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.

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.

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.

Submitted by LUCAS KENJY BAZAGLIA KURODA null (lucaskuroda@ibb.unesp.br) on 2016-04-15T18:44:12Z No. of bitstreams: 1 Dissertacao_Lucas.pdf: 6780478 bytes, checksum: 57d28e20a68fa0c65cd5e519b8657401 (MD5) / Approved for entry into archive by Felipe Augusto Arakaki (arakaki@reitoria.unesp.br) on 2016-04-18T19:59:31Z (GMT) No. of bitstreams: 1 kuroda_lkb_me_bot.pdf: 6780478 bytes, checksum: 57d28e20a68fa0c65cd5e519b8657401 (MD5) / Made available in DSpace on 2016-04-18T19:59:31Z (GMT). No. of bitstreams: 1 kuroda_lkb_me_bot.pdf: 6780478 bytes, checksum: 57d28e20a68fa0c65cd5e519b8657401 (MD5) Previous issue date: 2016-02-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Um dos problemas de saúde mais conhecidos e temidos atualmente e o câncer. Hoje em dia, existem diversos estudos e trabalhos acerca do tratamento e combate desta doença. Nesse sentido, este trabalho utiliza o Cálculo Fracionário (generalização do cálculo usual, com integração e diferenciação de ordens arbitrárias) para descrever o comportamento do número de células tumorais sob a ação do sistema imunológico e do tratamento quimioterápico. Para isso, dividimos a apresentação deste trabalho em três etapas. Primeiramente, é apresentado um estudo do Cálculo Fracionário, suas principais definições, transformadas de Laplace e funções especiais relacionadas. No segundo momento, é apresentado o método "Multi-Step Generalized Differential Transform Method" (MSGDTM), utilizado para resolver sistemas de equações diferenciais fracionárias. Por fim, a versão fracionária de um modelo de dinâmica tumoral e apresentado e discutido. É observado que uma mudança na ordem da derivada fracionária gera uma mudança no comportamento da dinâmica tumoral, apresentada pelo modelo clássico. / One of the most known and feared health problems is cancer. Nowadays, there are several studies and works about the treatment and combat to this disease. In this sense, this work uses the fractional calculus (generalization of the usual calculus, with integration and differentiation of arbitrary orders) to describe the behavior of the number of tumor cells under the action of the immune system and chemotherapy. To do this, we divide the presentation of this work in three stages. First, a study of Fractional Calculus, its main de nitions, Laplace transforms and special functions, are presented. In the second phase, the "Multi-Step Generalized Differential Transform Method" (MSGDTM) is displayed and used to solve fractional differential equations. Finally, the fractional version of a tumor dynamics model is presented and discussed. It is observed that a change in the order of fractional derivative generates a change in behavior of the tumor dynamics, presented by the classical model.

