Dissertations / Theses on the topic 'Fractional calculus'

Doutoramento em Matemática e Aplicações
O cálculo de ordem não inteira, mais conhecido por cálculo fracionário, consiste numa generalização do cálculo integral e diferencial de ordem inteira. Esta tese é dedicada ao estudo de operadores fracionários com ordem variável e problemas variacionais específicos, envolvendo também operadores de ordem variável. Apresentamos uma nova ferramenta numérica para resolver equações diferenciais envolvendo derivadas de Caputo de ordem fracionária variável. Consideram- -se três operadores fracionários do tipo Caputo, e para cada um deles é apresentada uma aproximação dependendo apenas de derivadas de ordem inteira. São ainda apresentadas estimativas para os erros de cada aproximação. Além disso, consideramos alguns problemas variacionais, sujeitos ou não a uma ou mais restrições, onde o funcional depende da derivada combinada de Caputo de ordem fracionária variável. Em particular, obtemos condições de otimalidade necessárias de Euler–Lagrange e sendo o ponto terminal do integral, bem como o seu correspondente valor, livres, foram ainda obtidas as condições de transversalidade para o problema fracionário.
The calculus of non–integer order, usual known as fractional calculus, consists in a generalization of integral and differential integer-order calculus. This thesis is devoted to the study of fractional operators with variable order and specific variational problems involving also variable order operators. We present a new numerical tool to solve differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them, an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided. Furthermore, we consider variational problems subject or not to one or more constraints, where the functional depends on a combined Caputo derivative of variable fractional order. In particular, we establish necessary optimality conditions of Euler–Lagrange. As the terminal point in the cost integral, as well the terminal state, are free, thus transversality conditions are obtained.

We shall present a theory of fractional calculus for generalised functions on (0,∞) and use this theory as a basis for extensions to some related areas. In the first section, appropriate spaces of test-functions and generalised functions on (0,∞) are introduced and the properties of operators of fractional calculus obtained relative to these spaces. Applications are given to hypergeometric integral equations, Hankel transforms and dual integral equations of Titchmarsh type. In the second section, the Mellin transform is used to define fractional powers of a very general class of operators. These definitions include standard operators as special cases. Of particular interest are powers of differential operators of Bessel or hyper-Bessel type which are related to integral operators with special functions, notably G-functions, as kernels. In the third section, we examine operators whose Mellin multipliers involve products and/or quotients of Γ-functions. There is a detailed study of the range and invertibility of such operators in weighted L^P-spaces and in appropriate spaces of smooth functions. The Laplace and Stieltjes transforms give two particular examples. In the final section, we show how our theory of fractional calculus on (0,∞) can be used to develop a corresponding theory on IRⁿ in the presence of radial symmetry. In this framework the mapping properties of multidimensional radial integrals and Riesz potentials are obtained very precisely.

Doutoramento em Matemática
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.
We study problems of the calculus of variations and optimal control within the framework of time scales. Specifically, we obtain Euler–Lagrange type equations for both Lagrangians depending on higher order delta derivatives and isoperimetric problems. We also develop some direct methods to solve certain classes of variational problems via dynamic inequalities. In the last chapter we introduce fractional difference operators and propose a new discrete-time fractional calculus of variations. Corresponding Euler–Lagrange and Legendre necessary optimality conditions are derived and some illustrative examples provided.

Fractional calculus is ”the theory of integrals and derivatives of arbitrary order, which unify and generalize the notions of integer-order differentiation and n-fold integration”. The idea of generalizing differential operators to a non-integer order, in particular to the order 1/2, first appears in the correspondence of Leibniz with L’Hopital (1695), Johann Bernoulli (1695), and John Wallis (1697) as a mere question or maybe even play of thoughts. In the following three hundred years a lot of mathematicians contributed to the fractional calculus: Laplace (1812), Lacroix (1812), Fourier (1822), Abel (1823-1826), Liouville (1832-1837), Riemann (1847), Grunwald (1867-1872), Letnikov (1868-1872), Sonin (1869), Laurent (1884), Heaviside (1892-1912), Weyl (1917), Davis (1936), Erde`lyi (1939-1965), Gelfand and Shilov (1959-1964), Dzherbashian (1966), Caputo (1969), and many others. Yet, it is only after the First Conference on Fractional Calculus and its applications that the fractional calculus becomes one of the most intensively developing areas of mathematical analysis. Recently, many mathematicians and applied researchers have tried to model real processes using the fractional calculus. This is because of the fact that the realistic modeling of a physical phenomenon does not depend only on the instant time, but also on the history of the previous time which can be successfully achieved by using fractional calculus. In other words, the nature of the definition of the fractional derivatives have provided an excellent instrument for the modeling of memory and hereditary properties of various materials and processes.
Il calcolo frazionario e` ”the theory of integrals and derivatives of arbitrary order, which unify and generalize the notions of integer-order differentiation and n-fold integration”. L’ idea di generalizzare operatori differenziali ad un ordine non intero, in particolare di ordine 1/2, compare per la prima volta in una corrispondenza di Leibniz con L’Hopital (1695), Johann Bernoulli (1695), e John Wallis (1697), come una semplice domanda o forse un gioco di pensieri. Nei successive trecento anni molti matematici hanno contribuito al calcolo frazionario: Laplace (1812), Lacroix (1812), di Fourier (1822), Abel (1823-1826), Liouville (1832-1837), Riemann (1847), Grunwald (1867-1872), Letnikov (1868-1872), Sonin (1869), Laurent (1884), Heaviside (1892-1912), Weyl (1917), Davis (1936), Erde`lyi (1939-1965), Gelfand e Shilov (1959-1964), Dzherbashian (1966), Caputo (1969), e molti altri. Eppure, è solo dopo la prima conferenza sul calcolo frazionario e le sue applicazioni che questo tema diventa una delle le aree più intensamente studiate dell’analisi matematica. Recentemente, molti matematici e ingegneri hanno cercato di modellare i processi reali utilizzando il calcolo frazionario. Questo a causa del fatto che spesso, la modellazione realistica di un fenomeno fisico non è locale nel tempo, ma dipende anche dalla storia, e questo comportamento può essere ben rappresentato attraverso modelli basati sul calcolo frazionario. In altre parole, la definizione dei derivata frazionaria fornisce un eccellente strumento per la modellazione della memoria e delle proprietà ereditarie di vari materiali e processi.

