Multiple Roots of Systems of Equations by Repulsion Merit Functions

In this paper we address the problem of computing multiple roots of a system of nonlinear equations through the global optimization of an appropriate merit function. The search procedure for a global minimizer of the merit function is carried out by a metaheuristic, known as harmony search, which does not require any derivative information. The multiple roots of the system are sequentially determined along several iterations of a single run, where the merit function is accordingly modified by penalty terms that aim to create repulsion areas around previously computed minimizers. A repulsion algorithm based on a multiplicative kind penalty function is proposed. Preliminary numerical experiments with a benchmark set of problems show the effectiveness of the proposed method.

Riesz potential versus fractional Laplacian

This paper starts by introducing the Grünwald–Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy–Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian.

New and low cost plastic membrane electrode with low detection limits for sulfadimethoxine determination in aquaculture waters

Sulfadimethoxine (SDM) is one of the drugs, often used in the aquaculture sector to prevent the spread of disease in freshwater fish aquaculture. Its spread through the soil and surface water can contribute to an increase in bacterial resistance. It is therefore important to control this product in the environment. This work proposes a simple and low-cost potentiometric device to monitor the levels of SDM in aquaculture waters, thus avoiding its unnecessary release throughout the environment. The device combines a micropipette tip with a PVC membrane selective to SDM, prepared from an appropriate cocktail, and an inner reference solution. The membrane includes 1% of a porphyrin derivative acting as ionophore and a small amount of a lipophilic cationic additive (corresponding to 0.2% in molar ratio). The composition of the inner solution was optimized with regard to the kind and/or concentration of primary ion, chelating agent and/or a specific interfering charged species, in different concentration ranges. Electrodes constructed with inner reference solutions of 1 × 10−8 mol/L SDM and 1 × 10−4 mol/L chromate ion showed the best analytical features. Near-Nernstian response was obtained with slopes of −54.1 mV/decade, an extraordinary detection limit of 7.5 ng/mL (2.4 × 10−8 mol/L) when compared with other electrodes of the same type. The reproducibility, stability and response time are good and even better than those obtained by liquid contact ISEs. Recovery values of 98.9% were obtained from the analysis of aquaculture water samples.

Fractional Dynamics of Computer Virus Propagation

We propose a fractional model for computer virus propagation. The model includes the interaction between computers and removable devices. We simulate numerically the model for distinct values of the order of the fractional derivative and for two sets of initial conditions adopted in the literature. We conclude that fractional order systems reveal richer dynamics than the classical integer order counterpart. Therefore, fractional dynamics leads to time responses with super-fast transients and super-slow evolutions towards the steady-state, effects not easily captured by the integer order models.

Riesz potential versus fractional Laplacian

This paper starts by introducing the Grünwald–Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy–Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian.

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.

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.

Introducing numerical analysis tools in engineering: a Scilab user case in electronics course

Software tools in education became popular since the widespread of personal computers. Engineering courses lead the way in this development and these tools became almost a standard. Engineering graduates are familiar with numerical analysis tools but also with simulators (e.g. electronic circuits), computer assisted design tools and others, depending on the degree. One of the main problems with these tools is when and how to start use them so that they can be beneficial to students and not mere substitutes for potentially difficult calculations or design. In this paper a software tool to be used by first year students in electronics/electricity courses is presented. The growing acknowledgement and acceptance of open source software lead to the choice of an open source software tool – Scilab, which is a numerical analysis tool – to develop a toolbox. The toolbox was developed to be used as standalone or integrated in an e-learning platform. The e-learning platform used was Moodle. The first approach was to assess the mathematical skills necessary to solve all the problems related to electronics and electricity courses. Analysing the existing circuit simulators software tools, it is clear that even though they are very helpful by showing the end result they are not so effective in the process of the students studying and self learning since they show results but not intermediate steps which are crucial in problems that involve derivatives or integrals. Also, they are not very effective in obtaining graphical results that could be used to elaborate reports and for an overall better comprehension of the results. The developed tool was based on the numerical analysis software Scilab and is a toolbox that gives their users the opportunity to obtain the end results of a circuit analysis but also the expressions obtained when derivative and integrals calculations, plot signals, obtain vector diagrams, etc. The toolbox runs entirely in the Moodle web platform and provides the same results as the standalone application. The students can use the toolbox through the web platform (in computers where they don't have installation privileges) or in their personal computers by installing both the Scilab software and the toolbox. This approach was designed for first year students from all engineering degrees that have electronics/electricity courses in their curricula.

Bipedal Locomotion: A Fractional CPG for Generating Patterns

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

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.

