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.

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.

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.

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.

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.

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.

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.

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.

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 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.

Novel materials based on chitosan, its derivatives and cellulose fibres

O presente trabalho tem como principal objectivo o desenvolvimento de novos materiais baseados em quitosano, seus derivados e celulose, na forma de nanofibras ou de papel. Em primeiro lugar procedeu-se à purificação das amostras comerciais de quitosano e à sua caracterização exaustiva em termos morfológicos e físicoquímicos. Devido a valores contraditórios encontrados na literatura relativamente à energia de superfície do quitosano, e tendo em conta a sua utilização como precursor de modificações químicas e a sua aplicação em misturas com outros materiais, realizou-se também um estudo sistemático da determinação da energia de superfície do quitosano, da quitina e seus respectivos homólogos monoméricos, por medição de ângulos de contacto Em todas as amostras comerciais destes polímeros identificaram-se impurezas não polares que estão associadas a erros na determinação da componente polar da energia de superfície. Após a remoção destas impurezas, o valor da energia total de superfície (gs), e em particular da sua componente polar, aumentou consideravelmente. Depois de purificadas e caracterizadas, algumas das amostras de quitosano foram então usadas na preparação de filmes nanocompósitos, nomeadamente dois quitosanos com diferentes graus de polimerização, correspondentes derivados solúveis em água (cloreto de N-(3-(N,N,N-trimetilamónio)-2- hidroxipropilo) de quitosano) e nanofibras de celulose como reforço (celulose nanofibrilada (NFC) e celulose bacteriana (BC). Estes filmes transparentes foram preparados através de um processo simples e com conotação ‘verde’ pela dispersão homogénea de diferentes teores de NFC (até 60%) e BC (até 40%) nas soluções de quitosano (1.5% w/v) seguida da evaporação do solvente. Os filmes obtidos foram depois caracterizados por diversas técnicas, tais como SEM, AFM, difracção de raio-X, TGA, DMA, ensaios de tracção e espectroscopia no visível. Estes filmes são altamente transparentes e apresentam melhores propriedades mecânicas e maior estabilidade térmica do que os correspondentes filmes sem reforço. Outra abordagem deste trabalho envolveu o revestimento de folhas de papel de E. globulus com quitosano e dois derivados, um derivado fluorescente e um derivado solúvel em água, numa máquina de revestimentos (‘máquina de colagem’) à escala piloto. Este estudo envolveu inicialmente a deposição de 1 a 5 camadas do derivado de quitosano fluorescente sobre as folhas de papel de forma a estudar a sua distribuição nas folhas em termos de espalhamento e penetração, através de medições de reflectância e luminescência. Os resultados mostraram que, por um lado, a distribuição do quitosano na superfície era homogénea e que, por outro lado, a sua penetração através dos poros do papel cessou após três deposições. Depois da terceira camada verificou-se a formação de um filme contínuo de quitosano sobre a superfície do papel. Estes resultados mostram que este derivado de quitosano fluorescente pode ser utilizado como marcador na optimização e compreensão de mecanismos de deposição de quitosano em papel e outros substratos. Depois de conhecida a distribuição do quitosano nas folhas de papel, estudou-se o efeito do revestimento de quitosano e do seu derivado solúvel em água nas propriedades finais do papel. As propriedades morfológicas, mecânicas, superficiais, ópticas, assim como a permeabilidade ao ar e ao vapor de água, a aptidão à impressão e o envelhecimento do papel, foram exaustivamente avaliadas. De uma forma geral, os revestimentos com quitosano e com o seu derivado solúvel em água tiveram um impacto positivo nas propriedades finais do papel, que se mostrou ser dependente do número de camadas depositadas. Os resultados também mostraram que os papéis revestidos com o derivado solúvel em água apresentaram melhores propriedades ópticas, aptidão à impressão e melhores resultados em relação ao envelhecimento do que os papéis revestidos com quitosano. Assim, o uso de derivados de quitosano solúveis em água em processos de revestimento de papel representa uma estratégia bastante interessante e sustentável para o desenvolvimento de novos materiais funcionais ou na melhoria das propriedades finais dos papéis. Por fim, tendo como objectivo valorizar os resíduos e fracções menos nobres da quitina e do quitosano provenientes da indústria transformadora, estes polímeros foram convertidos em polióis viscosos através de uma reacção simples de oxipropilação. Este processo tem também conotação "verde" uma vez que não requer solvente, não origina subprodutos e não exige nenhuma operação específica (separação, purificação, etc) para isolar o produto da reacção. As amostras de quitina e quitosano foram pré-activadas com KOH e depois modificadas com um excesso de óxido de propileno (PO) num reactor apropriado. Em todos os casos, o produto da reacção foi um líquido viscoso composto por quitina ou quitosano oxipropilados e homopolímero de PO. Estas duas fracções foram separadas e caracterizadas.