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.

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.

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.

Fractional calculus has been used as a research tool in the fields of pharmacology, biology, chemistry, and other areas [3]. The main purpose of this thesis is to calculate Green's functions of fractional difference equations, and to model problems in pharmacokinetics. We claim that the discrete fractional calculus yields the best prediction performance compared to the continuous fractional calculus in the application of a one-compartmental model of drug concentration. In Chapter 1, the Gamma function and its properties are discussed to establish a theoretical basis. Additionally, the basics of discrete fractional calculus are discussed using particular examples for further calculations. In Chapter 2, we use these basic results in the analysis of a linear fractional difference equation. Existence of solutions to this difference equation is then established for both initial conditions (IVP) and two-point boundary conditions (BVP). In Chapter 3, Green's functions are introduced and discussed, along with examples. Instead of using Cauchy functions, the technique of finding Green's functions by a traditional method is demonstrated and used throughout this chapter. The solutions of the BVP play an important role in analysis and construction of the Green's functions. Then, Green's functions for the discrete calculus case are calculated using particular problems, such as boundary value problems, discrete boundary value problems (DBVP) and fractional boundary value problems (FBVP). Finally, we demonstrate how the Green's functions of the FBVP generalize the existence results of the Green's functions of DVBP. In Chapter 4, different compartmental pharmacokinetic models are discussed. This thesis limits discussion to the one-compartmental model. The Mathematica FindFit command and the statistical computational techniques of mean square error (MSE) and cross-validation are discussed. Each of the four models (continuous, continuous fractional, discrete and discrete fractional) is used to compute the MSE numerically with the given data of drug concentration. Then, the best fit and the best model are obtained by inspection of the resulting MSE. In the last Chapter, the results are summarized, conclusions are drawn, and directions for future work are stated.

There is evidence that hydrologic systems exhibit memory processes that may be represented by fractional order systems. A new theory is developed in this work that generalises the classical unit hydrograph technique for the rainfall-runoff transformation. The theory is based upon a fractional order linear deterministic systems approach subject to an initial condition and is taken to apply to the entire rainfallstreamflow transformation (i.e. including baseflow). The general equation for a cascade of time-lagged linear reservoirs of fractional order subject to a constant initialisation function is derived, and is shown to be a form of fractional relaxation model. Dooge's (1959) general theory of the instantaneous unit hydrograph is shown to fit within the new theoretical framework. Similarly the relationship to the general storage equation of Chow and Kulandaiswamy (1971) is demonstrated. It is shown that the correct initialisation of cascade models requires a substantial number of initial conditions which may limit the viability of applying them in practice. Consequently, the differential formulation of the classical Nash cascade has been corrected and reinterpreted. The unbounded nature of the solution to the convolution integral form of the single fractional relaxation model is overcome by application of the Laplace transform of the pulse rainfall hyetograph following Wang and Wu (1983). The model parameters are fitted using the genetic algorithm. The fractional order cascade equations are tested for classical rainfall-runoff modelling using a set of 22 events for the River Nenagh. The cascade of 2 unequal fractionalorder reservoirs is shown to converge to that of the integer order case, whilst the cascade of equal reservoirs shows some differences. For the modelling of the total rainfall-streamflow process the single fractional order reservoir model with a constant initialisation function is tested on a selection of events for a range of UK catchment scales (22km^ to 510km ). A rainfall loss model is incorporated to account for infiltration and evapotranspiration. The results show that the new approach is viable for modelling the rainfall-streamflow transformation at the lumped catchment scale, although the parameter values are not constant for a given catchment. Further work is recommended on determining the nature of the initialisation function using field studies to improve the identification of the model parameters on an event-by-event basis.

Almost every theory of mathematics has its discrete counterpart that makes it conceptually easier to understand and practically easier to use in the modeling process of real world problems. For instance, one can take the "difference" of any function, from 1st order up to the n-th order with discrete calculus. However, it is also possible to extend this theory by means of discrete fractional calculus and make n- any real number such that the ½-th order difference is well defined. This thesis is comprised of five chapters that demonstrate some basic definitions and properties of discrete fractional calculus while developing the simplest discrete fractional variational theory. Some applications of the theory to tumor growth are also studied. The first chapter is a brief introduction to discrete fractional calculus that presents some important mathematical functions widely used in the theory. The second chapter shows the main fractional difference and sum operators as well as their important properties. In the third chapter, a new proof for Leibniz formula is given and summation by parts for discrete fractional calculus is stated and proved. The simplest variational problem in discrete calculus and the related Euler-Lagrange equation are developed in the fourth chapter. In the fifth chapter, the fractional Gompertz difference equation is introduced. First, the existence and uniqueness of the solution is shown and then the equation is solved by the method of successive approximation. Finally, applications of the theory to tumor and bacterial growth are presented.

The fractional calculus is a generalisation of the calculus of Newton and Leibniz. The substitution of fractional differential operators in ordinary differential equations substantially increases their modelling power. Fractional differential operators set exciting new challenges to the computational mathematician because the computational cost of approximating fractional differential operators is of a much higher order than that necessary for approximating the operators of classical calculus. 1. We present a new formulation of the fractional integral. 2. We use this to develop a new method for reducing the computational cost of approximating the solution of a fractional differential equation. 3. This method can be implemented with two levels of sophistication. We compare their rates of convergence, their algorithmic complexity, and their weight set sizes so that an optimal choice, for a particular application, can be made. 4. We show how linear multiterm fractional differential equations can be approximated as systems of fractional differential equations of order at most 1.

