Thesis (Ph. D.)--Colorado State University, 2004. / Includes bibliographical references.

Fractional calculus, which is a generalization of classical calculus, has been the subject of numerous applications in physics and engineering during the last decade. In this thesis, fractional calculus has been implemented for chemical engineering applications, namely in process control and in the modeling mass transfer in adsorption. With respect to process control, some researchers have proposed fractional PIλDμ controllers based on fractional calculus to replace classical PI and PID controllers. The closed-loop control of different benchmark dynamic systems using optimally-tuned fractional PIλDμ controllers were investigated to determine for which dynamic systems this more computationally-intensive controller would be beneficial. Four benchmark systems were used: first order plus dead time system, high order system, nonlinear system, and first order plus integrator system. The optimal tuning of the fractional PIλDμ controller for each system was performed using multi-objective optimization minimizing three performance criteria, namely the ITAE, OZ, and ISDU. Conspicuous advantages of using PIλDμ controllers were confirmed and compared with other types of controllers for these systems. In some cases, a PIλ controller was also a good alternative to the PIλDμ controller with the advantage of being less computationally intensive. For the optimal tuning of fractional controllers for each benchmark dynamic system, a new version of the non-dominated sorting genetic algorithm (NSGA-III) was used to circumscribe the Pareto domain. However, it was found that for the tuning of PIλDμ controllers, it was difficult to circumscribe the complete Pareto domain using NSGA-III. Indeed, the Pareto domain obtained was sometimes fragmentary, unstable and/or susceptible to user-defined parameters and operators of NSGA-III. To properly use NSGA-III and determine a reliable Pareto domain, an investigation on the effect of these user-defined operators and parameters of this algorithm was performed. It was determined that a reliable Pareto domain was obtained with a crossover operator with a significant extrapolation component, a Gaussian mutation operator, and a large population. The findings on the proper use of NSGA-III can also be used for the optimization of other systems. Fractional calculus was also implemented in the modeling of breakthrough curves in packed adsorption columns using finite differences. In this investigation, five models based on different assumptions were proposed for the adsorption of butanol on activated carbon. The first four models are based on integer order partial differential equations accounting for the convective mass transfer through the packed bed and the diffusion and adsorption of an adsorbate within adsorbent particles. The fifth model assumes that the diffusion inside adsorbent particles is potentially anomalous diffusion and expressed by a fractional partial differential equation. For all these models, the best model parameters were determined by nonlinear regression for different sets of experimental data for the adsorption of butanol on activated carbon. The recommended model to represent the breakthrough curves for the two different adsorbents is the model that includes diffusion within the adsorbent particles. For the breakthrough experiments for the adsorption of butanol on activated carbon F-400, it is recommended using a model which accounts for the inner diffusion within the adsorbent particles. It was found that instantaneous or non-instantaneous adsorption models can be used. Best predictions were obtained with fractional order diffusion with instantaneous adsorption. For the adsorption of butanol on activated carbon Norit ROW 0.8, it is recommended using an integer diffusion model with instantaneous adsorption. The gain of using fractional order diffusion equation, given the intensity in computation, was not sufficient to recommend its use.

Thesis (Ph. D.)--Illinois State University, 1995. / Title from title page screen, viewed May 2, 2006. Dissertation Committee: John A. Dossey (chair), Stephen H. Friedberg, Beverly S. Rich, Kenneth Strand, Jane O. Swafford. Includes bibliographical references (leaves 93-97) and abstract. Also available in print.

