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.

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.

Design of Digital Fractional-Order Integrators and Differentiators by Least Squares

First IFAC Workshop on Fractional Differentiation and Its Application - 19-21 July 2004, Enseirb, Bordeaux, France - FDA'04

Fractional Dynamics and Control of Distributed Parameter 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 chaos that revealed subtle relationships with the FC concepts. In the field of dynamical systems theory some work has been carried out but the proposed models and algorithms are still in a preliminary stage of establishment. Having these ideas in mind, the paper discusses a FC perspective in the study of the dynamics and control of some distributed parameter systems.

System Modeling and Control Through Fractional-Order Algorithms

The theory of fractional calculus goes back to the beginning of thr throry of differential calculus but its inherent complexity postponed the applications of the associated concepts. In the last decade the progress in the areas of chaos and fractals revealed subtle relationships with the fractional calculus leading to an increasing interest in the development of the new paradigm. In the area of automaticcontrol preliminary work has already been carried out but the proposed algorithms are restricted to the frequency domain. The paper discusses the design of fractional-order discrete-time controllers. The algorithms studied adopt the time domein, which makes them suited for z-transform analusis and discrete-time implementation. The performance of discrete-time fractional-order controllers with linear and non-linear systems is also investigated.

Quinoxaline, its derivatives and applications: a state of the art review

Quinoxaline derivatives are an important class of heterocycle compounds, where N replaces some carbon atoms in the ring of naphthalene. Its molecular formula is C8H6N2, formed by the fusion of two aromatic rings, benzene and pyrazine. It is rare in natural state, but their synthesis is easy to perform. In this review the State of the Art will be presented, which includes a summary of the progress made over the past years in the knowledge of the structure and mechanism of the quinoxaline and quinoxaline derivatives, associated medical and biomedical value as well as industrial value. Modifying quinoxaline structure it is possible to obtain a wide variety of biomedical applications, namely antimicrobial activities and chronic and metabolic diseases treatment.

Analysis of Natural and Artificial Phenomena Using Signal Processing and Fractional Calculus

In this paper we study several natural and man-made complex phenomena in the perspective of dynamical systems. For each class of phenomena, the system outputs are time-series records obtained in identical conditions. The time-series are viewed as manifestations of the system behavior and are processed for analyzing the system dynamics. First, we use the Fourier transform to process the data and we approximate the amplitude spectra by means of power law functions. We interpret the power law parameters as a phenomenological signature of the system dynamics. Second, we adopt the techniques of non-hierarchical clustering and multidimensional scaling to visualize hidden relationships between the complex phenomena. Third, we propose a vector field based analogy to interpret the patterns unveiled by the PL parameters.

ANTHELMINTIC ACTIVITY OF LAPACHOL, β-LAPACHONE AND ITS DERIVATIVES AGAINST Toxocara canis LARVAE

Anthelmintics used for intestinal helminthiasis treatment are generally effective; however, their effectiveness in tissue parasitosis (i.e. visceral toxocariasis) is moderate. The aim of this study was to evaluate the in vitroactivity of lapachol, β-lapachone and phenazines in relation to the viability of Toxocara canis larvae. A concentration of 2 mg/mL (in duplicate) of the compounds was tested using microculture plates containing Toxocara canis larvae in an RPMI-1640 environment, incubated at 37 °C in 5% CO2 tension for 48 hours. In the 2 mg/mL concentration, four phenazines, lapachol and three of its derivatives presented a larvicide/larvistatic activity of 100%. Then, the minimum larvicide/larvistatic concentration (MLC) test was conducted. The compounds that presented the best results were nor-lapachol (MLC, 1 mg/mL), lapachol (MLC 0.5 mg/mL), β-lapachone, and β-C-allyl-lawsone (MLC, 0.25 mg/mL). The larvae exposed to the compounds, at best MLC with 100% in vitro activity larvicide, were inoculated into healthy BALB/c mice and were not capable of causing infection, confirming the larvicide potential in vitro of these compounds.

Fractional bioheat equation

In this work we develop a new mathematical model for the Pennes’ bioheat equation assuming a fractional time derivative of single order. A numerical method for the solu- tion of such equations is proposed, and, the suitability of the new model for modelling real physical problems is studied and discussed

Comparison of different numerical methods for the solution of the time-fractional reaction-diffusion equation with variable diffusion coefficient

In this work we perform a comparison of two different numerical schemes for the solution of the time-fractional diffusion equation with variable diffusion coefficient and a nonlinear source term. The two methods are the implicit numerical scheme presented in [M.L. Morgado, M. Rebelo, Numerical approximation of distributed order reaction- diffusion equations, Journal of Computational and Applied Mathematics 275 (2015) 216-227] that is adapted to our type of equation, and a colocation method where Chebyshev polynomials are used to reduce the fractional differential equation to a system of ordinary differential equations