This thesis presents results in two apparently disparate mathematical fields which can both be examined -- and even united -- by means of pure analysis. Fractional calculus is the study of differentiation and integration to non-integer orders. Dating back to Leibniz, this idea was considered by many great mathematical figures, and in recent decades it has been used to model many real-world systems and processes, but a full development of the mathematical theory remains incomplete. Many techniques for partial differential equations (PDEs) can be extended to fractional PDEs too. Three chapters below cover my results in this area: establishing the elliptic regularity theorem, Malgrange-Ehrenpreis theorem, and unified transform method for fractional PDEs. Each one is analogous to a known result for classical PDEs, but the proof in the general fractional scenario requires new ideas and modifications. Fractional derivatives and integrals are not uniquely defined: there are many different formulae, each of which has its own advantages and disadvantages. The most commonly used is the classical Riemann-Liouville model, but others may be preferred in different situations, and now new fractional models are being proposed and developed each year. This creates many opportunities for new research, since each time a model is proposed, its mathematical fundamentals need to be examined and developed. Two chapters below investigate some of these new models. My results on the Atangana-Baleanu model proposed in 2016 have already had a noticeable impact on research in this area. Furthermore, this model and the results concerning it can be extended to more general fractional models which also have certain desirable properties of their own. Fractional calculus and zeta functions have rarely been united in research, but one chapter below covers a new formula expressing the Lerch zeta function as a fractional derivative of an elementary function. This result could have many ramifications in both fields, which are yet to be explored fully. Zeta functions are very important in analytic number theory: the Riemann zeta function relates to the distribution of the primes, and this field contains some of the most persistent open problems in mathematics. Since 2012, novel asymptotic techniques have been applied to derive new results on the growth of the Riemann zeta function. One chapter below modifies some of these techniques to prove asymptotics to all orders for the Hurwitz zeta function. Many new ideas are required, but the end result is more elegant than the original one for Riemann zeta, because some of the new methodologies enable different parts of the argument to be presented in a more unified way. Several related problems involve asymptotics arbitrarily near a stationary point. Ideally it should be possible to find uniform asymptotics which provide a smooth transition between the integration by parts and stationary phase methods. One chapter below solves this problem for a particular integral which arises in the analysis of zeta functions.

Doutoramento em Matemática
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.
We introduce a discrete-time fractional calculus of variations on the time scales ℤ and (ℎℤ)!. First and second order necessary optimality conditions are established. Some numerical examples illustrating the use of the new Euler— Lagrange and Legendre type conditions are given. We also give new definitions of fractional derivatives and integrals on time scales via the inverse generalized Laplace transform.

The main purpose of this thesis is to define the stability of a system of linear difference equations of the form, ∇y(t) = Ay(t), and to analyze the stability theory for such a system using the eigenvalues of the corresponding matrix A in nabla discrete calculus and nabla fractional discrete calculus. Discrete exponential functions and the Putzer algorithms are studied to examine the stability theorem. This thesis consists of five chapters and is organized as follows. In the first chapter, the Gamma function and its properties are studied. Additionally, basic definitions, properties and some main theorem of discrete calculus are discussed by using particular example. In the second chapter, we focus on solving the linear difference equations by using the undetermined coefficient method and the variation of constants formula. Moreover, we establish the matrix exponential function which is the solution of the initial value problems (IVP) by the Putzer algorithm.