Partial differential equations (PDEs) are used to describe multi-dimensional physical phenomena. However, some of these phenomena are described by a more general class of systems called fractional systems. Indeed, fractional calculus has emerged as a new tool for modeling complex phenomena thanks to the memory and hereditary properties of fraction derivatives. In this thesis, we explore a class of controllers and estimators that respond to some control and estimation challenges for both PDE and FPDE. We first propose a backstepping controller for the flow control of a first-order hyperbolic PDE modeling the heat transfer in parabolic solar collectors. While backstepping is a well-established method for boundary controlled PDEs, the process is less straightforward for in-domain controllers. One of the main contributions of this thesis is the development of a new integral transformation-based control algorithm for the study of reference tracking problems and observer designs for fractional PDEs using the extended backstepping approach. The main challenge consists of the proof of stability of the fractional target system, which utilizes either an alternative Lyapunov method for time FPDE or a fundamental solution for the error system for reference tracking, and observer design of space FPDE. Examples of applications involving reference tracking of FPDEs are gas production in fractured media and solute transport in porous media. The designed controllers, require knowledge of some system’s parameters or the state. However, these quantities may be not measurable, especially, for space-evolving PDEs. Therefore, we propose a non-asymptotic and robust estimation algorithm based on the so-called modulating functions. Unlike the observers-based methods, the proposed algorithm has the advantage that it converges in a finite time. This algorithm is extended for the state estimation of linear and non-linear PDEs with general non-linearity. This algorithm is also used for the estimation of parameters and disturbances for FPDEs. This thesis aims to design an integral transformation-based algorithm for the control and estimation of PDEs and FDEs. This transformation is defined through a suitably designed function that transforms the identification problem into an algebraic system for non-asymptotic estimation purposes. It also maps unstable systems to stable systems to achieve control goals.

The fractal operators discussed in this dissertation are introduced in the form originally proposed in an earlier book of the candidate, which proves to be very convenient for physicists, due to its heuristic and intuitive nature. This dissertation proves that these fractal operators are the most convenient tools to address a number of problems in condensed matter, in accordance with the point of view of many other authors, and with the earlier book of the candidate. The microscopic foundation of the fractal calculus on the basis of either classical or quantum mechanics is still unknown, and the second part of this dissertation aims at this important task. This dissertation proves that the adoption of a master equation approach, and so of probabilistic as well as dynamical argument yields a satisfactory solution of the problem, as shown in a work by the candidate already published. At the same time, this dissertation shows that the foundation of Levy statistics is compatible with ordinary statistical mechanics and thermodynamics. The problem of the connection with the Kolmogorov-Sinai entropy is a delicate problem that, however, can be successfully solved. The derivation from a microscopic Liouville-like approach based on densities, however, is shown to be impossible. This dissertation, in fact, establishes the existence of a striking conflict between densities and trajectories. The third part of this dissertation is devoted to establishing the consequences of the conflict between trajectories and densities in quantum mechanics, and triggers a search for the experimental assessment of spontaneous wave-function collapses. The research work of this dissertation has been the object of several papers and two books.

本論文會討論有關於二維的耗散準地轉方程，特別是有關於存在性及規律性的問題。有關的討論主要取決於該方程的分數冪。這份論文中將會介紹一些最近有關耗散準地轉方程的結果。 / This paper is discussing about problems in the 2-D Dissipative Quasi-Geostrophic equation, mainly the existence and regularity results, depending on the fractional power. We will introduce the recent results in this topics. / Detailed summary in vernacular field only. / Kwan, Danny. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 65-68). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.4 / Chapter 2 --- Main Results in QG --- p.9 / Chapter 2.1 --- Definitions --- p.9 / Chapter 2.2 --- Subcritical Case (γ>[1/2]) --- p.10 / Chapter 2.3 --- Critical Case (γ=[1/2]) --- p.10 / Chapter 2.4 --- Supercritical Case (γ<[1/2]) --- p.11 / Chapter 3 --- The Proofs of Main Results --- p.12 / Chapter 3.1 --- Some Previous Results --- p.12 / Chapter 3.2 --- Subcritical Case (γ>[1/2]) --- p.14 / Chapter 3.2.1 --- Proof of Theorem 1 --- p.14 / Chapter 3.2.2 --- Proof of Theorem 2 --- p.19 / Chapter 3.2.3 --- Proof of Corollary 1 --- p.25 / Chapter 3.2.4 --- Summary for Subcritical Case --- p.27 / Chapter 3.3 --- Critical Case (γ=[1/2]) --- p.28 / Chapter 3.3.1 --- Proof of Theorem 3 --- p.28 / Chapter 3.3.2 --- Proof of Theorem 4 --- p.36 / Chapter 3.3.3 --- Summary for Critical Case --- p.41 / Chapter 3.4 --- Supercritical Case (γ<[1/2]) --- p.41 / Chapter 3.4.1 --- Proof of Theorem 5 --- p.41 / Chapter 3.4.2 --- Proof of Theorem 6 --- p.50 / Chapter 3.4.3 --- Proof of Theorem 7 --- p.54 / Chapter 3.4.4 --- Summary for Supercritical Case --- p.64 / Chapter 4 --- Further Development --- p.65

