Modeling some real phenomena by fractional differential equations

This paper deals with fractional differential equations, with dependence on a Caputo fractional derivative of real order. The goal is to show, based on concrete examples and experimental data from several experiments, that fractional differential equations may model more efficiently certain problems than ordinary differential equations. A numerical optimization approach based on least squares approximation is used to determine the order of the fractional operator that better describes real data, as well as other related parameters.

Finite time extinction for nonlinear fractional evolution equations and related properties

The finite time extinction phenomenon (the solution reaches an equilibrium after a finite time) is peculiar to certain nonlinear problems whose solutions exhibit an asymptotic behavior entirely different from the typical behavior of solutions associated to linear problems. The main goal of this work is twofold. Firstly, we extend some of the results known in the literature to the case in which the ordinary time derivative is considered jointly with a fractional time differentiation. Secondly, we consider the limit case when only the fractional derivative remains. The latter is the most extraordinary case, since we prove that the finite time extinction phenomenon still appears, even with a non-smooth profile near the extinction time. Some concrete examples of quasi-linear partial differential operators are proposed. Our results can also be applied in the framework of suitable nonlinear Volterra integro-differential equations.

Finite time extinction for nonlinear fractional evolution equations and related properties.

The finite time extinction phenomenon (the solution reaches an equilibrium after a finite time) is peculiar to certain nonlinear problems whose solutions exhibit an asymptotic behavior entirely different from the typical behavior of solutions associated to linear problems. The main goal of this work is twofold. Firstly, we extend some of the results known in the literature to the case in which the ordinary time derivative is considered jointly with a fractional time differentiation. Secondly, we consider the limit case when only the fractional derivative remains. The latter is the most extraordinary case, since we prove that the finite time extinction phenomenon still appears, even with a non-smooth profile near the extinction time. Some concrete examples of quasi-linear partial differential operators are proposed. Our results can also be applied in the framework of suitable nonlinear Volterra integro-differential equations.

Truncation errors in finite difference models for solute transport equation with first-order reaction

The truncation errors associated with finite difference solutions of the advection-dispersion equation with first-order reaction are formulated from a Taylor analysis. The error expressions are based on a general form of the corresponding difference equation and a temporally and spatially weighted parametric approach is used for differentiating among the various finite difference schemes. The numerical truncation errors are defined using Peclet and Courant numbers and a new Sink/Source dimensionless number. It is shown that all of the finite difference schemes suffer from truncation errors. Tn particular it is shown that the Crank-Nicolson approximation scheme does not have second order accuracy for this case. The effects of these truncation errors on the solution of an advection-dispersion equation with a first order reaction term are demonstrated by comparison with an analytical solution. The results show that these errors are not negligible and that correcting the finite difference scheme for them results in a more accurate solution. (C) 1999 Elsevier Science B.V. All rights reserved.

Determination of pesticides in fruit and fruit juices by chromatographic methods. An overview

In order to combat a variety of pests, pesticides are widely used in fruits. Several extraction procedures (liquid extraction, single drop microextraction, microwave-assisted extraction, pressurized liquid extraction, supercritical fluid extraction, solid-phase extraction, solid-phase microextraction, matrix solid-phase dispersion, and stir bar sorptive extraction) have been reported to determine pesticide residues in fruits and fruit juices. The significant change in recent years is the introduction of the Quick, Easy, Cheap, Effective, Rugged, and Safe (QuEChERS) methods in these matrices analysis. A combination of techniques reported the use of new extraction methods and chromatography to provide better quantitative recoveries at low levels. The use of mass spectrometric detectors in combination with liquid and gas chromatography has played a vital role to solve many problems related to food safety. The main attention in this review is on the achievements that have been possible because of the progress in extraction methods and the latest advances and novelties in mass spectrometry, and how these progresses have influenced the best control of food, allowing for an increase in the food safety and quality standards.

Complex order biped rhythms

Animal locomotion is a complex process, involving the central pattern generators (neural networks, located in the spinal cord, that produce rhythmic patterns), the brainstem command systems, the steering and posture control systems and the top layer structures that decide which motor primitive is activated at a given time. Pinto and Golubitsky studied an integer CPG model for legs rhythms in bipeds. It is a four-coupled identical oscillators' network with dihedral symmetry. This paper considers a new complex order central pattern generator (CPG) model for locomotion in bipeds. A complex derivative Dα±jβ, with α, β ∈ ℜ+, j = √-1, is a generalization of the concept of an integer derivative, where α = 1, β = 0. Parameter regions where periodic solutions, identified with legs' rhythms in bipeds, occur, are analyzed. Also observed is the variation of the amplitude and period of periodic solutions with the complex order derivative.