Essays on energy derivatives pricing and financial risk management

This thesis consists of an introductory chapter (essay I) and five more empirical essays on electricity markets and CO2 spot price behaviour, derivatives pricing analysis and hedging. Essay I presents the structure of the thesis and electricity markets functioning and characteristics, as well as the type of products traded, to be analyzed on the following essays. In the second essay we conduct an empirical study on co-movements in electricity markets resorting to wavelet analysis, discussing long-term dynamics and markets integration. Essay three is about hedging performance and multiscale relationships in the German electricity spot and futures markets, also using wavelet analysis. We concentrate the investigation on the relationship between coherence evolution and hedge ratio analysis, on a time-frequency-scale approach, between spot and futures which conditions the effectiveness of the hedging strategy. Essays four, five and six are interrelated between them and with the other two previous essays given the nature of the commodity analyzed, CO2 emission allowances, traded in electricity markets. Relationships between electricity prices, primary energy fuel prices and carbon dioxide permits are analyzed on essay four. The efficiency of the European market for allowances is examined taking into account markets heterogeneity. Essay five analyzes stylized statistical properties of the recent traded asset CO2 emission allowances, for spot and futures returns, examining also the relation linking convenience yield and risk premium, for the German European Energy Exchange (EEX) between October 2005 and October 2009. The study was conducted through empirical estimations of CO2 allowances risk premium, convenience yield, and their relation. Future prices from an ex-post perspective are examined to show evidence for significant negative risk premium, or else a positive forward premium. Finally, essay six analyzes emission allowances futures hedging effectiveness, providing evidence for utility gains increases with investor’s preference over risk. Deregulation of electricity markets has led to higher uncertainty in electricity prices and by presenting these essays we try to shed new lights about structuring, pricing and hedging in this type of markets.

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.

Analysis of the organic matter associated to sea salt : definition of potential molecular markers based on the volatile composition and presence of glycosidic derivatives and polysaccharides

Sea salt is a natural product obtained from the evaporation of seawater in saltpans due to the combined effect of wind and sunlight. Nowadays, there is a growing interest for protection and re-valorisation of saltpans intrinsically associated to the quality of sea salt that can be evaluated by its physico-chemical properties. These man-made systems can be located in different geographical areas presenting different environmental surroundings. During the crystallization process, organic compounds coming from these surroundings can be incorporated into sea salt crystals, influencing their final composition. The organic matter associated to sea salt arises from three main sources: algae, surrounding bacterial community, and anthropogenic activity. Based on the hypothesis that sea salt contains associated organic compounds that can be used as markers of the product, including saltpans surrounding environment, the aim of this PhD thesis was to identify these compounds. With this purpose, this work comprised: 1) a deep characterisation of the volatile composition of sea salt by headspace solid phase microextraction combined with comprehensive two-dimensional gas chromatography time-of-flight mass spectrometry (HS-SPME/GCGC–ToFMS) methodology, in search of potential sea salt volatile markers; 2) the development of a methodology to isolate the polymeric material potentially present in sea salt, in amounts that allow its characterisation in terms of polysaccharides and protein; and 3) to explore the possible presence of triacylglycerides. The high chromatographic resolution and sensitivity of GC×GC–ToFMS enabled the separation and identification of a higher number of volatile compounds from sea salt, about three folds, compared to unidimentional chromatography (GC–qMS). The chromatographic contour plots obtained revealed the complexity of marine salt volatile composition and confirmed the relevance of GC×GC–ToFMS for this type of analysis. The structured bidimentional chromatographic profile arising from 1D volatility and 2D polarity was demonstrated, allowing more reliable identifications. Results obtained for analysis of salt from two locations in Aveiro and harvested over three years suggest the loss of volatile compounds along the time of storage of the salt. From Atlantic Ocean salts of seven different geographical origins, all produced in 2007, it was possible to identify a sub-set of ten compounds present in all salts, namely 6-methyl-5-hepten-2-one, 2,2,6-trimethylcyclohexanone, isophorone, ketoisophorone, β-ionone-5,6-epoxide, dihydroactinidiolide, 6,10,14-trimethyl-2-pentadecanone, 3-hydroxy-2,4,4-trimethylpentyl 2-methylpropanoate, 2,4,4-trimethylpentane-1,3-diyl bis(2-methylpropanoate), and 2-ethyl-1-hexanol. These ten compounds were considered potential volatile markers of sea salt. Seven of these compounds are carotenoid-derived compounds, and the other three may result from the integration of compounds from anthropogenic activity as metabolites of marine organisms. The present PhD work also allowed the isolation and characterisation, for the first time, of polymeric material from sea salt, using 16 Atlantic Ocean salts. A dialysis-based methodology was developed to isolate the polymeric material from sea salt in amounts that allowed its characterisation. The median content of polymeric material isolated from the 16 salts was 144 mg per kg of salt, e.g. 0.014% (w/w). Mid-infrared spectroscopy and thermogravimetry revealed the main occurrence of sulfated polysaccharides, as well as the presence of protein in the polymeric material from sea salt. Sea salt polysaccharides were found to be rich in uronic acid residues (21 mol%), glucose (18), galactose (16), and fucose (13). Sulfate content represented a median of 45 mol%, being the median content of sulfated polysaccharides 461 mg/g of polymeric material, which accounted for 66 mg/kg of dry salt. Glycosidic linkage composition indicates that the main sugar residues that could carry one or more sulfate groups were identified as fucose and galactose. This fact allowed to infer that the polysaccharides from sea salt arise mainly from algae, due to their abundance and composition. The amino acid profile of the polymeric material from the 16 Atlantic Ocean salts showed as main residues, as medians, alanine (25 mol%), leucine (14), and valine (14), which are hydrophobic, being the median protein content 35 mg/g, i.e. 4,9 mg per kg of dry salt. Beside the occurrence of hydrophobic volatile compounds in sea salt, hydrophobic non-volatile compounds were also detected. Triacylglycerides were obtained from sea salt by soxhlet extraction with n-hexane. Fatty acid composition revealed palmitic acid as the major residue (43 mol%), followed by stearic (13), linolenic (13), oleic (12), and linoleic (9). Sea salt triacylglycerides median content was 1.5 mg per kg of dry salt. Both protein and triacylglycerides seem to arise from macro and microalgae, phytoplankton and cyanobacteria, due to their abundance and composition. Despite the variability resulting from saltpans surrounding environment, this PhD thesis allowed the identification of a sea salt characteristic organic compounds profile based on volatile compounds, polysaccharides, protein, and triacylglycerides.