Computational methods in the fractional calculus of variations and optimal control

The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the lack of analytic methods to solve such fractional problems, numerical techniques are developed. Here, we mainly investigate the approximation of fractional operators by means of series of integer-order derivatives and generalized finite differences. We give upper bounds for the error of proposed approximations and study their efficiency. Direct and indirect methods in solving fractional variational problems are studied in detail. Furthermore, optimality conditions are discussed for different types of unconstrained and constrained variational problems and for fractional optimal control problems. The introduced numerical methods are employed to solve some illustrative examples.

A Discretization of the Hadamard fractional derivative

We present a new discretization for the Hadamard fractional derivative, that simplifies the computations. We then apply the method to solve a fractional differential equation and a fractional variational problem with dependence on the Hadamard fractional derivative.

Balanceamento de linhas de produção

O presente relatório de projeto descreve o trabalho desenvolvido na empresa F, relativo ao balanceamento de linhas de produção. Foi estudado, tendo por base a literatura disponível, em que consiste o problema de balanceamento de uma linha de produção, assim como as limitações deste problema, e que métodos podem ser utilizados na sua resolução. Relativamente ao caso de estudo são expostos os objetivos a atingir, apresentando os dados recolhidos e respetivo tratamento dos mesmos. Para cada caso é efetuada a descrição do mesmo, descrevendo a situação como a linha de produção se encontrava e a respetiva melhoria. Para o primeiro caso de estudo foi realizada uma simulação da mudança na linha de produção através do Software Arena. Finalmente são apresentadas algumas conclusões que dizem respeito às vantagens do balanceamento da linha de produção, revelando ainda perspetivas de desenvolvimento futuro.

A scale variational principle of Herglotz

The Herglotz problem is a generalization of the fundamental problem of the calculus of variations. In this paper, we consider a class of non-differentiable functions, where the dynamics is described by a scale derivative. Necessary conditions are derived to determine the optimal solution for the problem. Some other problems are considered, like transversality conditions, the multi-dimensional case, higher-order derivatives and for several independent variables.

A numerical method to solve higher-order fractional differential equations

In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order derivatives only. With this, we can rewrite FDEs in terms of a classical one and then apply any known technique. With some examples, we show the accuracy of the method.

Eurípides, Ifigénia em Áulide. Introdução e versão de Carlos Alberto Pais de Almeida. Notas e revisão de Maria Fátima Silva. Coimbra, Fundação Calouste Gulbenkian, J.N.I.C.T. 1998, 2ª ed

Sem resumo disponível.

Fractional variational problems depending on indefinite integrals and with delay

The aim of this paper is to exhibit a necessary and sufficient condition of optimality for functionals depending on fractional integrals and derivatives, on indefinite integrals and on presence of time delay. We exemplify with one example, where we nd analytically the minimizer.

A Caputo fractional derivative of a function with respect to another function

In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor’s Theorem, Fermat’s Theorem, etc., are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Population Growth Model, and showing that we can model more accurately the process using different kernels for the fractional operator is provided.

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.