A signal processing view of differintegration

Proceedings of the European Control Conference, ECC’01, Porto, Portugal, September 2001

New findings on the dynamics of HIV and TB coinfection models

In this paper we study a model for HIV and TB coinfection. We consider the integer order and the fractional order versions of the model. Let α∈[0.78,1.0] be the order of the fractional derivative, then the integer order model is obtained for α=1.0. The model includes vertical transmission for HIV and treatment for both diseases. We compute the reproduction number of the integer order model and HIV and TB submodels, and the stability of the disease free equilibrium. We sketch the bifurcation diagrams of the integer order model, for variation of the average number of sexual partners per person and per unit time, and the tuberculosis transmission rate. We analyze numerical results of the fractional order model for different values of α, including α=1. The results show distinct types of transients, for variation of α. Moreover, we speculate, from observation of the numerical results, that the order of the fractional derivative may behave as a bifurcation parameter for the model. We conclude that the dynamics of the integer and the fractional order versions of the model are very rich and that together these versions may provide a better understanding of the dynamics of HIV and TB coinfection.

Simple Kinematic Design for Evading the Forced Oscillation of a Car-Wheel Suspension System

An adaptive control damping the forced vibration of a car while passing along a bumpy road is investigated. It is based on a simple kinematic description of the desired behavior of the damped system. A modified PID controller containing an approximation of Caputo’s fractional derivative suppresses the high-frequency components related to the bumps and dips, while the low frequency part of passing hills/valleys are strictly traced. Neither a complete dynamic model of the car nor ’a priori’ information on the surface of the road is needed. The adaptive control realizes this kinematic design in spite of the existence of dynamically coupled, excitable internal degrees of freedom. The method is investigated via Scicos-based simulation in the case of a paradigm. It was found that both adaptivity and fractional order derivatives are essential parts of the control that can keep the vibration of the load at bay without directly controlling its motion.

Materials nanocomposits a partir de whiskers de cel•lulosa i matriu polimèrica de cautxú

De acuerdo con los objetivos generales del proyecto y plan de trabajo previsto, para esta anualidad, se obtuvieron fibras y microfibras de celulosa a partir de dos fuentes: celulosa vegetal de pino y eucalipto y celulosa bacterial. Las microfibrillas han sido utilizadas como material de refuerzo para la fabricación de materiales compuestos a partir de caucho natural, policaprolactona y polivinil alcohol. Las muestras se fabricaron mediante la técnica de "casting" en medio acuoso y temperatura ambiente. Las muestras fueron caracterizados en sus propiedades mecánicas, físicas y térmicas. Se observó que, en general, la adición de las microfibrillas de celulosa en las matrices poliméricas provoca una mejora sustancial en las propiedades mecánicas del material en comparación con el polímero sin reforzar. Los resultados pueden resumirse de la siguiente manera: 1.Fabricación de materiales compuestos a base de caucho natural y fibras de celulosa. Se obtuvieron fibras y nanofibras de celulosa que fueron modificadas químicamente y usadas como refuerzo en matriz de caucho. Los resultados mostraron mejora de propiedades mecánicas del material, principalmente en los materiales compuestos reforzados con nanofibras. 2. Obtención de whiskers de celulosa y su utilización como material de refuerzo en una matriz de policaprolactona. Se obtuvieron whiskers de celulosa a partir de pasta blanqueada. La adición en una matriz de policaprolactona produjo materiales compuestos con propiedades mecánicas superiores a la matriz, con buena dispersión de los whiskers. 3. Obtención de fibras de celulosa bacterial y nanofibras de celulosa, aislamiento y utilización sobre una matriz de polivinil alcohol. Se obtuvo celulosa bacterial a partir de la bacteria Gluconacetobacter xylinum. Además se fabricaron nanofibras de celulosa a partir eucalipto blanqueado. La celulosa bacterial como material de refuerzo no produjo importantes mejoras en las propiedades mecánicas de la matriz; en cambio se observaron mejoras destacables con la nanofibra como refuerzo.

