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.

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

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

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

In this dissertation, a new numerical method for solving fractional calculus problems is presented. The method is based upon the fractional Taylor vector approximations. The operational matrix of the fractional integration for the fractional Taylor vector is introduced. This matrix is then utilized to reduce the solution of the fractional calculus problems to the solution of a system of algebraic equations. This method is used to solve fractional differential equations, Bagley-Torvik equations, fractional integro-differential equations, and fractional duffing problems. Illustrative examples are included to demonstrate the validity and applicability of this technique.

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

Orientadores: Edmundo Capelas de Oliveira, Ary Orozimbo Chiacchio / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-12T21:42:46Z (GMT). No. of bitstreams: 1 Camargo_RubensdeFigueiredo_D.pdf: 9956358 bytes, checksum: 45d7b7d76ae44d9b713d341ffc7a1ad5 (MD5) Previous issue date: 2009 / 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

AN ABSTRACT OF THE DISSERTATION OFSaleh Aljurbua, for the Doctor of Philosophy degree in APPLIED MATHEMATICS, presented on January 27th, 2021, at Southern Illinois University Carbondale. TITLE: THE EXISTENCE OF SOLUTIONS FOR FRACTIONAL DIFFERENTIAL EQUATIONS FOR ORDER q ∈ (n − 1, n], n ∈ N, WITH ANTIPERIODIC BOUNDARY CONDITIONS MAJOR PROFESSOR: Dr. Mingqing Xiao Differential equations play a major role in natural science, physics and technology. Fractional differential equations (FDE) gained a lot of popularity in the past three decades and they became very important in economics, physics and chemistry. In fact, fractional integrals and derivatives became essential and made a significant contribution in dynamical systems which simulate it. They fill the gaps between the integer-types of integrations and derivatives in the classical settings. This work consists of four Chapters. The first Chapter will be covering background, preliminary and fundamental tools used in our dissertation topic. The second Chapter consists of the existence of solutions for nonlinear fractional differential equations of some specific orders with antiperiodic boundary conditions followed by the main topic which is the existence of solutions for nonlinear fractional differential equations of order q ∈ (n−1, n], n ∈ N with antiperiodic boundary conditions of a continuous function f(t, x(t)). Moreover, definitions, theorems and some lemmas will be provided. v In the third Chapter, we offer some examples to illustrate our approach in the main topic. Finally, the fourth Chapter includes the summary and perspective researches.

Motivated by the huge success of the applications of the abstract fractional equations in many areas of science and engineering, and by the unsolved question in this theory, in this work we study several matters related to abstract fractional Cauchy problems of order \'alpha\' \'it belongs\' (0, 1). We search to answer some questions that were open: for instance, we analyze the existence of local mild solutions for the problem, and its possible continuation to a maximal interval of existence. The case of critical nonlinearities and corresponding regular mild solutions is also studied. Finally, by establishing some general comparison results, we apply them to conclude the global well-posedness of a fractional partial differential equation coming from heat conduction theory / Motivados pelo êxito das aplicações nas equações abstratas em muitas áreas da ciência e da engenharia, e pelas perguntas ainda abertas, neste trabalho estudamos questões relativas aos problemas fracionários abstratos de Cauchy de ordem \'alpha\' \'pertence a\' (0, 1). Buscamos responder algumas perguntas: por exemplo, analisamos a existência de soluções locais fracas do problema e sua possível continuação em um intervalo maximal de existência. O caso da não-linearidade crítica e sua correspondente solução regular fraca também é abordado. Por último, mediante o estabelecimento de alguns resultados gerais de comparação, chegamos a conclusão de que as soluções de uma equação diferencial parcial fracionária, proveniente da teoria de condução de calor, existe globalmente

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.