Fractional calculus on time scales - Cálculo fraccional em escalas temporais

Introduzimos um cálculo das variações fraccional nas escalas temporais ℤ e (hℤ)!. Estabelecemos a primeira e a segunda condição necessária de optimalidade. São dados alguns exemplos numéricos que ilustram o uso quer da nova condição de Euler–Lagrange quer da nova condição do tipo de Legendre. Introduzimos também novas definições de derivada fraccional e de integral fraccional numa escala temporal com recurso à transformada inversa generalizada de Laplace.

A transmissão de conceitos matemáticos para Portugal: integrais e funções elípticas

Esta tese considera a transmissão de conceitos matemáticos para Portugal no século XIX, particularmente no campo dos Integrais Elípticos e das Funções Elípticas, tal como foi realizado no trabalho de António Zeferino Cândido. Depois de uma introdução histórica geral ao assunto no capítulo 1, o capítulo 2 estuda a vida de António Zeferino Cândido da Piedade. Ele foi, talvez, o primeiro matemático português a publicar uma tese sobre este assunto. A parte principal, isto é, o capítulo 3, é dedicada à análise do seu trabalho “Integraes e Funcções Ellipticas”. Mostra detalhes da sua abordagem baseada, não só, no livro dos autores Franceses Briot e Bouquet, mas também do autor alemão Schloemilch, o que reflecte as mudanças que ocorreram naquela época na liderança matemática na Europa.

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.

Cálculo das variações fraccional generalizado

Nesta tese de doutoramento apresentamos um cálculo das variações fraccional generalizado. Consideramos problemas variacionais com derivadas e integrais fraccionais generalizados e estudamo-los usando métodos directos e indirectos. Em particular, obtemos condições necessárias de optimalidade de Euler-Lagrange para o problema fundamental e isoperimétrico, condições de transversalidade e teoremas de Noether. Demonstramos a existência de soluções, num espaço de funções apropriado, sob condições do tipo de Tonelli. Terminamos mostrando a existência de valores próprios, e correspondentes funções próprias ortogonais, para problemas de Sturm- Liouville.

Calculus of variations on time scales and applications to economics

We consider some problems of the calculus of variations on time scales. On the beginning our attention is paid on two inverse extremal problems on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variation functional that attains a local minimum at a given point of the vector space. Furthermore, we prove a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. Afterwards, we prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems. Next we investigate the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange equations in integral form, transversality conditions, and necessary optimality conditions for isoperimetric problems, on an arbitrary time scale, are proved. In the end, two main issues of application of time scales in economic, with interesting results, are presented. In the former case we consider a firm that wants to program its production and investment policies to reach a given production rate and to maximize its future market competitiveness. The model which describes firm activities is studied in two different ways: using classical discretizations; and applying discrete versions of our result on time scales. In the end we compare the cost functional values obtained from those two approaches. The latter problem is more complex and relates to rate of inflation, p, and rate of unemployment, u, which inflict a social loss. Using known relations between p, u, and the expected rate of inflation π, we rewrite the social loss function as a function of π. We present this model in the time scale framework and find an optimal path π that minimizes the total social loss over a given time interval.

Fractional dynamics and MDS visualization of earthquake phenomena

This paper analyses earthquake data in the perspective of dynamical systems and fractional calculus (FC). This new standpoint uses Multidimensional Scaling (MDS) as a powerful clustering and visualization tool. FC extends the concepts of integrals and derivatives to non-integer and complex orders. MDS is a technique that produces spatial or geometric representations of complex objects, such that those objects that are perceived to be similar in some sense are placed on the MDS maps forming clusters. In this study, over three million seismic occurrences, covering the period from January 1, 1904 up to March 14, 2012 are analysed. The events are characterized by their magnitude and spatiotemporal distributions and are divided into fifty groups, according to the Flinn–Engdahl (F–E) seismic regions of Earth. Several correlation indices are proposed to quantify the similarities among regions. MDS maps are proven as an intuitive and useful visual representation of the complex relationships that are present among seismic events, which may not be perceived on traditional geographic maps. Therefore, MDS constitutes a valid alternative to classic visualization tools for understanding the global behaviour of earthquakes.

On development of fractional calculus during the last fifty years

Fractional calculus generalizes integer order derivatives and integrals. During the last half century a considerable progress took place in this scientific area. This paper addresses the evolution and establishes an assertive measure of the research development.

Fractional generalization of memristor and higher order elements

Fractional calculus generalizes integer order derivatives and integrals. Memristor systems generalize the notion of electrical elements. Both concepts were shown to model important classes of phenomena. This paper goes a step further by embedding both tools in a generalization considering complex-order objects. Two complex operators leading to real-valued results are proposed. The proposed class of models generate a broad universe of elements. Several combinations of values are tested and the corresponding dynamical behavior is analyzed.