Submitted by Arianne Vellasco Gomes (ariannevellasco@gmail.com) on 2018-06-15T18:52:22Z No. of bitstreams: 1 Arianne_Vellasco_Gomes_TESE_POSMAT_2018.pdf: 4211125 bytes, checksum: 16221f3149817fbc6e4db2f2026f2f14 (MD5) / Approved for entry into archive by Lucilene Cordeiro da Silva Messias null (lubiblio@bauru.unesp.br) on 2018-06-18T17:39:32Z (GMT) No. of bitstreams: 1 gomes_av_dr_bauru.pdf: 3510911 bytes, checksum: 2abe98b4f93107bb6dc267a184ebef70 (MD5) / Made available in DSpace on 2018-06-18T17:39:32Z (GMT). No. of bitstreams: 1 gomes_av_dr_bauru.pdf: 3510911 bytes, checksum: 2abe98b4f93107bb6dc267a184ebef70 (MD5) Previous issue date: 2018-04-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Tópicos fundamentais da estrutura eletrônica de materiais cristalinos, são investigados de forma generalizada mediante o Cálculo Fracionário. São calculadas as bandas de energia, as funções de Bloch e as funções de Wannier, para a equação de Schrödinger fracionária com derivada de Riesz. É apresentado um estudo detalhado do caráter não local desse tipo de derivada fracionária. Resolve-se a equação de Schrödinger fracionária para o modelo de Kronig-Penney e estuda-se os efeitos da ordem da derivada e da intensidade do potencial. Verificou-se que, ao passar da derivada de segunda ordem para derivadas fracionárias, o comportamento assintótico das funções de Wannier muda apreciavelmente. Elas perdem o decaimento exponencial, e exibem um decaimento acentuado em forma de potência. Fórmulas simples foram dadas para as caudas das funções de Wannier. A banda de energia mais baixa mostrou-se estar relacionada ao estado ligado de um único poço quântico. Sua função de onda também apresentou decaimento em lei de potência. As bandas de energia superiores mudam de comportamento em função da intensidade do potencial. No caso inteiro, a largura de cada uma dessas bandas diminui. No caso fracionário, diminui inicialmente e depois volta a aumentar, aproximando-se de um valor infinito à medida que a intensidade do potencial tende ao infinito. O grau de localização das funções de Wannier, expresso pelo desvio padrão da posição, mostra um comportamento similar ao da largura das bandas de energia. Além dos cristais perfeitos a Ciência de Materiais estuda cristais com defeito. Os defeitos são responsáveis por muitas propriedades de interesse tecnológico e podem induzir estados localizados. Neste trabalho, calculado o estado localizado de menor energia no modelo de Kronig-Penney fracionário com defeito, mediante método das transformadas de Fourier e das funções de Wannier. Verificou-se que este estado também decai em forma de lei de potência. / Basics topics on the electronic structure of crystalline materials are investigated in a generalized fashion through Fractional Calculus. The energy bands, the Bloch and Wannier functions for the fractional Schr odinger equation with Riesz derivative are calculated. The non-locality of the Riesz fractional derivative is analyzed. The fractional Schr odinger equation is solved for the Kronig-Penney model and the e ects of the derivative order and the potential intensity are studied. It was shown that moving from the integer to the fractional order strongly a ects the asymptotic behavior of the Wannier functions. They lose the exponential decay, gaining a strong power-law decay. Simple formulas have been given for the tails of the Wannier functions. A close relatim between the lowest energy band and the bound state of a single quantum well was found. The wavefunction of the latter decays as a power law. Higher energy bands change their behavior as the periodic potential gets stronger. In the integer case, the width of each one of those bands decreases. In the fractional case, it initially decreases and then increases. The width approaching a nite value as the strength tends to in nity. The degree of localization of the Wannier functions, as expressed by the position standard deviation, behaves similarly to the width of the energy bands. In addition to perfect crystals, Materials Science studies defective crystals. Defects are responsible for many properties of technological interest and can induce localized states. In this work, the localized state of lowest energy in the fractional Kronig-Penney model with defect is calculated through of the Fourier transform method and the Wannier functions. It was shown that is decays as a power law.

Orientador: Rubens de Figueiredo Camargo / Resumo: Um dos problemas de saúde mais conhecidos e temidos atualmente e o câncer. Hoje em dia, existem diversos estudos e trabalhos acerca do tratamento e combate desta doença. Nesse sentido, este trabalho utiliza o Cálculo Fracionário (generalização do cálculo usual, com integração e diferenciação de ordens arbitrárias) para descrever o comportamento do número de células tumorais sob a ação do sistema imunológico e do tratamento quimioterápico. Para isso, dividimos a apresentação deste trabalho em três etapas. Primeiramente, é apresentado um estudo do Cálculo Fracionário, suas principais definições, transformadas de Laplace e funções especiais relacionadas. No segundo momento, é apresentado o método "Multi-Step Generalized Differential Transform Method" (MSGDTM), utilizado para resolver sistemas de equações diferenciais fracionárias. Por fim, a versão fracionária de um modelo de dinâmica tumoral e apresentado e discutido. É observado que uma mudança na ordem da derivada fracionária gera uma mudança no comportamento da dinâmica tumoral, apresentada pelo modelo clássico. / Abstract: One of the most known and feared health problems is cancer. Nowadays, there are several studies and works about the treatment and combat to this disease. In this sense, this work uses the fractional calculus (generalization of the usual calculus, with integration and differentiation of arbitrary orders) to describe the behavior of the number of tumor cells under the action of the immune system and chemotherapy. To do this, we divide the presentation of this work in three stages. First, a study of Fractional Calculus, its main de nitions, Laplace transforms and special functions, are presented. In the second phase, the "Multi-Step Generalized Differential Transform Method" (MSGDTM) is displayed and used to solve fractional differential equations. Finally, the fractional version of a tumor dynamics model is presented and discussed. It is observed that a change in the order of fractional derivative generates a change in behavior of the tumor dynamics, presented by the classical model. / Mestre