Fractional calculus enables the possibility of using real number powers or complex number powers of the differentiation operator. The fundamental connection between fractional calculus and subordination processes is explored and affords a physical interpretation for a fractional trajectory, that being an average over an ensemble of stochastic trajectories. With an ensemble average perspective, the explanation of the behavior of fractional chaotic systems changes dramatically. Before now what has been interpreted as intrinsic friction is actually a form of non-Markovian dissipation that automatically arises from adopting the fractional calculus, is shown to be a manifestation of decorrelations between trajectories. Nonlinear Langevin equation describes the mean field of a finite size complex network at criticality. Critical phenomena and temporal complexity are two very important issues of modern nonlinear dynamics and the link between them found by the author can significantly improve the understanding behavior of dynamical systems at criticality. The subject of temporal complexity addresses the challenging and especially helpful in addressing fundamental physical science issues beyond the limits of reductionism.

Ο κλασματικός λογισμός είναι ο κλάδος της μαθηματικής ανάλυσης που μελετά παραγώγους και ολοκληρώματα κλασματικής τάξης, και επομένως επιτρέπει την διατύπωση κλασματικών διαφορικών εξισώσεων (FDEs). Aν και ο κλασματικός λογισμός εισήχθη για πρώτη φορά από τον Leibniz περισσότερα από 300 χρόνια πριν, εν τούτοις η εφαρμογή του σε προβλήματα της μαθηματικής φυσικής ξεκίνησε τις τελευταίες δεκαετίες. Συγκεκριμένα, η κλασματική ανάλυση ξεκίνησε βρίσκοντας εφαρμογή σε πολλούς τομείς των φυσικών επιστημών και της επιστήμης της μηχανικής, και μόλις το 2009 εισήχθη για πρώτη φορά στον τομέα της φαρμακοκινητικής. Η φαρμακοκινητική είναι η επιστήμη η οποία μελετά την κινητική της απορρόφησης, της κατανομής και της απομάκρυνσης των φαρμάκων, δηλαδή περιγράφει τη χρονική εξέλιξη του φαρμάκου στον ανθρώπινο οργανισμό και χρησιμοποιεί κυρίως διαμερισματικά μοντέλα. Έχει αποδειχθεί ότι συγκεκριμένα είδη φαρμάκων, μετά τη χορήγησή τους στο ανθρώπινο σώμα, ακολουθούν κινητική η οποία περιγράφεται καλύτερα με τη χρήση κλασματικών διαφορικών εξισώσεων. Ο κλασματικός λογισμός και οι εφαρμογές του είναι ένας αναπτυσσόμενος τομέας ενεργούς έρευνας. Σε ό,τι αφορά τη φαρμακοκινητική, πρόκειται για ένα πολλά υποσχόμενο εργαλείο και η αντίστοιχη βιβλιογραφία αυξάνεται ολοένα και περισσότερο. Στην παρούσα εργασία μελετάται η εφαρμογή του κλασματικού λογισμού στη φαρμακοκινητική. Συγκεκριμένα, δίνουμε αναλυτική λύση σε γραμμικά συστήματα κλασματικών διαφορικών εξισώσεων, τα οποία αντιπροσωπεύουν φαρμακοκινητικά μοντέλα που έχουν προκύψει από την έως τώρα βιβλιογραφία. Όλα τα φαρμακοκινητικά μοντέλα που έχουν μελετηθεί δίνουν μόνο αριθμητικές λύσεις. Aυτό που επιχειρείται για πρώτη φορά στην παρούσα εργασία, είναι να δοθούν οι αναλυτικές λύσεις των μοντέλων αυτών, έστω και αν η μορφή τους είναι πολύπλοκη. Αναλυτικότερα, το πρώτο κεφάλαιο της εργασίας περιέχει μια ανασκόπηση των βασικότερων στοιχείων της θεωρίας της κλασματικής ανάλυσης που θα χρησιμοποιήσουμε, όπως: συναρτήσεις Mittag-Leffler, βασικές ιδιότητες αυτών και υπολογισμός μετασχηματισμού Laplace συγκεκριμένων μορφών αυτών των συναρτήσεων, καθώς επίσης και ορισμός του κλασματικού ολοκληρώματος και της κλασματικής παραγώγου συναρτήσεων. Στο δεύτερο κεφάλαιο αναλύεται η σύνδεση της διαμερισματικής ανάλυσης με την φαρμακοκινητική. Στο τρίτο κεφάλαιο περιγράφεται η σύνδεση του κλασματικού λογισμού με τη φαρμακοκινητική, καθώς και οι λόγοι για τους οποίους υπερτερεί η προσέγγιση αυτή έναντι των προσεγγίσεων που χρησιμοποιούνταν έως και το 2009. Tο τέταρτο κεφάλαιο αφορά εφαρμογές του κλασματικού λογισμού, ενώ δίνονται οι αναλυτικές λύσεις των γραμμικών συστημάτων κλασματικών διαφορικών εξισώσεων που προκύπτουν. Ακόμη, στο Παράρτημα Α αναφέρονται κάποια στοιχεία που αφορούν στο ισοζύγιο μάζας, στο Παράρτημα Β δίνονται τα αποτελέσματα και οι γραφικές παραστάσεις των εφαρμογών που μελετήθηκαν στο