Aplicação da técnica de dispersão da matriz em fase sólida (DMFS) na análise de pesticidas em quiabo por CG-EM

A matrix solid phase dispersion and gas chromatography-mass selective detection method for the simultaneous determination of monocrotophos, methyl parathion, cypermethrin and deltamethrin in okra is described. Analyses of 2 g of fortified okra (0.05-0.75 mg kg-1) showed an average recovery of 96.2% (71.4-128.4%) and average relative standard deviation of 11.7% (1.4-37.1%). The cypermethrin recovery at the lower level was above 130%. The limit of detection ranged from 0.02 to 0.15 mg kg-1. The procedure was applied to the okra samples and has found 0.56 mg kg-1 of cypermethrin-cis, 0.75 mg kg-1 of cypermethrin-trans and 2.71 mg kg-1 of deltamethrin.

Minimization of the experimental workload for the prediction of pollutants propagation in rivers. Mathematical modelling and knowledge re-use

The present thesis in focused on the minimization of experimental efforts for the prediction of pollutant propagation in rivers by mathematical modelling and knowledge re-use. Mathematical modelling is based on the well known advection-dispersion equation, while the knowledge re-use approach employs the methods of case based reasoning, graphical analysis and text mining. The thesis contribution to the pollutant transport research field consists of: (1) analytical and numerical models for pollutant transport prediction; (2) two novel techniques which enable the use of variable parameters along rivers in analytical models; (3) models for the estimation of pollutant transport characteristic parameters (velocity, dispersion coefficient and nutrient transformation rates) as functions of water flow, channel characteristics and/or seasonality; (4) the graphical analysis method to be used for the identification of pollution sources along rivers; (5) a case based reasoning tool for the identification of crucial information related to the pollutant transport modelling; (6) and the application of a software tool for the reuse of information during pollutants transport modelling research. These support tools are applicable in the water quality research field and in practice as well, as they can be involved in multiple activities. The models are capable of predicting pollutant propagation along rivers in case of both ordinary pollution and accidents. They can also be applied for other similar rivers in modelling of pollutant transport in rivers with low availability of experimental data concerning concentration. This is because models for parameter estimation developed in the present thesis enable the calculation of transport characteristic parameters as functions of river hydraulic parameters and/or seasonality. The similarity between rivers is assessed using case based reasoning tools, and additional necessary information can be identified by using the software for the information reuse. Such systems represent support for users and open up possibilities for new modelling methods, monitoring facilities and for better river water quality management tools. They are useful also for the estimation of environmental impact of possible technological changes and can be applied in the pre-design stage or/and in the practical use of processes as well.

Análise de resíduos de agrotóxicos em tomates empregando dispersão da matriz em fase sólida (DMFS) e cromatografia gasosa

This work was aimed on optimization of the matrix solid-phase dispersion (MSPD) technique using gas chromatography for analyzing residues of chlorpyriphos, λ-cyhalothrin, cypermethrin and deltamethrin in tomatoes. The results showed that silica was more efficient for the clean up of extracts, but florisil provided the highest recovery rates. A 2³ complete factorial design was carried out to evaluate the absorbent/sample ratio, presence of co-column (silica) and ultrasonic bath on the extraction rate. The percentage of extraction of the pesticides chlorpyriphos, λ-cyhalothrin, cypermethrin and deltamethrin were 64.7, 88.3, 99.2 and 89.2%, respectively, with relative standard deviations below 5%.

Determinação de resíduos de pesticidas em plasma bovino por cromatografia gasosa-espectrometria de massas

An analytical method for the isolation based on matrix solid-phase dispersion technique and gas chromatographic determination of pesticides in cattle plasma is presented. It was fortified 0.25 g of plasma with pesticides and blended with 1 g each C18 and Na2SO4. The homogenized matter was transferred to a SPE cartridge, which contained 1 g of activated florisil with 5 mL acetonitrile. The analites were eluted under vaccum with 15 mL acetonitrile, the extract was analyzed by gas chromatography-mass spectrometry. The limit of quantification of the method was 0.04 mg L-1 for chlorphenvinfos and fipronil and 0.02 mg L-1 for cypermethrin..

MSPD procedure combined with GC-MS for the determination of procymidone, bifenthrin, malathion and pirimicarb in honey

A method based on matrix solid-phase dispersion and gas chromatography-mass spectrometry to determine procymidone, malathion, bifenthrin and pirimicarb in honey is described. The best results were obtained using 1.0 g of honey, 1.0 g of silica-gel as dispersant sorbent and acetonitrile as eluting solvent. The method was validated by fortified honey samples at three concentration levels (0.2, 0.5 to 1.0 mg kg-1). Average recoveries (n=7) ranged from 54 to 84%, with relative standard deviations between 3.7 and 8.5%. Detection and quantification limits attained by the developed method ranged from 0.02 to 0.08 mg kg-1 and 0.07 to 0.25 mg kg-1 for the honey, respectively.