Calculus of variations on time scales and discrete fractional calculus

Estudamos problemas do cálculo das variações e controlo óptimo no contexto das escalas temporais. Especificamente, obtemos condições necessárias de optimalidade do tipo de Euler–Lagrange tanto para lagrangianos dependendo de derivadas delta de ordem superior como para problemas isoperimétricos. Desenvolvemos também alguns métodos directos que permitem resolver determinadas classes de problemas variacionais através de desigualdades em escalas temporais. No último capítulo apresentamos operadores de diferença fraccionários e propomos um novo cálculo das variações fraccionário em tempo discreto. Obtemos as correspondentes condições necessárias de Euler– Lagrange e Legendre, ilustrando depois a teoria com alguns exemplos.

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.

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.

Fractional and time-scales differential equations

Resumo indisponível.

Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case

In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator $\Delta_+^{(\alpha,\beta,\gamma)}:= D_{x_0^+}^{1+\alpha} +D_{y_0^+}^{1+\beta} +D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$, and the fractional derivatives $D_{x_0^+}^{1+\alpha}$, $D_{y_0^+}^{1+\beta}$, $D_{z_0^+}^{1+\gamma}$ are in the Riemann-Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator $\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. Making use of the Mittag-Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions.

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.

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.

Clifford-Hermite polynomials in fractional Clifford analysis

In this paper we generalize radial and standard Clifford-Hermite polynomials to the new framework of fractional Clifford analysis with respect to the Riemann-Liouville derivative in a symbolic way. As main consequence of this approach, one does not require an a priori integration theory. Basic properties such as orthogonality relations, differential equations, and recursion formulas, are proven.

Eigenfunctions and fundamental solutions of the Caputo fractional Laplace and Dirac operators

In this paper, by using the method of separation of variables, we obtain eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator defined via fractional Caputo derivatives. The solutions are expressed using the Mittag-Leffler function and we show some graphical representations for some parameters. A family of fundamental solutions of the corresponding fractional Dirac operator is also obtained. Particular cases are considered in both cases.

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.

Fractional differential equations with dependence on the Caputo-Katugampola derivative

In this paper we present a new type of fractional operator, the Caputo–Katugampola derivative. The Caputo and the Caputo–Hadamard fractional derivatives are special cases of this new operator. An existence and uniqueness theorem for a fractional Cauchy type problem, with dependence on the Caputo–Katugampola derivative, is proven. A decomposition formula for the Caputo–Katugampola derivative is obtained. This formula allows us to provide a simple numerical procedure to solve the fractional differential equation.

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.

Geoquímica dos granitóides de Aguiar da Beira, norte de Portugal

A área de Aguiar da Beira está integrada nos terrenos autóctones da Zona Centro-ibérica e é constituída essencialmente por rochas granitóides variscas instaladas durante e após a terceira fase de deformação (D3). As relações de campo mostram que estes granitóides intruíram formações metassedimentares de idade proterozóica superior-câmbrica e as sequências do Ordovícico e do Carbónico do sinclinal Porto-Sátão, cujo extremo SE aflora na área de estudo. Com base na cartografia publicada e nos dados de campo colhidos no âmbito deste trabalho, foi possível individualizar oito intrusões distintas: o granodiorito -granito biotítico de Sernancelhe, o granito gnaissoso de duas micas, o granito moscovítico-biotítico de Vila Nova de Paiva, o granodiorito-granito biotítico-moscovítico de Lagares e os granitos de Touro (biotítico-moscovítico), Aguiar da Beira (moscovítico-biotítico), Pera Velha / Vila da Ponte (biotítico-moscovítico) e Rei Mouro (moscovítico-biotítico). A presença de encraves microgranulares em cinco dos granitóides estudados sugere que os processos de mistura de magmas desempenharam um papel importante na sua petrogénese. As datações U-Pb obtidas em zircões e monazites durante o presente estudo permitiram subdividir os granitóides de Aguiar da Beira em três grupos, de acordo com as suas relações com a terceira fase de deformação (D3): granitóides sin-tectónicos (Sernancelhe e granito gnaissoso; 322-317 Ma), tardi-tectónicos (Vila Nova de Paiva, Lagares e Touro; 308-306 Ma), e tardi- a pós-tectónicos (Aguiar da Beira, Pera Velha / Vila da Ponte e Rei Mouro; 303297 Ma). As assinaturas geoquímicas de elementos maiores e traço dos granitóides estudados, em conjunto com os dados isotópicos Sr-Nd e δ18 (rocha total e zircão) apontam para uma contribuição significativa de protólitos crustais na génese destes magmas. Á excepção do granito gnaissoso, todos os granitóides possuem um carácter transicional entre os granitos do tipo I e do tipo S, o que apoiado pelos dados de geoquímica de rocha total e isotópica, e pela presença de encraves microgranulares de composição mais máfica presentes em muitos deles, indicia uma forte intervenção de processos de hibridização de líquidos de proveniência distinta (crustais e mantélicos), em diferentes proporções, na sua origem. Pelo contrário, as características geoquímicas e isotópicas do granito gnaissoso revelam claras afinidades com os granitos do tipo S, e sugerem que tenha derivado da anatexia de fontes exclusivamente supracrustais. No entanto, parte da variabilidade geoquímica e isotópica observada em todos os granitóides estudados só poderá ser explicada pela actuação de processos de cristalização fraccionada, especialmente intensos no caso do granito gnaissoso e dos granitos tardi- a PÓS-D3.