2015 - 2016
The thesis collects the outcomes of the author’s research carried out in the research group Probability Theory and Mathematical Statistics at the Department of Mathematics, University of Salerno, during the doctoral programme “Mathematics, Physics and Applications”. The results are at the interface between Fractional Calculus and Probability Theory. While research in probability and applied fields is now well established and enthusiastically supported, the subject of fractional calculus, i.e. the study of an extension of derivatives and integrals to any arbitrary real or complex order, has achieved widespread popularity only during the past four decades or so, because of its applications in several fields of science, engineering and finance. Moreover, the application of the fractional paradigm to probability theory has been carefully but partially explored over the years, especially from the point of view of stochastic processes. The aim of the thesis is to prove some new theorems at the interface between Mathematical Analysis and Probability Theory, and to study rigorously certain new stochastic processes and statistical models constructed on top of some well-known classical results and then generalized by means of fractional calculus. The dissertation is organized as follows. In Chapter 1 we give an overview about the main ideas that inspire fractional calculus and about the mathematical techniques for dealing with fractional operators and the related special functions and probability distributions. In order to develop certain fractional probabilistic analogues of Taylor’s theorem 1 and mean value theorem, in Chapter 2 we introduce the nth-order fractional equilibrium distribution in terms of the Weyl fractional integral and investigate its main properties. Specifically, we show a characterization result by which the nth-order fractional equilibrium distribution is identical to the starting distribution if and only if it is exponential. The nth-order fractional equilibrium density is then used to prove a fractional probabilistic Taylor’s theorem based on derivatives of Riemann-Liouville type. A fractional analogue of the probabilistic mean value theorem is thus developed for pairs of nonnegative random variables ordered according to the survival bounded stochastic order. We also provide some related results, both involving the normalized moments and a fractional extension of the variance, and a formula of interest to actuarial science. In conclusion, we discuss the probabilistic Taylor’s theorem based on fractional Caputo derivatives. In Chapter 3 we consider a fractional counting process with jumps of integer amplitude 1,2,...,k, whose probabilities satisfy a suitable system of fractional differencedifferential equations. We obtain the moment generating function and the probability law of the resulting process in terms of generalized Mittag-Leffler functions. We also discuss two equivalent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known property of the Poisson process. When k = 2 we also express the distribution of the first-passage time of the fractional counting process in an integral form. We then show that the ratios given by the powers of the fractional Poisson process and of the counting process over their means tend to 1 in probability. In Chapter 4 we propose a generalization of the alternating Poisson process from the point of view of fractional calculus. We consider the system of differential equations governing the state probabilities of the alternating Poisson process and replace the ordinary derivative with a fractional one (in the Caputo sense). This produces a fractional 2-state point process, whose probability mass is expressed in terms of the (two-parameter) Mittag-Leffler function. We then show that it can be recovered also by means of renewal theory arguments. We study the limit state probability, and certain proportions involving the fractional moments of the sub-renewal periods of the process. In order to derive new Mittag-Leffler-like distributions related to the considered process, we then exploit a transformation acting on pairs of stochastically ordered random variables, which is an extension of the equilibrium operator and deserves interest in the analysis of alternating stochastic processes. In Chapter 5 we analyse a jump-telegraph process by replacing the classical exponential distribution of the interarrival times which separate consecutive velocity changes (and jumps) with a generalized Mittag-Leffler distribution. Such interarrival times constitute the random times of a fractional alternating Poisson process. By means of renewal theory-based arguments, we obtain the forward and backward transition 2 densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. We conclude the chapter by investigating the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds. Chapter 6 is dedicated to a stochastic model for competing risks involving the MittagLeffler distribution, inspired by fractional random growth phenomena. We prove the independence between the time to failure and the cause of failure, and investigate some properties of the related hazard rates and ageing notions. We also face the general problem of identifying the underlying distribution of latent failure times when their joint distribution is expressed in terms of copulas and the time transformed exponential model. The special case concerning the Mittag-Leffler distribution is approached by means of numerical treatment. We finally adapt the proposed model to the case of a random number of independent competing risks. This leads to certain mixtures of Mittag-Leffler distributions, whose parameters are estimated through the method of moments for fractional moments. [edited by author]
La tesi raccoglie i risultati dell’attivit`a di ricerca condotta dall’autore nel gruppo di ricerca Calcolo delle Probabilit`a e Statistica Matematica, presso il Dipartimento di Matematica dell’Universita` di Salerno, nell’ambito del Corso di Dottorato in “Matematica, Fisica e Applicazioni”, XXIX ciclo. I risultati si collocano all’interfaccia tra Calcolo delle Probabilit`a e Calcolo Frazionario. Mentre la ricerca in probabilit`a `e oggi ben consolidata e supportata, il calcolo frazionario, cio`e lo studio della possibilita` di generalizzare il calcolo integrale e il calcolo differenziale classici ad un ordine arbitrario, reale o complesso, ha acquisito notevole popolarita` e importanza nel corso degli ultimi quattro decenni, soprattutto in virtu` delle sue applicazioni in numerosi campi delle scienze e dell’ingegneria. Inoltre, le intersezioni tra calcolo delle probabilita` e calcolo frazionario sono state esplorate con attenzione, ma parzialmente, nel corso degli anni, soprattutto dal punto di vista dei processi stocastici. Lo scopo della tesi `e quello di dimostrare alcuni nuovi teoremi che si collocano all’interfaccia tra l’Analisi Matematica e il Calcolo delle Probabilita`, e di studiare con rigore certi nuovi processi stocastici e modelli statistici costruiti a partire da risultati classici ben noti e poi modificati mediante le tecniche del calcolo frazionario. La tesi `e strutturata come segue. Nel primo capitolo si richiamano alcune nozioni di base e le proprieta` dei principali operatori e delle funzioni del calcolo frazionario, l’integrale di Riemann-Liouville, le derivate di Riemann-Liouville e di Caputo, la funzione di Mittag-Leffler. 1 Nel capitolo 2, al fine di ricavare alcuni analoghi probabilistici di tipo frazionario dei teoremi di Taylor e di Lagrange, `e stata introdotta la distribuzione di equilibrio frazionaria di ordine n definita in termini dell’integrale di Weyl e ne sono state indagate le propriet`a principali. In particolare, si dimostra che la distribuzione di equilibrio frazionaria di ordine n costruita a partire da un’assegnata distribuzione di probabilita`, coincide con questa se e solo se essa `e esponenziale. La distribuzione introdotta viene utilizzata per dimostrare una versione frazionaria dei teoremi di Taylor e del valore medio probabilistici, quest’ultimo applicabile a coppie di variabili aleatorie opportunamente ordinate. Inoltre, si forniscono sia risultati che coinvolgono i momenti normalizzati e un’estensione frazionaria della varianza, sia una formula di interesse nelle scienze attuariali. In conclusione, si discute il teorema di Taylor probabilistico basato sulla derivata frazionaria nel senso di Caputo. Nel terzo capitolo `e stata considerata una generalizzazione frazionaria del processo di Poisson con salti di ampiezza arbitraria, esprimendo la legge di probabilita` mediante funzioni di tipo Mittag- Leffler. L’evoluzione del processo `e guidata da equazioni differenziali e alle differenze finite frazionarie. Dopo aver studiato due rappresentazioni equivalenti del processo considerato, particolare attenzione `e stata posta al problema del tempo di primo passaggio, alla determinazione dei tempi di attesa ed a problemi di tipo asintotico. Tra le altre cose, si `e mostrato che il tempo di prima occorrenza di un salto di ampiezza i, i ∈{1,2,...,k}, k ∈N, `e distribuito come il tempo di prima occorrenza di un evento di un processo di Poisson frazionario di parametro λi > 0, generalizzando, quindi, una importante proprieta` valida nel caso classico. Nel quarto capitolo si propone una generalizzazione del processo di Poisson alternante dal punto di vista del calcolo frazionario, ottenuta sostituendo nel sistema di equazioni differenziali che governa la funzione di probabilita` del processo di Poisson alternante la derivata ordinaria con la derivata frazionaria (nel senso di Caputo) o, equivalentemente, mediante argomenti di teoria del rinnovo. La massa di probabilit`a del nuovo processo `e espressa in termini della funzione di Mittag-Leffler con due parametri. Abbiamo studiato il comportamento asintotico delle probabilit`a di stato e alcune proporzioni che coinvolgono i momenti frazionari dei periodi di rinnovo del processo. Infine, sono state ricavate nuove distribuzioni di tipo Mittag-Leffler relative al processo considerato sfruttando una trasformazione agente su coppie di variabili casuali ordinate stocasticamente, che estende l’operatore equilibrio, di interesse per l’analisi di processi stocastici alternanti. Nel Capitolo 5 si studia un processo stocastico unidimensionale che descrive un moto aleatorio caratterizzato dall’alternarsi di due diverse velocit`a in direzioni opposte. Il processo che regola i cambi di velocit`a (e di direzione) `e il processo di Poisson alternante di tipo frazionario studiato nel capitolo 4. In particolare, nell’istante in cui si verifica un evento di tale processo si compie un salto di ampiezza non aleatoria e quindi il cambiamento di direzione. Pertanto, il processo in esame `e una generalizzazione del processo del telegrafo integrato con salti. Le densit`a di transizione in avanti e all’indietro del moto sono espresse come serie uniformemente convergenti 2 di funzioni di Mittag-Leffler. Particolare attenzione `e stata dedicata al caso di salti di ampiezza costante e uguale distribuzione dei tempi di rinnovo. La distribuzione del tempo di primo passaggio attraverso una barriera costante `e espressa in modo implicito. Tuttavia, in alcuni casi `e data la forma esplicita. L’analisi viene eseguita anche mediante un approccio computazionale. Partendo da fenomeni di crescita di tipo frazionario, nel capitolo 6 abbiamo costruito un modello statistico a rischi competitivi che coinvolge la distribuzione di MittagLeffler. Abbiamo dimostrato l’indipendenza tra il tempo e la causa del fallimento, ed abbiamo indagato alcune propriet`a dei tassi di rischio e delle nozioni di invecchiamento relativi. Abbiamo trattato il problema dell’individuazione della distribuzione sottostante dei tempi di guasto latenti quando la loro distribuzione congiunta `e espressa in termini di copule e mediante il modello TTE (Time Transformed Exponential). Il caso particolare riguardante la distribuzione Mittag-Leffler `e stato trattato numericamente. Il modello proposto `e stato adattato al caso di un numero casuale di rischi in competizione indipendenti. Questo porta ad alcune misture di distribuzioni di tipo Mittag-Leffler, i cui parametri sono stati stimati mediante il metodo dei momenti per momenti frazionari. [a cura dell'autore]
XV n.s.

