A column generation based approach is proposed to solve scheduling problems for fractional airlines efficiently and return near optimal schedules. Crew tours are building blocks of our approach, and our approach is focused on exploring more feasible tours than other approaches. In particular, all elements of a crew tour are optimized during the preparation and tour generation procedures. Moreover, time windows of customer-requested flights are handled exactly, and generalized to time window and crew time window of duties and tours. Furthermore, time windows of tours are contained in the MIP formulation to ensure more feasible connections between tours. In the pricing subproblem, an efficient constrained shortest path algorithm is proposed, which is necessary for our model and also provides extensibility for incorporating more complex constraints in the future. Computational results of our model show very small optimality gaps and consistent improvements over the model used in practice. Moreover, restricted versions of our model that have fast running time are provided, thus very desired in the case that running time has more priority than solution quality. In order to understand the demand, data mining of demand data is presented and analyzed. Moreover, a recovery model is proposed to deal with unscheduled maintenance in practice, by reserving airplanes and crews in the model. Computational experiments show the advantage of the recovery model, in the case of simulated unscheduled maintenance and comparing to models without recovery considerations.

Η παρούσα εργασία αποτελεί μια ανασκόπηση των βασικότερων στοιχείων της θεωρίας της κλασματικής ανάλυσης, των γραμμικών συνήθων διαφορικών εξισώσεων κλασματικής τάξης, καθώς και εφαρμογές αυτών. Η εργασία αυτή αποτελείται από τρία μέρη: Στο πρώτο μέρος αναφέρουμε ειδικές συναρτήσεις (Γάμμα συνάρτηση, Βήτα συνάρτηση και συνάρτηση Mittag – Leffler) που χρησιμοποιούνται στην κλασματική ανάλυση, καθώς και ιδιότητες αυτών. Επιπλέον, ορίζεται το κλασματικό ολοκλήρωμα, οι κλασματικές παράγωγοι Riemann – Liouville και Caputo καθώς και οι σειριακές (sequential) κλασματικές παράγωγοι και δίνονται ιδιότητες αυτών. Το δεύτερο μέρος περιλαμβάνει εισαγωγικά ιστορικά στοιχεία μελέτης των συνήθων διαφορικών εξισώσεων κλασματικής τάξης. Αναφέρεται το θεώρημα ύπαρξης και μοναδικότητας της λύσης ενός προβλήματος αρχικών τιμών και δίνονται κάποιοι τρόποι επίλυσης γραμμικών διαφορικών εξισώσεων κλασματικής τάξης με σταθερούς συντελεστές. Το τρίτο μέρος αφορά σε εφαρμογές των συνήθων διαφορικών εξισώσεων κλασματικής τάξης. Αρχικά, παραθέτουμε κάποιες εφαρμογές σε διάφορους κλάδους των επιστημών και προσεγγίζουμε τη γραμμική βισκοελαστικότητα διαμέσου της κλασματικής ανάλυσης. Στη συνέχεια πιο αναλυτικά με τη βοήθεια των κλασματικών διαφορικών εξισώσεων μελετάμε το πρόβλημα του Basset και ταλαντωτικές διαδικασίες με κλασματική απόσβεση. / This dissertation is a review of the fractional analysis theory for linear ordinary differential equations (ODE)of fractional order. The first part of our work is a review of some special functions (Gamma, Beta and Mittag - Leffler) which are used in the fractional analysis as well as their properties. We also define the fractional integral, the Riemann - Liouville and Caputo fractional derivatives, the sequential derivative of fractional order and their properties. In the second part, we introduce the basic theory of fractional order ODE's. We present the theorem of existence and uniqueness of the solution of an initial values problem and we give some algorithms for solving linear fractional order ODE's with constant coefficients. In the last part we present some applications of fractional order ODE's. Some of these are: viscoelasticity, Basset's problem and oscillatory processes of fractional damping.

The attitude determination and control subsystem (ADCS) is a critical part of any satellite conducting scientific experiments that require accurate positioning (such as Earth observation and solar spectroscopy). The engineering design process of this subsystem has a long heritage; yet, it is surrounded by several limitations due to the stringent physical constraints imposed on small satellites. These limitations (e.g., limited computational capabilities, power, and volume) require an improved approach for the purpose of attitude determination (AD) and control. Previous space missions relied mostly on the extended Kalman filter (EKF) to estimate the relative orientation of the spacecraft because it yields an optimal estimator under the assumption that the measurement and process models are white Gaussian processes. However, this filter suffers from several limitations such as a high computational cost. This thesis addresses all the limitations found in small satellites by introducing a computationally efficient algorithm for AD based on a fuzzy inference system with a gradient decent optimization technique to calculate and optimize the bounds of the membership functions. Also, an optimal controller based on a fractional proportional-integral-derivative controller has been implemented to provide an energy-efficient control scheme. The AD algorithm presented in this thesis relies on the residual information of the Earth magnetic field. In contrast to current approaches, the new algorithm is immune to several limitations such as sensitivity to initial conditions and divergence problems. Additionally, its computational cost has been reduced. Simulation results illustrate a higher pointing stability, while maintaining satisfying levels of pointing accuracy and increasing reliability. Moreover, the optimal controller designed provides a shorter time delay, settling time, and steady-state error. This demonstrates that accurate attitude determination and control can be conducted in small spacecraft.

