependent fashion, were able to inhibit monocyte chemoattractant protein-3 synthesis/release, with 4,5-DCQ being the most potent at the highest tested concentration. These results add important information on the effects of plant natural polyphenols, namely vicenin-2 and caffeoylquinic acid derivatives, on the production of inflammatory mediators by cultured cells.

Xanthenedione derivatives, new promising antioxidant and acetylcholinesterase inhibitor agents

Natural and synthetic xanthone derivatives are well-known for their ability to act as antioxidants and/or enzyme inhibitors. This paper aims to present a successful synthetic methodology towards xanthenedione derivatives and the study of their aromatization to xanthones. Additionally their ability to reduce Fe(III), to scavenge DPPH radicals and to inhibit AChE was evaluated. The results demonstrated that xanthenedione derivative 5e, bearing a catechol unit, showed higher reduction capacity than BHT and similar to quercetin, strong DPPH scavenging activity (EC50 = 3.79 ± 0.06 μM) and it was also showed to be a potent AChEI (IC50 = 31.0 ± 0.09 μM) when compared to galantamine (IC50 = 211.8 ± 9.5 μM).

An introduction to the fractional continuous-time linear systems

A brief introduction to the fractional continuous-time linear systems is presented. It will be done without needing a deep study of the fractional derivatives. We will show that the computation of the impulse and step responses is very similar to the classic. The main difference lies in the substitution of the exponential by the Mittag-Leffler function. We will present also the main formulae defining the fractional derivatives.

Optimal tuning of fractional controllers using genetic algorithms

This study addresses the optimization of fractional algorithms for the discrete-time control of linear and non-linear systems. The paper starts by analyzing the fundamentals of fractional control systems and genetic algorithms. In a second phase the paper evaluates the problem in an optimization perspective. The results demonstrate the feasibility of the evolutionary strategy and the adaptability to distinct types of systems.

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

IEEE CIRCUITS AND SYSTEMS MAGAZINE, Third Quarter

Microencapsulation of herbicide MCPA with native beta-cyclodextrin and its methyl and hydroxypropyl derivatives: An experimental and theoretical investigation

When a pesticide is released into the environment, most of it is lost before it reaches its target. An effective way to reduce environmental losses of pesticides is by using controlled release technology. Microencapsulation becomes a promising technique for the production of controlled release agricultural formulations. In this work, the microencapsulation of chlorophenoxy herbicide MCPA with native b-cyclodextrin and its methyl and hydroxypropyl derivatives was investigated. The phase solubility study showed that both native and b-CD derivatives increased the water solubility of the herbicide and inclusion complexes are formed in a stoichiometric ratio of 1:1. The stability constants describing the extent of formation of the complexes have been determined by phase solubility studies. 1H NMR experiments were also accomplished for the prepared solid systems and the data gathered confirm the formation of the inclusion complexes. 1H NMR data obtained for the MCPA/CDs complexes disclosed noticeable proton shift displacements for OCH2 group and H6 aromatic proton of MCPA provided clear evidence of inclusion complexation process, suggesting that the phenyl moiety of the herbicide was included in the hydrophobic cavity of CDs. Free energy molecular mechanics calculations confirm all these findings. The gathered results can be regarded as an essential step to the development of controlled release agricultural formulations containing herbicide MCPA.

Local Fractional Variational Iteration Method for Local Fractional Poisson Equations in Two Independent Variables

The local fractional Poisson equations in two independent variables that appear in mathematical physics involving the local fractional derivatives are investigated in this paper. The approximate solutions with the nondifferentiable functions are obtained by using the local fractional variational iteration method.

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.