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.

Doutoramento em Matemática
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.
O cálculo das variações e controlo óptimo fraccionais são generalizações das correspondentes teorias clássicas, que permitem formulações e modelar problemas com derivadas e integrais de ordem arbitrária. Devido à carência de métodos analíticos para resolver tais problemas fraccionais, técnicas numéricas são desenvolvidas. Nesta tese, investigamos a aproximação de operadores fraccionais recorrendo a séries de derivadas de ordem inteira e diferenças finitas generalizadas. Obtemos majorantes para o erro das aproximações propostas e estudamos a sua eficiência. Métodos directos e indirectos para a resolução de problemas variacionais fraccionais são estudados em detalhe. Discutimos também condições de optimalidade para diferentes tipos de problemas variacionais, sem e com restrições, e para problemas de controlo óptimo fraccionais. As técnicas numéricas introduzidas são ilustradas recorrendo a exemplos.

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.

This thesis concerns the development of parallel algorithms to solve fractional differential equations using a numerical approach. The methodology adopted is to adapt existing numerical schemes and to develop prototype parallel programs using the MatLab Parallel Computing Toolbox (MPCT). The approach is to build on existing insights from parallel implementation of ordinary differential equations methods and to test a range of potential candidates for parallel implementation in the fractional case. As a consequence of the work, new insights on the use of MPCT for prototyping are presented, alongside conclusions and algorithms for the effective implementation of parallel methods for the fractional calculus. The principal parallel approaches considered in the work include: - A Runge-Kutta Method for Ordinary Differential Equations including the application of an adapted Richardson Extrapolation Scheme - An implementation of the Diethelm-Chern Algorithm for Fractional Differential Equations - A parallel version of the well-established Fractional Adams Method for Fractional Differential Equations - The adaptation for parallel implementation of Lubich's Fractional Multistep Method for Fractional Differential Equations An important aspect of the work is an improved understanding of the comparative diffi culty of using MPCT for obtaining fair comparisons of parallel implementation. We present details of experimental results which are not satisfactory, and we explain how the problems may be overcome to give meaningful experimental results. Therefore, an important aspect of the conclusions of this work is the advice for other users of MPCT who may be planning to use the package as a prototyping tool for parallel algorithm development: by understanding how implicit multithreading operates, controls can be put in place to allow like-for-like performance comparisons between sequential and parallel programs.

This thesis is comprised of two main parts: Monotonicity results on discrete fractional operators and discrete fractional rheological constitutive equations. In the first part of the thesis, we introduce and prove new monotonicity concepts in discrete fractional calculus. In the remainder, we carry previous results about fractional rheological models to the discrete fractional case. The discrete method is expected to provide a better understanding of the concept than the continuous case as this has been the case in the past. In the first chapter, we give brief information about the main results. In the second chapter, we present some fundamental definitions and formulas in discrete fractional calculus. In the third chapter, we introduce two new monotonicity concepts for nonnegative or nonpositive valued functions defined on discrete domains, and then we prove some monotonicity criteria based on the sign of the fractional difference operator of a function. In the fourth chapter, we emphasize the rheological models: We start by giving a brief introduction to rheological models such as Maxwell and Kelvin-Voigt, and then we construct and solve discrete fractional rheological constitutive equations. Finally, we finish this thesis by describing the conclusion and future work.

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.

The aim of this PhD Thesis was to build and develop a stochastic calculus (in particular a stochastic integral) with respect to multifractional Brownian motion (mBm). Since the choice of the theory and the tools to use was not fixed a priori, we chose the White Noise theory which generalizes, in the case of fractional Brownian motion (fBm) , the Malliavin calculus. The first chapter of this thesis presents several notions we will use in the sequel.In the second chapter we present a construction as well as the main properties of stochastic integral with respect to harmonizable mBm.We also give Ito formulas and a Tanaka formula with respect to this mBm. In the third chapter we give a new definition, simplier and generalier of multifractional Brownian motion. We then show that mBm appears naturally as a limit of a sequence of fractional Brownian motions of different Hurst index.We then use this idea to build an integral with respect to mBm as a limit of sum of integrals with respect ot fBm. This being done we particularize this definition to the case of Malliavin calculus and White Noise theory. In this last case we compare the integral hence defined to the one we got in chapter 2. The fourth and last chapter propose a multifractional stochastic volatility model where the process of volatility is driven by a mBm. The interest lies in the fact that we can hence take into account, in the same time, the long range dependence of increments of volatility process and the fact that regularity vary along the time.Using the functional quantization theory in order to, among other things, approximate the solution of stochastic differential equations, we can compute the price of forward start options and then get and plot the implied volatility nappe that we graphically represent.