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.

An application of cell-cluster theory to a rare gas crystal /|n[by] K. Westera. - 260 St. Catharines [Ont.] : Dept. of Physics, Brock University,

The Lennard-Jones Devonshire 1 (LJD) single particle theory for liquids is extended and applied to the anharmonic solid in a high temperature limit. The exact free energy for the crystal is expressed as a convergent series of terms involving larger and larger sets of contiguous particles called cell-clusters. The motions of all the particles within cell-clusters are correlated to each other and lead to non-trivial integrals of orders 3, 6, 9, ... 3N. For the first time the six dimensional integral has been calculated to high accuracy using a Lennard-Jones (6-12) pair interaction between nearest neighbours only for the f.c.c. lattice. The thermodynamic properties predicted by this model agree well with experimental results for solid Xenon.

Exchange energy calculation of He-He with supermolecular one-matrix

Exch~nge energy of the He-He system is calculated using the one-density matrix which has been modified according to the supermolecular density formula quoted by Kolos. The exchange energy integrals are computed analytically and by the Monte Carlo method. The results obtained from both ways compared favourably,with the results obtained from the SCF program HONDO

On the path integral formulation and the evaluation of quantum statistical averages

Four problems of physical interest have been solved in this thesis using the path integral formalism. Using the trigonometric expansion method of Burton and de Borde (1955), we found the kernel for two interacting one dimensional oscillators• The result is the same as one would obtain using a normal coordinate transformation, We next introduced the method of Papadopolous (1969), which is a systematic perturbation type method specifically geared to finding the partition function Z, or equivalently, the Helmholtz free energy F, of a system of interacting oscillators. We applied this method to the next three problems considered• First, by summing the perturbation expansion, we found F for a system of N interacting Einstein oscillators^ The result obtained is the same as the usual result obtained by Shukla and Muller (1972) • Next, we found F to 0(Xi)f where A is the usual Tan Hove ordering parameter* The results obtained are the same as those of Shukla and Oowley (1971), who have used a diagrammatic procedure, and did the necessary sums in Fourier space* We performed the work in temperature space• Finally, slightly modifying the method of Papadopolous, we found the finite temperature expressions for the Debyecaller factor in Bravais lattices, to 0(AZ) and u(/K/ j,where K is the scattering vector* The high temperature limit of the expressions obtained here, are in complete agreement with the classical results of Maradudin and Flinn (1963) .

Efficient estimation using the characteristic function : theory and applications with high frequency data

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Classification analytique de systèmes différentiels linéaires déployant une singularité irrégulière de rang de Poincaré 1

Cette thèse traite de la classification analytique du déploiement de systèmes différentiels linéaires ayant une singularité irrégulière. Elle est composée de deux articles sur le sujet: le premier présente des résultats obtenus lors de l'étude de la confluence de l'équation hypergéométrique et peut être considéré comme un cas particulier du second; le deuxième contient les théorèmes et résultats principaux. Dans les deux articles, nous considérons la confluence de deux points singuliers réguliers en un point singulier irrégulier et nous étudions les conséquences de la divergence des solutions au point singulier irrégulier sur le comportement des solutions du système déployé. Pour ce faire, nous recouvrons un voisinage de l'origine (de manière ramifiée) dans l'espace du paramètre de déploiement $\epsilon$. La monodromie d'une base de solutions bien choisie est directement reliée aux matrices de Stokes déployées. Ces dernières donnent une interprétation géométrique aux matrices de Stokes, incluant le lien (existant au moins pour les cas génériques) entre la divergence des solutions à $\epsilon=0$ et la présence de solutions logarithmiques autour des points singuliers réguliers lors de la résonance. La monodromie d'intégrales premières de systèmes de Riccati correspondants est aussi interprétée en fonction des éléments des matrices de Stokes déployées. De plus, dans le second article, nous donnons le système complet d'invariants analytiques pour le déploiement de systèmes différentiels linéaires $x^2y'=A(x)y$ ayant une singularité irrégulière de rang de Poincaré $1$ à l'origine au-dessus d'un voisinage fixé $\mathbb{D}_r$ dans la variable $x$. Ce système est constitué d'une partie formelle, donnée par des polynômes, et d'une partie analytique, donnée par une classe d'équivalence de matrices de Stokes déployées. Pour chaque valeur du paramètre $\epsilon$ dans un secteur pointé à l'origine d'ouverture plus grande que $2\pi$, nous recouvrons l'espace de la variable, $\mathbb{D}_r$, avec deux secteurs et, au-dessus de chacun, nous choisissons une base de solutions du système déployé. Cette base sert à définir les matrices de Stokes déployées. Finalement, nous prouvons un théorème de réalisation des invariants qui satisfont une condition nécessaire et suffisante, identifiant ainsi l'ensemble des modules.