In this dissertation we focus on fractional-order dynamical systems and classify these problems as optimal control of system described by fractional derivative, fractional-order nonlinear differential equations, optimal control of systems described by variable-order differential equations and delay fractional optimal control problems. These problems are solved by using the spectral method and reducing the problem to a system of algebraic equations. In fact for the optimal control problems described by fractional and variable-order equations, the variables are approximated by chosen wavelets with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem is converted to an optimization problem, which can be solved numerically. We have applied the new generalized wavelets to approximate the fractional-order nonlinear differential equations such as Riccati and Bagley-Torvik equations. Then, the solution of this kind of problem is found using the collocation method. For solving the fractional optimal control described by fractional delay system, a new set of hybrid functions have been constructed. Also, a general and exact formulation for the fractional-order integral operator of these functions has been achieved. Then we utilized it to solve delay fractional optimal control problems directly. The convergence of the present method is discussed. For all cases, some numerical examples are presented and compared with the existing results, which show the efficiency and accuracy of the present method.

This thesis is comprised of three main parts: The Hermite-Hadamard inequality on discrete time scales, the fractional Hermite-Hadamard inequality, and Karush-Kuhn- Tucker conditions on higher dimensional discrete domains. In the first part of the thesis, Chapters 2 & 3, we define a convex function on a special time scale T where all the time points are not uniformly distributed on a time line. With the use of the substitution rules of integration we prove the Hermite-Hadamard inequality for convex functions defined on T. In the fourth chapter, we introduce fractional order Hermite-Hadamard inequality and characterize convexity in terms of this inequality. In the fifth chapter, we discuss convexity on n{dimensional discrete time scales T = T1 × T2 × ... × Tn where Ti ⊂ R , i = 1; 2,…,n are discrete time scales which are not necessarily periodic. We introduce the discrete analogues of the fundamental concepts of real convex optimization such as convexity of a function, subgradients, and the Karush-Kuhn-Tucker conditions. We close this thesis by two remarks for the future direction of the research in this area.

O método das matrizes operacionais é expandido para o corpo complexo a ordens arbitrárias pela abordagem de Riemann-Liouville e Caputo com ênfase nas séries de Fourier complexas. Elabora-se uma adaptação do formalismo bra-ket de Dirac à linguagem tensorial no espaço de Hilbert de funções com expansões finitas para uso específico na teoria de equações diferenciais e matrizes operacionais, denominado \\Formalismo Coadjunto\". Estende-se o tratamento aos operadores fracionais de Weyl para períodos genéricos a fim de determinar as matrizes operacionais de derivação e integração de ordem arbitrária na base complexa de Fourier. Introduz-se um novo método de resolução de equações diferenciais ordinárias lineares e fracionais não-homogêneas, denominado \\Modelagem Operacional\", que permite a obtenção de soluções de equações de alta ordem com grande precisão sem a necessidade de imposição de condições iniciais ou de contorno. O método apresentado é aperfeiçoado por meio de um novo tipo de expansão, que denominamos \"Séries Associadas de Fourier\", a qual apresenta convergência mais rápida que a série de Fourier original numa restrição de domínio, possibilitando soluções de EDOs e EDFs de alta ordem com maior precis~ao e ampliando a esfera de casos passíveis de resolução. / Operational matrices method is expanded to complex field and arbitrary orders by using the Riemann-Liouville and Caputo approach with emphasis on complex Fourier series. Dirac\'s bra-ket notation is associated to tensor procedures in Hilbert spaces for finite function expansions to be applied specifically to dfferential equations and operational matrices, being called \\Coadjoint Formalism\". This treatment is extended to Weyl fractional operators for generic periods in order to establish the integral and derivative operational matrices of fractional order to complex Fourier basis. A new method to solve linear non-homogeneous ODEs and FDEs, called \\Operational Modelling\"is introduced. It yields high precision solutions on high order dfferential equations without assumption of boundary or initial conditions. The presented method is improved by a new kind of function expansion, called \\Fourier Associated Series\", which yields a faster convergence than original Fourier in a restrict domain, enabling to obtain solutions of high order ODEs and FDEs with excellent precision and broadening the set of solvable equations.