Fractional Calculus can be thought of as a generalisation of conventional calculus in the sense that it extends the concept of a derivative (integral) to include non-integer orders. Effective mathematical modelling using Fractional Differential Equations (FDEs) requires the development of reliable flexible numerical methods. The thesis begins by reviewing a selection of numerical methods for the solution of Single-term and Multi-term FDEs. We then present: 1. a graphical technique for comparing the efficiency of numerical methods. We use this to compare Single-term and Multi-term methods and give recommendations for which method is best for any given FDE. 2. a new method for the solution of a non-linear Multi-term Fractional Dif¬ferential Equation. 3. a sequence of methods for the numerical solution of a Distributed Order Differential Equation. 4. a discussion of the problems associated with producing a computer program for obtaining the optimum numerical method for any given FDE.

Orientador: Edmundo Capelas de Oliveira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T02:22:44Z (GMT). No. of bitstreams: 1 Contharteze_Eliana_D.pdf: 2292843 bytes, checksum: c606ccefade98acff6e3f2b74c2ac021 (MD5) Previous issue date: 2014 / Resumo: Apresentamos operadores de integração e derivação fracionárias, que em particular, podem ser utilizados para descrever um processo difusivo anômalo através de uma equação diferencial fracionária. Como aplicação, discutimos uma equação diferencial fracionária associada ao processo de desaceleração de nêutrons, utilizando as transformadas integrais de Laplace e Fourier e através de uma conveniente implementação computacional, obtemos gráficos associados à solução dessa equação. Algumas propriedades dos operadores de integração e derivação fracionárias são mencionadas e utilizadas para escrever o teorema fundamental do cálculo fracionário. A clássica função de Mittag-Leffler, envolvendo um parâmetro e a função de Mittag-Leffler com dois parâmetros desempenham um papel importante no estudo das equações diferenciais fracionárias. A chamada função de Mittag-Leffler com três parâmetros, que generaliza as duas anteriores, emerge naturalmente no estudo da equação diferencial fracionária associada ao problema do telégrafo. Novas representações para as funções de Mittag-Leffler foram obtidas em termos de integrais impróprias de funções trigonométricas, a partir do cálculo da transformada de Laplace inversa sem usar um contorno de integração e como aplicação, encontramos algumas integrais impróprias interessantes que, geralmente, são demonstradas por aproximação com o uso de análise de Fourier ou teoria dos resíduos / Abstract: We present the operators of fractional integration and differentiation, which can be used to describe an anomalous diffusion process by means of a fractional differential equation. As an application we discuss a fractional differential equation associated with the slowing-down of neutrons using Laplace and Fourier transforms. With the help of a convenient computational implementation we obtain graphs of the solutions of this equation. Some properties of the operators of fractional integration and differentiation are mentioned and used to demonstrate the fundamental theorem of fractional calculus. The classical Mittag-Leffler function with one parameter and the Mittag-Leffler function with two parameters play an important role in the study of fractional differential equations. The so-called Mittag-Leffler function with three parameters, which generalizes the previous two functions, naturally arises in the study of the fractional differential equation associated with the telegraph problem. By calculating the inverse Laplace transform without using contour integration we obtain new representations for the Mittag-Leffler functions in terms of improper integrals of trigonometric functions; as an application we obtain some interesting improper integrals which are usually proved by approximation using Fourier analysis or residue theory / Doutorado / Matematica Aplicada / Doutora em Matemática Aplicada

Orientador: Edmundo Capelas de Oliveira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T23:00:41Z (GMT). No. of bitstreams: 1 Rodrigues_FabioGrangeiro_D.pdf: 1185818 bytes, checksum: 96f82c6ff4622e4ecdd3ccae79803dae (MD5) Previous issue date: 2015 / Resumo: Neste trabalho é apresentado um modo de se obter soluções de um caso particular da equação hipergeométrica confluente, a equação de Bessel de ordem p, utilizando-se da teoria do cálculo de ordem arbitrária, também conhecido popularmente por cálculo fracionário. Em particular, discutimos alguns equívocos identificados na literatura e levantamos questionamentos sobre algumas interpretações a respeito dos operadores formulados segundo Riemann-Liouville quando aplicados a certos tipos de funções. Para tanto, apresentamos inicialmente os operadores de integração e diferenciação fracionárias segundo as formulações mais clássicas (Riemann-Liouville, Caputo e Grünwald-Letnikov) e, em seguida, apresentamos o operador de integrodiferenciação fracionária que é a tentativa de unificar as operações de integração e diferenciação sob um único operador. Ao longo do texto indicamos as principais propriedades destes operadores e citamos algumas das suas aplicações comumente encontrados na Matemática, Física e Engenharias / Abstract: In this thesis we discuss the solvability of the Bessel's differential equation of order p, which is a particular case of the confluent hypergeometric equation, from the perspective of the theory of calculus of arbitrary order, also commonly known as fractional calculus. In particular, we expose some misconceptions encountered in the literature and we raise some questions about interpretations of the Riemann-Liouville operators when acting on certain types of functions. In order to do so, we present the main fractional operators (Riemann-Liouville, Caputo and Grünwald-Letnikov) as well as the fractional integrodifferential operator, which is an unified view of both integration and differentiation under a single operator. We also show the main properties of these operators and mention some of its applications in Mathematics, Physics and Engeneering / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

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In this dissertation, we use the Natural transform decomposition method to obtain approximate analytical solution of fractional differential equations. This technique is a combination of decomposition methods and natural transform method. We use the Adomian decomposition, the homotopy perturbation and the Daftardar-Jafari methods as our decomposition methods. The fractional derivatives are considered in the Caputo and Caputo- Fabrizio sense. / Mathematical Sciences / M. Sc. (Applied Mathematics)