This thesis combines the concepts of fractional calculus, norms and means of vectors and machine condition monitoring. Fractional calculus is a branch of calculus that studies the concept of generalising differentiation and integration to arbitrary order. It is as old a discipline as classical calculus although not as widely known. Norm is a generalisation of the length or size of a mathematical object and it provides a single positive value to measure it. Machine condition monitoring is a discipline which focuses on extracting information mainly from industrial machines with noninvasive methods during their operation. This information is used to deduce the condition of the machine accurately. In this thesis, the mathematical background for fractional calculus is built rigorously. Main focus is on application of Fourier analysis to fractional calculus. In addition to functions, distributions are utilised in the theory. Novel results are obtained on the continuity properties of some fractional derivatives and also on equivalences of different definitions. An effective numerical algorithm for calculating fractional derivatives and integrals in the frequency domain is presented. The theory and numerical algorithm is utilised in calculating fractional derivatives and integrals of vibration signals. These signals are collected from the front axle of a load haul dumper working underground in the Pyhäsalmi mine. Generalised norms are calculated from the fractional derivatives and integrals and effective values for condition monitoring of the axle are found. Derivatives and integrals of complex order are also shown to change as the condition of the axle deteriorates during the 368 day long measurement period
Differentiaalilaskennan perusoperaatiot derivointi ja integrointi ovat yleistettävissä jopa kompleksilukuasteisiksi. Näitä yleistyksiä tutkitaan fraktionaalisessa analyysissä, joka on löytänyt myös yllättäviä sovelluskohteita fysiikan ja tekniikan aloilta. Näitä ovat esimerkiksi koneiden kunnonvalvonta ja kunnon diagnostikka, joissa teollisuuden koneiden kuntoa seurataan ja tutkitaan niiden elinkaaren aikana niitä vahingoittamatta. Tässä tutkielmassa esittelen fraktionaalisen analyysin perusteita kattavasti. Tutkimukseni tärkeimpiä osa-alueita ovat Fourier-analyysi ja distribuutioteoria. Esittelen uusia tuloksia fraktionaalisten derivaattojen jatkuvuusominaisuuksista sekä eri määritelmien yhtenevyydestä. Lisäksi johdan tehokkaan algoritmin fraktionaalisten derivaattojen ja integraalien numeerista laskentaa varten. Laskenta suoritetaan taajuustasossa, jolloin voidaan hyödyntää myös FFT-algoritmin nopeutta. Tutkielman kokeellisessa osuudessa käsittelen kiihtyvyyssignaaleja, jotka on mitattu Pyhäsalmen kaivoksessa malminlastausajoneuvon etuakselistosta. Signaaleista lasketaan reaaliasteisia derivaattoja ja integraaleja sekä niistä edelleen yleistettyjä vektorinormeja. Laskettujen arvojen joukosta etsitään herkimpiä indikaattoreita lastauslaitteen akseliston kunnolle. Myös kompleksiderivaatat paljastavat havainnollisesti akseliston kunnon heikkenemisen 368 päivää kestävän mittausjakson aikana

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.

This doctoral thesis endeavors to extend probability and statistical models using stochastic differential equations. The described models capture essential features from data that are not explained by classical diffusion models driven by Brownian motion. New results obtained by the author are presented in five articles. These are divided into two parts. The first part involves three articles on statistical inference and simulation of a family of processes related to fractional Brownian motion and Ornstein-Uhlenbeck process, the so-called fractional Ornstein-Uhlenbeck process of the second kind (fOU2). In two of the articles, we show how to simulate fOU2 by means of circulant embedding method and memoryless transformations. In the other one, we construct a least squares consistent estimator of the drift parameter and prove the central limit theorem using techniques from Stochastic Calculus for Gaussian processes and Malliavin Calculus. The second phase of my research consists of two articles about jump market models and arbitrage portfolio strategies for an insider trader. One of the articles describes two arbitrage free markets according to their risk neutral valuation formula and an arbitrage strategy by switching the markets. The key aspect is the difference in volatility between the markets. Statistical evidence of this situation is shown from a sequential data set. In the other one, we analyze the arbitrage strategies of an strong insider in a pure jump Markov chain financial market by means of a likelihood process. This is constructed in an enlarged filtration using Itô calculus and general theory of stochastic processes.
Föreliggande doktorsavhandling strävar efter att utöka sannolikhetsbaserade och statistiska modeller med stokastiska differentialekvationer. De beskrivna modellerna fångar väsentliga egenskaper i data som inte förklaras av klassiska diffusionsmodeller för brownsk rörelse.  Nya resultat, som författaren har härlett, presenteras i fem uppsatser. De är ordnade i två delar. Del 1 innehåller tre uppsatser om statistisk inferens och simulering av en familj av stokastiska processer som är relaterade till fraktionell brownsk rörelse och Ornstein-Uhlenbeckprocessen, så kallade andra ordningens fraktionella Ornstein-Uhlenbeckprocesser (fOU2). I två av uppsatserna visar vi hur vi kan simulera fOU2-processer med hjälp av cyklisk inbäddning och minneslös transformering. I den tredje uppsatsen konstruerar vi en minsta-kvadratestimator som ger konsistent skattning av driftparametern och bevisar centrala gränsvärdessatsen med tekniker från statistisk analys för gaussiska processer och malliavinsk analys.  Del 2 av min forskning består av två uppsatser om marknadsmodeller med plötsliga hopp och portföljstrategier med arbitrage för en insiderhandlare. En av uppsatserna beskriver två arbitragefria marknader med riskneutrala värderingsformeln och en arbitragestrategi som består i växla mellan marknaderna. Den väsentliga komponenten är skillnaden mellan marknadernas volatilitet. Statistisk evidens i den här situationen visas utifrån ett sekventiellt datamaterial. I den andra uppsatsen analyserar vi arbitragestrategier hos en insiderhandlare i en finansiell marknad som förändrar sig enligt en Markovkedja där alla förändringar i tillstånd består av plötsliga hopp. Det gör vi med en likelihoodprocess. Vi konstruerar detta med utökad filtrering med hjälp av Itôanalys och allmän teori för stokastiska processer.

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Manuscript. Paper 5: Manuscript.