Li, Ming Hin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 42-46). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Inference for Fractional Cointegration --- p.5 / Chapter 2.1 --- Concept of Fractional Cointegration --- p.5 / Chapter 2.1.1 --- Fractional Integration --- p.5 / Chapter 2.1.2 --- Fractional Cointegration --- p.8 / Chapter 2.2 --- Fractional Cointegration Modeling --- p.9 / Chapter 2.2.1 --- Engle-Granger's Methodology --- p.9 / Chapter 2.2.2 --- Johansen's Methodology --- p.10 / Chapter 2.2.2.1 --- Maximum Likelihood Estimators --- p.12 / Chapter 2.2.2.2 --- Cofractional Rank Test --- p.16 / Chapter 3 --- Pairs Trading Strategy --- p.19 / Chapter 3.1 --- Statistical Arbitrage --- p.19 / Chapter 3.2 --- Fractional Cointegration Pairs Trading --- p.20 / Chapter 3.2.1 --- Trading Procedures --- p.22 / Chapter 4 --- Empirical Study --- p.27 / Chapter 4.1 --- Backgrounds --- p.27 / Chapter 4.2 --- Settings --- p.28 / Chapter 4.3 --- Empirical Results --- p.29 / Chapter 5 --- Conclusions and Further Research --- p.39 / Bibliography --- p.42

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This thesis consists of six chapters. In the first chapter, we review some basic definitions and concepts of fractional calculus. Then we introduce fractional difference equations involving the Riemann-Liouville operator of real number order between zero and one. In the second chapter, we apply the Brouwer fixed point and Contraction Mapping Theorems to prove that there exists a solution for up to the first order nabla fractional difference equation with an initial condition. In chapter three, we define a lower and an upper solution for up to the first order nabla fractional difference equation with an initial condition. Under certain assumptions we prove that a lower solution stays less than an upper solution. Some examples are given to illustrate our findings in this chapter. Then we give constructive proofs of existence of a solution by defining monotone sequences. In the fourth chapter, we derive a continuous form of the Mittag-Leffler function. Then we use successive approximations method to calculate a discrete form of the Mittag-Leffler function. In the fifth chapter, we focus on finding the model which fits best for the data of tumor growth for twenty-eight mice. The models contain either three parameters (Gompertz, Logistic) or four parameters (Weibull, Richards). For each model, we consider continuous, discrete, continuous fractional and discrete fractional forms. Nihan Acar who is a former graduate student in mathematics department has already worked on Gompertz and Logistic models [1]. Here we continue and work on Richards curve. The difference between Acar’s work and ours is the number of parameters in each model. Gompertz and Logistic models contain three parameters and an alpha parameter. The Richards model has four parameters and an alpha parameter. In addition, we use statistical computation techniques such as residual sum of squares and cross-validation to compare fitting and predictive performance of these models. In conclusion, we put three models together to conclude which model is fitting best for the data of tumor growth for twenty-eight mice. In the last chapter, we conclude this thesis and state our future work.