Orientador: Edmundo Capelas de Oliveira / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-16T18:34:10Z (GMT). No. of bitstreams: 1 Oliveira_HeronSilva_M.pdf: 1078106 bytes, checksum: 9eb6e7bdc70150b5e616010bdfc9ab58 (MD5) Previous issue date: 2010 / Resumo: Efetuamos um levantamento histórico concernente ao cálculo integral e diferencial de ordem arbitrária, também conhecido como cálculo de ordem fracionária ou ainda cálculo fracionário, com o intuito de justificar sua importância, nos dias de hoje, a partir de uma audaciosa e profética frase proferida por Leibniz. A partir das várias definições para derivada de ordem arbitrária, em particular, as definições de Riemann, Liouville, Riemann-Liouville, Grünwald-Letnikov, Weyl e Caputo, elucidamos e justificamos a importância de cada uma delas, nas aplicações, quando associadas ao estudo de uma equação diferencial parcial de ordem arbitrária. Justificamos que, para problemas modelados pelas assim chamadas equações diferenciais de ordem arbitrária, o enfoque conforme proposto por Caputo parece ser o mais conveniente / Abstract: We propose a hystorical review associated with the integral and differential calculus of arbitrary order, known as calculus of fractional order or also fractional calculus with the objective to justify its importance nowadays as of an audacious and profetic phrasis said by Leibniz. By means of several definitions associated with the derivative of fractional order, specifically, the definitions of Riemann, Liouville, Riemann-Liouville, Grünwald-Letnikov,Weyl and Caputo, we discuss and justify the importance of each one, in the applications, when associated with the study to the so-called differential equations of arbitrary order. We also justify that the derivative as proposed by Caputo is the most convenient in problems modelled by a fractional differential equation / Mestrado / Mestre em Matemática

Orientador: Edmundo Capelas de Oliveira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T19:21:10Z (GMT). No. of bitstreams: 1 Costa_FelixSilva_D.pdf: 1599119 bytes, checksum: dddbc1cbaa34b9a87f2c20ebcaddd8fa (MD5) Previous issue date: 2011 / Resumo: Neste trabalho é apresentado um estudo sistemático da função H de Fox e aplicações no cálculo fracionário. Inicialmente é feito um estudo da função hipergeométrica e suas possíveis generalizações, logo em seguida é definida a integral de Mellin-Barnes e a função G de Meijer, em conjunto com suas propriedades e seus casos particulares. Depois é definida a função H de Fox, objetivo principal do trabalho, e seu atual campo de aplicação, que é o cálculo fracionário. Finalmente, apresentam-se as aplicações envolvendo a função H de Fox e o cálculo fracionário. Das três aplicações, os dois primeiros resultados correspondem a duas generalizações: uma da equação do telégrafo e a outra da equação de Schrödinger. Enfim, é discutida uma generalização da equação de onda-difusão no caso em que as condições iniciais são periódicas / Abstract: This work presents a systematic study of the Fox H function and its possible applications in fractional calculus. It begins with a study about the hypergeometric function and its possible generalizations; after that, the Mellin-Barnes integral and the Meijer G function are defined and their properties and particular cases are presented. The Fox H function is then defined and its current field of application, fractional calculus, is discussed. In the sequence some applications involving the Fox H function and fractional calculus are presented, which constitute its main results; the two first results involve the telegraph equation and the Schrödinger equation in their generalized sense. Finally, one discusses a generalization of the wave-diffusion equation in the case in which the initial conditions are periodic / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