Fractional State Space Analysis of Temperature Time Series

Atmospheric temperatures characterize Earth as a slow dynamics spatiotemporal system, revealing long-memory and complex behavior. Temperature time series of 54 worldwide geographic locations are considered as representative of the Earth weather dynamics. These data are then interpreted as the time evolution of a set of state space variables describing a complex system. The data are analyzed by means of multidimensional scaling (MDS), and the fractional state space portrait (fSSP). A centennial perspective covering the period from 1910 to 2012 allows MDS to identify similarities among different Earth’s locations. The multivariate mutual information is proposed to determine the “optimal” order of the time derivative for the fSSP representation. The fSSP emerges as a valuable alternative for visualizing system dynamics.

Adaptive Penalty and Barrier function based on Fuzzy Logic

Optimization methods have been used in many areas of knowledge, such as Engineering, Statistics, Chemistry, among others, to solve optimization problems. In many cases it is not possible to use derivative methods, due to the characteristics of the problem to be solved and/or its constraints, for example if the involved functions are non-smooth and/or their derivatives are not know. To solve this type of problems a Java based API has been implemented, which includes only derivative-free optimization methods, and that can be used to solve both constrained and unconstrained problems. For solving constrained problems, the classic Penalty and Barrier functions were included in the API. In this paper a new approach to Penalty and Barrier functions, based on Fuzzy Logic, is proposed. Two penalty functions, that impose a progressive penalization to solutions that violate the constraints, are discussed. The implemented functions impose a low penalization when the violation of the constraints is low and a heavy penalty when the violation is high. Numerical results, obtained using twenty-eight test problems, comparing the proposed Fuzzy Logic based functions to six of the classic Penalty and Barrier functions are presented. Considering the achieved results, it can be concluded that the proposed penalty functions besides being very robust also have a very good performance.

Produção de Biodiesel em reator de ultrassons

A contínua subida dos preços dos combustíveis fósseis tradicionais aliada à crescente pressão por parte de várias instituições mundiais para uma política “verde” no que diz respeito aos combustíveis, levam a um aumento da procura dos biocombustíveis e é neste contexto que surge o biodiesel como um dos principais intervenientes. O biodiesel pode ser definido como um derivado éster monoalquílico de ácidos gordos de cadeia longa proveniente de fontes renováveis como óleos vegetais ou gorduras animais e que apresenta características semelhantes ao diesel de petróleo, podendo ser utilizado sem qualquer problema em motores de ignição por compressão. Este trabalho apresenta como principal objetivo o estudo da aplicação da tecnologia de ultrassons na produção de biodiesel. Foi utilizado neste trabalho como matéria-prima um óleo doméstico usado. Este óleo foi previamente filtrado sendo depois analisado o seu índice de acidez para avaliar o seu teor em ácidos gordos livres. O valor obtido para o índice de acidez do óleo foi de 1,91 mg KOH/g, um valor relativamente baixo permitindo a sua utilização sem ser necessário um tratamento inicial via esterificação para diminuir a acidez do mesmo. Foram realizados três ensaios de reação independentes, o primeiro recorrendo ao método tradicional de produção de biodiesel através de transesterificação e recorrendo a agitação mecânica e aquecimento, o segundo utilizando uma sonda de ultrassons com a potência de 500 W e um terceiro ensaio de reação utilizando uma sonda de ultrassons de 2000 W. Em todas as reações foi utilizada uma proporção de 1:5 de óleo usado e metanol e 0,5 % (em relação á massa de óleo utilizada) de catalisador metilato de sódio. Todas as alíquotas recolhidas durante os ensaios foram analisadas através de cromatografia gasosa de modo a determinar o conteúdo em ésteres presente em cada uma delas. A reação convencional teve uma duração total de 150 minutos e decorreu a uma temperatura de 65ºC e a agitação constante de 500 rpm. Ao longo da reação foram retiradas alíquotas de cerca de 25 ml, que foram tratadas de imediato e posteriormente analisadas de modo a estudar-se o comportamento da reação ao longo do tempo. A percentagem de ésteres metílicos no biodiesel obtida ao fim de 90 minutos foi de 81,3%. Em seguida realizou-se uma reação utilizando uma sonda de ultrassons de 500 W de potência mergulhada num recipiente reacional devidamente isolado com uma rolha de cortiça de modo a minimizar as perdas de metanol por evaporação. O tempo total de reação foi de 90 minutos e foram-se retirando alíquotas de cerca de 25 ml para acompanhar o desenrolar da reação, tendo-se obtido uma percentagem de ésteres metílicos de 85,9% ao fim dos 90 minutos. Foi realizada por fim um terceiro ensaio de reação utilizando uma sonda de 2000 W com uma duração total de 90 minutos, tendo-se obtido resultados pouco satisfatórios (77,7%), provavelmente devido a algum problema operacional relacionado com a sonda de ultrassons utilizada ou devido a uma geometria do reator pouco eficiente. Os produtos resultantes da reação convencional e da reação utilizando a sonda de ultrassons de 500 W, assim como o óleo utilizado como matéria-prima foram caracterizados em termos de índice de acidez, densidade a 15ºC e viscosidade a 40ºC.