Previous decades of cartilage research have predominantly focused on decoupling the solid and fluid interactions of the mechanical response. The resulting biphasic and triphasic models are widely used in the biomechanics community. However, a simple viscoelastic model is able to account for the stress-relaxation behavior of cartilage, without the added complexity of solid and fluid interactions. Using a viscoelastic model, cartilage is considered a single material with elastic and dissipative properties. A mechanical characterization is made with fewer material parameters than are required by the conventional biphasic and triphasic models. This approach has tremendous utility when comparing cartilage of different species and varying healths. Additionally, the viscoelastic models can be easily extended in dynamic analysis and FEA programs. Cartilage primarily experiences compressive motion during exercise. Therefore, to mimic biological function, a mechanical test should also compress the cartilage. One such test is a viscoelastic stress-relaxation experiment. The Prony and fractional calculus viscoelastic models have shown promise in modeling stress-relaxation of equine articular cartilage. The elastic-viscoelastic correspondence principle is used to extend linear viscoelasticity to the frequency domain. This provides a comparison of articular cartilage based on stored and dissipated moduli. The storage and loss moduli metrics are hypothesized to serve as benchmarks for evaluating osteoarthritic cartilage, and provide guidelines for newly engineered bio-materials. The main goal of the current study is to test the applicability of modeling articular cartilage with viscoelastic models. A secondary goal is to establish a rigorous set of harvesting techniques that allows access to fresh explants with minimal environmental exposure. With a complex substance like cartilage, it is crucial to avoid unnecessary emph{in vitro} environmental exposure. Additional areas of study include: determining the strain-dependency of the mechanical response, exploring the response of cartilage in different fluid mediums such as saline, synovial fluid, and synthetic substitutes, and studying the time-dependent properties of cartilage during stress-relaxation experiments. Equine stifle joints, which are mechanically analogous to human knees, are harvested and used for analysis in this study. It is believed that the proposed viscoelastic models can model other articulating joints as well. If viscoelastic theory can be used to characterize cartilage, then comparisons can be drawn between real and artificial cartilage, leading to better joint replacement technology.

Soft biological tissue material models in the literature are frequently limited to integer order constitutive relations where the order of differentiation of stress and/or strain is integer-valued. However, it has been demonstrated that fractional calculus theory applied in soft tissue material model formulation yields more accurate and reliable soft tissue material models. In this study, firstly a fractional order (where the order of differentation of stress in the constitutive relation is non-integer-valued) linear viscoelastic material model for soft tissues is fitted to force-displacement-time indentation test data and compared with two different integer order linear viscoelastic material models by using MATLAB&reg / optimization toolbox. After the superiority of the fractional order material model compared to integer order material models has been shown, the linear fractional order material model is extended to its nonlinear counterpart in finite deformation regime. The material model developed is assumed to be isotropic and homogeneous. v A user-subroutine is developed for the material model formulated to implement it into the commercial finite element software Msc.Marc 2010. The user-subroutine developed is verified by performing a small strain finite element analysis and comparing the results obtained with linear viscoelastic counterpart of the model on MATLAB&reg / . Finally, the unknown coefficients of the fractional order material model are identified by employing the inverse finite element method. A material parameter set with an amount of accuracy is determined and the material model with the parameters identified is capable of simulating the three different indentation test protocols, i.e., &ldquo / relaxation&rdquo / , &ldquo / creep&rdquo / and &ldquo / cyclic loading&rdquo / protocols with a good accuracy.

This was a qualitative study carried out with learners from a grade twelve Standard Grade mathematics class from a South Durban school in the province of KwaZulu-Natal, South Africa. The main purpose of this study was to explore learners‟ understanding of the concept of the derivative. The participants comprised one class of twenty seven learners who were enrolled for Standard Grade mathematics at grade twelve level. Learners‟ responses to May and August examinations were examined. The examination questions that were highlighted were those based on the concept of the derivative. Additionally semi-structured interviews were carried out with a smaller sample of four of the twenty seven learners to gauge their perceptions of the derivative. The learners‟ responses to the examination questions and semi-structured interviews were exhaustively analysed. Themes that ran across the data were identified and further categorised in a bid to provide answers to the main research question. It was found that most learners‟ difficulties with the test items were grounded in their difficulties with algebraic manipulation skills. A further finding was that learners overwhelmingly preferred working out items that involved applying the rules. Although the Higher and Standard grade system of assessing learners‟ mathematical abilities has been phased out, with the advent of the new curriculum, the findings of this study is still important for learners, teachers, curriculum developers and mathematics educators because calculus forms a large component of the new mathematics curriculum. / Thesis (M.Ed.)-University of KwaZulu-Natal, Durban, 2008.

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.