Petrologic and geochemical characterization of São Jorge island volcanism, Azores

The island of São Jorge (38º 45’ 24’’ N - 28º 20’ 44’’W and 38º 33’ 00’’ N - 27º 44’ 32’’ W) is one of the nine islands of the Azores Archipelago that is rooted in the Azores Plateau, a wide and complex region which encompasses the triple junction between the American, Eurasia and Nubia plates. São Jorge Island has grown by fissural volcanic activity along fractures with the regional WNW-ESE trend, unveiling the importance of the regional tectonics during volcanic activity. The combination of the volcanostratigraphy (Forjaz & Fernandes, 1975; and Madeira, 1998) with geochronological data evidences that the island developed during two main volcanic phases. The first subaerial phase that occurred between 1.32 and 1.21 Ma ago (Hildenbrand et al. 2008) is recorded on the lava sequence forming the cliff at Fajã de São João, while the second phase started at 757 ka ago, is still active, and edified the rest of the island. This second phase edified the east side of the island that corresponds to Topo Volcanic Complex, in the period between 757 and 543 ka ago, while the west side named Rosais Volcanic Complex, started at 368 ka ago (Hildenbrand et al. 2008) and was still active at 117 ka ago. After the onset of Rosais, volcanic activity migrates to the center of São Jorge edifying Manadas Volcanic Complex. The volcanism on São Jorge is dominantly alkaline, with a narrow lithological composition ranging between the basanites/tefrites through the basaltic trachyandesites, in spite of this the two volcanic phases show distinct mineralogical, petrographic and geochemical characteristics that should be related with different petrogenetic conditions and growth rates of the island. Abstract viii During the first volcanic phase, growth rates are faster (≈3.4 m/ka), the lavas are slightly less alkaline and plagioclase-richer, pointing to the existence of a relative shallow and dynamic magma chamber where fractional crystallization associated with gravitational segregation and accumulation processes, produced the lavas of Fajã de São João sequence. The average growth rates during the second volcanic phase are lower (≈1.9 m/ka) and the lavas are mainly alkaline sodic, with a mineralogy composed by olivine, pyroxene, plagioclase and oxide phenocrysts, in a crystalline groundmass. The lavas are characterized by enrichment in incompatible trace element and light REE, but show differences for close-spaced lavas that unveil, in some cases, slight different degrees of fertilization of the mantle source along the island. These differences might also result from higher degrees of partial melting, as observed in the early stages of Topo and Rosais volcanic complexes, of a mantle source with residual garnet and amphibole, and/or from changing melting conditions of the mantle source as pressure. The subtle geochemical differences of the lavas contrast with the isotopic signatures, obtained from Sr-Nd-Pb-Hf isotopes, that São Jorge Island volcanism exhibit along its volcanic complexes. The lavas from Topo Volcanic Complex and from the submarine flank, i.e. the lavas located east of Ribeira Seca Fault, sample a mantle source with similar isotopic signature that, in terms of lead, overlaps Terceira Island. The lavas from Rosais and Manadas volcanic complexes, the western lavas, sample a mantle source that becomes progressively more distinct towards the west end of the island and that, in terms of lead isotopes, trends towards the isotopic composition of Faial Island. The two isotopic signatures of São Jorge, observed from the combination of lead isotopes with the other three systems, seem to result from the mixing of three distinct end-members. These end-members are (1) the common component related with the Azores Plateau and the MAR, (2) the eastern component with a FOZO signature and possibly related with the Azores plume located beneath Terceira, and (3) the western component, similar to Faial, where the lithosphere could have been entrained by an ancient magmatic liquid, isolated for a period longer than 2Ga. The two trends observed in the island reinforce the idea of small-scale mantle heterogeneities beneath the Azores region, as it has been proposed to explain the isotopic diversity observed in the Archipelago.