A new and efficient synthesis of benzoimidazopurine derivatives

[Excerpt] Purine nucleobases are essential biomolecules in living organisms. Playing several key roles in the cell, they have been a significant inspiration for drug design.1 Benzimidazole nucleus is an important pharmacophore in the development of molecules with pharmaceutical or biological interest. Benzimidazoles have been reported to display significant pharmacological activities such as antiulcer, antifungal, antiparkinson, anticancer and antibiotic.2 Fused structures incorporating these two scaffolds might be important for medicinal chemistry and, to the best of our knowledge, there are no reports of these systems in the literature. In particular, benzo[4,5]imidazo[2,1]purines seem to be novel and must be important target molecules in the heterocyclic synthesis. (...)

The contribution of phenolic acids to the anti-inflammatory activity of mushrooms: Screening in phenolic extracts, individual parent molecules and synthesized glucuronated and methylated derivatives

In the present study, the ethanolic extracts of fourteen edible mushrooms were investigated for their anti-inflammatory potential in LPS (lipopolysaccharide) activated RAW 264.7 macrophages. Furthermore the extracts were chemically characterized in terms of phenolic acids and related compounds. The identified molecules (p-hydroxybenzoic, p-coumaric and cinnamic acids) and their glucuronated and methylated derivatives obtained by chemical synthesis were also evaluated for the same bioactivity, in order to establish structure-activity relationships and to comprehend the effects of in vivo metabolism reactions in the activity of the compounds. The extracts of Pleurotus ostreatus, Macrolepiota procera, Boletus impolitus and Agaricus bisporus revealed the strongest anti-inflammatory potential (EC50 values 96 ± 1 to 190 ± 6 µg/mL, and also the highest concentration of cinnamic acid (656 to 156 µg/g), which was also the individual compound with the highest anti-inflammatory activity. The derivatives of p-coumaric acid revealed the strongest properties, specially the derivative methylated in the carboxylic group (CoA-M1) that exhibited similar activity to the one showed by dexamethaxone used as anti-inflammatory standard; by contrast, the derivatives of p-hydroxybenzoic revealed the lowest inhibition of NO production. All in all, whereas the conjugation reactions change the chemical structure of phenolic acids and may increase or decrease their activity, the glucuronated and methylated derivatives of the studied compounds are still displaying anti-inflammatory activity.

On the boundedness on inhomogeneous Lipschitz spaces of fractional integrals, singular integrals and hypersingular integrals associated to non-doubling measures on metric spaces

In this paper we prove T1 type necessary and sufficient conditions for the boundedness on inhomogeneous Lipschitz spaces of fractional integrals and singular integrals defined on a measure metric space whose measure satisfies a n-dimensional growth. We also show that hypersingular integrals are bounded on these spaces.

HSU-Robbins and Spitzer's theorems for the variations of fractional Brownian motion

Using recent results on the behavior of multiple Wiener-Itô integrals based on Stein's method, we prove Hsu-Robbins and Spitzer's theorems for sequences of correlated random variables related to the increments of the fractional Brownian motion.

Fractional hepatic de novo lipogenesis in healthy subjects during near-continuous oral nutrition and bed rest: a comparison with published data in artificially fed, critically ill patients.

BACKGROUND AND AIMS: In critically ill patients, fractional hepatic de novo lipogenesis increases in proportion to carbohydrate administration during isoenergetic nutrition. In this study, we sought to determine whether this increase may be the consequence of continuous enteral nutrition and bed rest. We, therefore, measured fractional hepatic de novo lipogenesis in a group of 12 healthy subjects during near-continuous oral feeding (hourly isoenergetic meals with a liquid formula containing 55% carbohydrate). In eight subjects, near-continuous enteral nutrition and bed rest were applied over a 10 h period. In the other four subjects, it was extended to 34 h. Fractional hepatic de novo lipogenesis was measured by infusing(13) C-labeled acetate and monitoring VLDL-(13)C palmitate enrichment with mass isotopomer distribution analysis. Fractional hepatic de novo lipogenesis was 3.2% (range 1.5-7.5%) in the eight subjects after 10 h of near continuous nutrition and 1.6% (range 1.3-2.0%) in the four subjects after 34 h of near-continuous nutrition and bed rest. This indicates that continuous nutrition and physical inactivity do not increase hepatic de novo lipogenesis. Fractional hepatic de novo lipogenesis previously reported in critically ill patients under similar nutritional conditions (9.3%) (range 5.3-15.8%) was markedly higher than in healthy subjects (P<0.001). These data from healthy subjects indicate that fractional hepatic de novo lipogenesis is increased in critically ill patients.