In this work, we study the time evolution of systems containing a large number of particles interacting via a $1/r$ binary interaction potential, such as Coulombian and self-gravitating systems. In particular, we study the effect on the dynamics of the Holtsmark-Chandrasekhar theory, which describes the static fluctuations of the total force field around the Vlasov mean-field. We derive these effects by developing a new perturbative theory using the fundamental representation of Statistical Mechanics :The BBGKY hierarchy. This leads to a modification of the Vlasov equation by an additional term involving a fractional Laplacian to the power $3/4$ in velocity space, and a fractional iterated time integral of order $1/2$. We show that one of the consequences of this new term for spatially homogeneous systems is the appearance in the velocity distribution of long tails in $v^{-5/2}$. By extension, similar behaviors can be expected for weakly inhomogeneous systems. These long tails correspond to a universal mechanism related to the divergence of the interaction potential in $1/r$. More specifically, they are induced by the long tails of the total force field distribution as described by the Holtsmark-Chandrasekhar theory. Such a result cannot be obtained from theories based on the weak-coupling between particles, which lead to the Vlasov term, and, the Landau collision operator at the next order. We verify numerically these results by means of molecular dynamics simulations. We study the evolution of the velocity distributions for times very short compared to the violent relaxation time. In this particular time regime, we find, as expected, power laws in $v^{alpha}$ for the velocity distribution tail. In particular, when the regularization parameter of the interaction potential tends to $0$, the exponent in the power law indeed tends from below toward the theoretically predicted value $alpha=-5/2$. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Cette thèse concerne l’étude qualitative (existence locale, existence globale, explosion en temps fini) de quelques équations de Schrödinger non-linéaires non-locales. Dans le cas où les solutions explosent en temps fini, l’estimation du temps maximal d’existence des solutions sera présentée. Le chapitre 1 concerne l’étude d’une équation de Schrödinger non-linéaire sur RN. On s’intéresse à l’existence locale d’une solution pour toute condition initiale donnée dans L2(RN). De plus, on montre que la norme-L2 de la solution explose en temps fini T < 1. Les démonstrations reposent essentiellement sur le théorème de point fixe de Banach et les estimations de Strichartz, et aussi sur le choix convenable de la fonction test dans la formulation faible du problème. Dans le chapitre 2, on considère une équation de Schrödinger non-linéaire non-locale en temps, et on démontre que les solutions de notre problème explosent en temps fini ; ensuite on obtient des conditions nécessaires d’existence globale. Finalement, on obtient une borne inférieure du temps maximal d’existence de la solution. Le chapitre 3 porte sur la non-existence de solutions d’une équation de Schrödinger non-linéaire posée dans RN. Dans un premier temps, sous certaines conditions sur la donnée initiale, on montre qu’il n’existe pas de solution faible globale ; puis on donne une estimation du temps maximal d’existence de la solution. Enfin, on établit des conditions d’existence locale, ou globale de l’équation considérée. En plus, on généralise les résultats précédents au cas d’un système 2 _ 2. Le dernier chapitre traite une équation de Schrödinger non-linéaire non-locale en temps sur le groupe de Heisenberg H. En utilisant la méthode de la fonction test, on démontre que l’équation n’admet pas de solution faible globale. De plus, on obtient, sous certaines conditions sur les données initiales, une estimation inférieure du temps maximal d’existence de la solution. / Our objective in this thesis is to study the existence of local solutions, existence global and blow up of solutions at a finite time to some nonlinear nonlocal Schrödinger equations. In the case when a solution blows-up at a finite time T < 1, we obtain an upper estimate of the life span of solutions. In the first chapter, we consider a nonlinear Schrödinger equation on RN. We first prove local existence of solution for any initial condition in L2 space. Then we prove nonexistence of a nontrivial global weak solution. Furthermore, we prove that the L2-norm of the local intime L2-solution blows up at a finite time. The second chapter is dedicated to study an initial value problem for the nonlocal intime nonlinear Schrödinger equation. Using the test function method, we derive a blow-up result. Then based on integral inequalities, we estimate the life span of blowing-up solutions. In the chapter 3, we prove nonexistence result of a space higher-order nonlinear Schrödinger equation. Then, we obtain an upper bound of the life span of solutions. Furthermore, the necessary conditions for the existence of local or global solutions are provided. Next, we extend our results to the 2 _ 2-system. Our method of proof rests on a judicious choice of the test function in the weak formulation of the equation. Finally, we consider a nonlinear nonlocal in time Schrödinger equation on the Heisenberg group. We prove nonexistence of non-trivial global weak solution of our problem. Furthermore, we give an upper bound of the life span of blowing up solutions.