In physics, process involving the phenomena of diffusion and wave propagation have great relevance; these physical process are governed, from a mathematical point of view, by differential equations of order 1 and 2 in time. By introducing a fractional derivatives of order $\alpha$ in time, with $0 < \alpha < 1$ or $1 <= \alpha <= 2$, we lead to process in mathematical physics which we may refer to as fractional phenomena; this is not merely a phenomenological procedure providing an additional fit parameter. The aim of this thesis is to provide a description of such phenomena adopting a mathematical approach to the fractional calculus. The use of Fourier-Laplace transform in the analysis of the problem leads to certain special functions, scilicet transcendental functions of the Wright type, nowadays known as M-Wright functions. We will distinguish slow-diffusion processes ($0 < \alpha < 1$) from intermediate processes ($1 <=\alpha <= 2$), and we point out the attention to the applications of fractional calculus in certain problems of physical interest, such as the Neuronal Cable Theory.

The paper deals with a three variable analogues of certain fractional integral operators introduced by M. Saigo. Resides giving three variable analogues of earlier known fractional integral operators of one variable as a special cases of newly defined operators, the paper establishes certain results in the form of theorems including integration by parts.

The main object of this thesis is to obtain numerous applications of fractional derivative operator concerning analytic and ρ-valent (or multivalent) functions in the open unit disk by introducing new classes and deriving new properties. Our finding will provide interesting new results and indicate extensions of a number of known results. In this thesis we investigate a wide class of problems. First, by making use of certain fractional derivative operator, we define various new classes of ρ-valent functions with negative coefficients in the open unit disk such as classes of ρ-valent starlike functions involving results of (Owa, 1985a), classes of ρ-valent starlike and convex functions involving the Hadamard product (or convolution) and classes of κ-uniformly ρ-valent starlike and convex functions, in obtaining, coefficient estimates, distortion properties, extreme points, closure theorems, modified Hadmard products and inclusion properties. Also, we obtain radii of convexity, starlikeness and close-to-convexity for functions belonging to those classes. Moreover, we derive several new sufficient conditions for starlikeness and convexity of the fractional derivative operator by using certain results of (Owa, 1985a), convolution, Jack's lemma and Nunokakawa' Lemma. In addition, we obtain coefficient bounds for the functional |α_ρ+2-θα²_ρ+1| of functions belonging to certain classes of p-valent functions of complex order which generalized the concepts of starlike, Bazilevič and non-Bazilevič functions. We use the method of differential subordination and superordination for analytic functions in the open unit disk in order to derive various new subordination, superordination and sandwich results involving the fractional derivative operator. Finally, we obtain some new strong differential subordination, superordination, sandwich results for ρ-valent functions associated with the fractional derivative operator by investigating appropriate classes of admissible functions. First order linear strong differential subordination properties are studied. Further results including strong differential subordination and superordination based on the fact that the coefficients of the functions associated with the fractional derivative operator are not constants but complex-valued functions are also studied.

The main object of this thesis is to obtain numerous applications of fractional derivative operator concerning analytic and -valent (or multivalent) functions in the open unit disk by introducing new classes and deriving new properties. Our finding will provide interesting new results and indicate extensions of a number of known results. In this thesis we investigate a wide class of problems. First, by making use of certain fractional derivative operator, we define various new classes of -valent functions with negative coefficients in the open unit disk such as classes of -valent starlike functions involving results of (Owa, 1985a), classes of -valent starlike and convex functions involving the Hadamard product (or convolution) and classes of -uniformly -valent starlike and convex functions, in obtaining, coefficient estimates, distortion properties, extreme points, closure theorems, modified Hadmard products and inclusion properties. Also, we obtain radii of convexity, starlikeness and close-to-convexity for functions belonging to those classes. Moreover, we derive several new sufficient conditions for starlikeness and convexity of the fractional derivative operator by using certain results of (Owa, 1985a), convolution, Jack¿s lemma and Nunokakawa¿ Lemma. In addition, we obtain coefficient bounds for the functional of functions belonging to certain classes of -valent functions of complex order which generalized the concepts of starlike, Bazilevi¿ and non-Bazilevi¿ functions. We use the method of differential subordination and superordination for analytic functions in the open unit disk in order to derive various new subordination, superordination and sandwich results involving the fractional derivative operator. Finally, we obtain some new strong differential subordination, superordination, sandwich results for -valent functions associated with the fractional derivative operator by investigating appropriate classes of admissible functions. First order linear strong differential subordination properties are studied. Further results including strong differential subordination and superordination based on the fact that the coefficients of the functions associated with the fractional derivative operator are not constants but complex-valued functions are also studied.

The first objective of this project is to develop new efficient numerical methods and supporting error and convergence analysis for solving fractional partial differential equations to study anomalous diffusion in biological tissue such as the human brain. The second objective is to develop a new efficient fractional differential-based approach for texture enhancement in image processing. The results of the thesis highlight that the fractional order analysis captured important features of nuclear magnetic resonance (NMR) relaxation and can be used to improve the quality of medical imaging.

This work revolves around the use of fractional order calculus in control science. Techniques such as fractional order universal adaptive stabilization (FO-UAS), and the fascinating results of their application to real-world systems, are presented initially. A major portion of this work deals with fractional order modeling and control of real-life systems like heat flow, fan and plate, and coupled tank systems. The fractional order models and controllers are not only simulated, they are also emulated using analog hardware. The main aim of all the above experimentation is to develop a fractional order controller for plasma position control of the Saskatchewan torus-1, modified (STOR-1M) tokamak at the Utah State University (USU) campus. A new method for plasma position estimation has been formulated. The results of hardware emulation of plasma position and its control are also presented. This work performs a small scale test measuring controller performance, so that it serves as a platform for future research efforts leading to real-life implementation of a plasma position controller for the tokamak.

Zlomkový kalkulus je matematická disciplína zabývající se vlastnostmi derivací a integrálů neceločíselných řádů (nazývaných zlomkové derivace a integrály, zkráceně diferintegrály) a metodami řešení diferenciálních rovnic obsahujících zlomkové derivace neznámé funkce (tzv. zlomkovými diferenciálními rovnicemi). V této práci představujeme standardní přístupy k definicím zlomkového kalkulu a důkazy některých základních vlastností diferintegrálů. Dále uvádíme krátký přehled metod řešení některých lineárních zlomkových diferenciálních rovnic a vymezujeme hranice jejich použitelnosti. Na závěr si všímáme některých fyzikálních aplikací zlomkového kalkulu.