τέταρτο κεφάλαιο, και, τέλος, στο Παράρτημα Γ δίνονται οι εντολές του Mathematica που χρησιμοποιήθηκαν για την απεικόνιση των αναλυτικών λύσεων. / Fractional calculus is the sector of mathematical analysis that deals with derivatives and fractional order integrals, resulting the derivation of Fractional Differential Equations (FDEs). Fractional calculus was first introduced by Leibniz more than 300 years ago. Nevertheless, its application on mathematical physics problems has just started the last few decades. In particular, fractional analysis started being applied on sciences of physics and mechanics . Furthermore, fractional analysis was introduced in the field of pharmacokinetics only a few years ago (2009). Pharmacokinetics is the science that deals with the kinetics of the absorption, the distribution and the excretion of drugs. In other words, it describes the time course of the drug inside the human body. Pharmacokinetics mostly uses compartmental models . It has been demonstrated that several types of drugs, follow a kinetic operation after entering in the human body, which is better described by Fractional Differential Equations. Fractional calculus and its applications is a developing sector of active research. Pharmacokinetics, in particular, is a promising tool and the corresponding literature is increasingly growing. The present thesis deals with the application of fractional calculus in pharmacokinetics. In particular, we provide an analytical solution in fractional differential equations linear systems, which represent pharmacokinetic models that have emerged of the existing literature. All the pharmacokinetic models that have been studied provide only arithmetical solutions. The new aspect of the present thesis is an attempt to provide the analytical solutions of these models, even if their form is complicated. In more detail, the first chapter of the study contains a review of the most fundamental fractional-analysis-theory elements that we will use, such as: Mittag-Leffler functions, their basic properties, calculation of Laplace transformation for specific forms of these functions, definition of the fractional integral and the fractional derivative of functions. In the second chapter the binding of compartmental analysis with pharmacokinetics is analyzed. In the third chapter the binding of fractional calculus with pharmacokinetics is described, as well as the reasons why this approach is superior to the previous approaches that were used until 2009. The fourth chapter contains applications of fractional calculus. The analytical solutions for the fractional differential equations linear systems that arise are also given. Furthermore, Appendix A includes some elements related to the mass balance, while Appendix B contains the results and graphs of the applications that were studied in the fourth chapter. Finally, Appendix C provides the Mathematica code that were used for the illustration of the analytical solutions.

No description available.

The standard Black-Scholes model is under the assumption of geometric Brownian motion, and the log-returns for Black-Scholes model are independent and Gaussian. However, most of the recent literature on the statistical properties of the log-returns makes this hypothesis not always consistent. One of the ongoing research topics is to nd a better nancial pricing model instead of the Black-Scholes model. In the present work, we concentrate on two typical 1-D option pricing models under the general exponential L evy processes, namely the nite moment log-stable (FMLS) model and the the Carr-Geman-Madan-Yor-eta (CGMYe) model, and we also propose a multivariate CGMYe model. Both the frameworks, and the numerical estimations and simulations are studied in this thesis. In the future work, we shall continue to study the fractional partial di erential equations (FPDEs) of the nancial models, and seek for the e cient numerical algorithms of the American pricing problems. Keywords: fractional partial di erential equation; option pricing models; exponential L evy process; approximate solution.