The aim of this thesis is to develop discrete fractional models of tumor growth for a given data and to estimate parameters of these models in order to have better data tting. We use discrete nabla fractional calculus because we believe the discrete counterpart of this mathematical theory will give us a better and more accurate outcome. This thesis consists of ve chapters. In the rst chapter, we give the history of the fractional calculus, and we present some basic de nitions and properties that are used in this theory. We de ne nabla fractional exponential and then nabla fractional trigonometric functions. In the second chapter, we concentrate on completely monotonic functions on R, and we introduce completely monotonic functions on discrete domain. The third chapter presents discrete Laplace N-transform table which is a great tool to nd solutions of -th order nabla fractional di erence equations. Furthermore, we nd the solution of nonhomogeneous up to rst order nabla fractional di erence equation using N-transform. In the fourth chapter, rst we give the de nition of Casoration for the set of solutions up to n-th order nabla fractional equation. Then, we state and prove some basic theorems about linear independence of the set of solutions. We focus on the solutions of up to second order nabla fractional di erence equation. We examine these solutions case by case namely, for the real and distinct characteristic roots, real and same, and complex ones. The fth chapter emphasizes the aim of this thesis. First, we give a vi brief introduction to parameter estimation with Gomperts and Logistic curves. In addition, we recall a statistical method called cross-validation for prediction. We state continuous, discrete, continuous fractional and discrete fractional forms of Gompertz and Logistic curves. We use the tumor growth data for twenty-eight mice for the comparison. These control mice were inoculated with tumors but did not receive any succeeding treatment. We claim that the discrete fractional type of sigmoidal curves have the best data tting results when they are compared to the other types of models.

Orientador: Edmundo Capelas de Oliveira
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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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

Orientadores: Edmundo Capelas de Oliveira, Ary Orozimbo Chiacchio
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Apresentamos neste trabalho um estudo sistemático e detalhado sobre integrais e derivadas de ordens arbitrárias, o assim chamado cálculo de ordem não-inteira, popularizado com o nome de Cálculo Fracionário. Em particular, discutimos e resolvemos equações diferenciais e integrodiferenciais de ordem não-inteira e suas aplicações em diversas áreas do conhecimento, bem como apresentamos resultados inéditos, isto é, teoremas de adição, envolvendo as funções de Mittag-Leffler. Após abordar as diferentes definições para a derivada de ordem não-inteira, justificamos o fato de utilizarmos, em nossas aplicações, a definição de derivada conforme proposta por Caputo, mais restritiva, e não a definição segundo Riemann-Liouville, embora seja esta a mais difundida. Nas aplicações apresentamos uma generalização para a equação diferencial associada ao problema do telégrafo na versão fracionária, cuja solução, obtida de duas maneiras distintas, deu origem a dois novos teoremas de adição envolvendo as funções de Mittag-Leffler. Numa segunda aplicação, discutimos o conhecido sistema de Lotka-Volterra na versão fracionária; por fim, introduzimos e resolvemos uma equação integrodiferencial fracionária, a assim chamada, equação de Langevin generalizada fracionária.
Abstract: At this work we present a systematic and detailed study about integrals and derivatives of arbitrary order, the so-called non-integer order calculus, popularized with the name Fractional Calculus. Particularly, we discuss and solve non-integer order differential and integrodifferential equations and its applications into several areas of the knowledge, as well as introduce some new results, i.e., addition theorems, involving the Mittag-Leffler functions. After approaching the different definitions to the non-integer order derivative, we justify the fact that we use, in our applications, the definition proposed by Caputo to the fractional derivative, which is more restrictive, instead of the Riemann-Liouville ones, although this one is best known. Into the applications we presented a fractional generalization to the equation associated with the telegraph's problem, whose solution, obtained by two different ways, was the origin of two new addition theorems to the Mittag-Leffler functions. As a second application, we present the fractional version of the Lotka-Volterra system; finally, we introduce and solve the fractional generalized Langevin equation.
Doutorado
Doutor em Matemática

Heart failure is one of the most common causes of death in the western world. Many heart problems are linked to disturbances in cardiac electrical activity. Further understanding of how electrical impulses propagate through the heart may lead to new diagnosis and treatment options. Using our novel numerical scheme, we are able to conduct preliminary investigations into the effect of fixed and variable order fractional Laplacian operators for modelling propagation of electrical impulses through the heart. We implement our numerical framework to solve the coupled monodomain, Beeler-Reuter model. Preliminary results confirm the effectiveness of our numerical scheme, and pave the way to exciting areas of future research.

This master's thesis deals with fractional differential equations. One of the aims of this thesis is to mention summary of basic types of fractional differential equations. It is very difficult to find their exact solution, hence we will analyze the main qualitative properties of solution, which are stability and asymptotics. Part of the text will be devoted to fractional difference equations, i.e. discussion of numerical solution. At the end of thesis the Bagley-Torvik model will be described with respect to qualitative properties and numerical solution.

Orientador: Edmundo Capelas de Oliveira
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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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: Rubens de Figueiredo Camargo
Resumo: O presente trabalho apresenta a nova modelagem fracionária, que considera propriedades hereditárias e efeitos de memória, no modelo de Gompertz, para descrever a evolução do câncer causado pela infecção do HPV 16. Devido a variabilidade do desenvolvimento do câncer em humanos, utiliza-se o crescimento in vivo do tumor em camundongo transgênico que expressam os oncogenes E6 e E7 tratados com DMBA / TPA (inicializador e promotor do HPV 16) para capturar as características gerais dessa variabilidade. Resultados mostram que a inserção de um novo parâmetro na correção dimensional da modelagem fracionária, descreve, em comparação ao modelo clássico, o progresso do volume tumoral em maior conformidade com os conjuntos de dados reais.
Abstract: The present work presents the fractional modeling, which considers hereditary properties and memory effects, to describe through the Gompertz model, the evolution of cancer caused by HPV 16 infection. Due to the variability of the development of cancer in humans, we used the in vivo growth of the transgenic mouse tumor expressing DMBA / TPA-treated E6 and E7 oncogenes (HPV 16 initiator and promoter) to capture the general characteristics of this variability. Results show that the insertion of a new parameter in the dimensional correction of fractional modeling describes, compared to the classical model, the progress of tumor volume in greater concorda with the actual data sets.
Doutor