S pokroky v teorii počtu neceločíselného řádu a také s rozšířením inženýrských aplikací systémů neceločíselného řádu byla značná pozornost věnována analogové implementaci integrátorů a derivátorů neceločíselného řádu. Je to dáno tím, že tento mocný matematický nástroj nám umožňuje přesněji popsat a modelovat fenomén reálného světa ve srovnání s klasickými „celočíselnými“ metodami. Navíc nám jejich dodatečný stupeň volnosti umožňuje navrhovat přesnější a robustnější systémy, které by s konvenčními kondenzátory bylo nepraktické nebo nemožné realizovat. V předložené disertační práci je věnována pozornost širokému spektru problémů spojených s návrhem analogových obvodů systémů neceločíselného řádu: optimalizace rezistivně-kapacitních a rezistivně-induktivních typů prvků neceločíselného řádu, realizace aktivních kapacitorů neceločíselného řádu, analogová implementace integrátoru neceločíselného řádů, robustní návrh proporcionálně-integračního regulátoru neceločíselného řádu, výzkum různých materiálů pro výrobu kapacitorů neceločíselného řádu s ultraširokým kmitočtovým pásmem a malou fázovou chybou, možná realizace nízkofrekvenčních a vysokofrekvenčních oscilátorů neceločíselného řádu v analogové oblasti, matematická a experimentální studie kapacitorů s pevným dielektrikem neceločíselného řádu v sériových, paralelních a složených obvodech. Navrhované přístupy v této práci jsou důležitými faktory v rámci budoucích studií dynamických systémů neceločíselného řádu.

<p>Doktorska disertacija je posvećena re&scaron;avanju Ko&scaron;ijevog problema odabranih klasa frakcionih diferencijalnih jednačina u okviru Kolomboovih prostora uop&scaron;tenih funkcija. U prvom delu disertacije razmatrane su nehomogene evolucione jednačine sa prostorno frakcionim diferencijalnim operatorima reda 0 &lt; &alpha; &lt; 2 i koeficijentima koji zavise od x i t. Ova klasa jednačina je aproksimativno re&scaron;avana, tako &scaron;to je umesto početne jednačine razmatrana aproksimativna jednačina data preko regularizovanih frakcionih izvoda, odnosno, njihovih regularizovanih množitelja. Za re&scaron;avanje smo koristili dobro poznate uop&scaron;tene uniformno neprekidne polugrupe operatora. U drugom delu disertacije aproksimativno su re&scaron;avane nehomogene frakcione evolucione jednačine sa Kaputovim<br />frakcionim izvodom reda 0 &lt; &alpha; &lt; 2, linearnim, zatvorenim i gusto definisanim<br />operatorom na prostoru Soboljeva celobrojnog reda i koeficijentima koji zavise<br />od x. Odgovarajuća aproksimativna jednačina sadrži uop&scaron;teni operator asociran sa polaznim operatorom, dok su re&scaron;enja dobijena primenom, za tu svrhu&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br />u disertaciji konstruisanih, uop&scaron;tenih uniformno neprekidnih operatora re&scaron;enja.<br />U oba slučaja ispitivani su uslovi koji obezbeduju egzistenciju i jedinstvenost<br />re&scaron;enja Ko&scaron;ijevog problema na odgovarajućem Kolomboovom prostoru.</p> / <p>Colombeau spaces of generalized functions. In the firs part, we studied inhomogeneous evolution equations with space fractional differential operators of order 0 &lt; &alpha; &lt; 2 and variable coefficients depending on x and t. This class of equations is solved&nbsp; approximately, in such a way that instead of the originate equation we considered the corresponding approximate equation given by regularized fractional derivatives, i.e. their&nbsp; regularized multipliers. In the solving procedure we used a well-known generalized uniformly continuous semigroups of operators. In the second part, we solved approximately inhomogeneous fractional evolution equations with Caputo fractional derivative of order 0 &lt; &alpha; &lt; 2, linear, closed and densely defined operator in Sobolev space of integer order and variable coefficients depending on x. The corresponding approximate equation&nbsp;&nbsp; is a given by the generalized operator associated to the originate&nbsp; operator, while the solutions are obtained by using generalized uniformly continuous solution operators, introduced and developed for that purpose. In both cases, we provided the conditions that ensure the existence and uniqueness solutions of the Cauchy problem in some Colombeau spaces.</p>

Made available in DSpace on 2016-04-27T16:57:36Z (GMT). No. of bitstreams: 1 Antonio Sergio dos Santos Oliveira.pdf: 3420485 bytes, checksum: a83bd8f0798c199d29cbfdc9affea731 (MD5) Previous issue date: 2015-03-18 / This paper had the objective of making a group of students from the fifth grade to build meaning to the fundamental operatorial rules with fractional numbers by using scientific calculators with fractional representation. With this in mind, we developed a sequence of teaching with four students of a public school located in the suburb of Belém-PA. We based our theories on the Theory of Didactic Situations (TDS) (Teoria das Situações Didáticas-TSD) and the Theory of Record of Semiotic Representation and the Didactic Engineering as methodology. The TDS helped us to elaborate, experiment and analyze the results of the sequence, while the Theory of Record of Semiotic Representation helped us to articulate among the figure and numeric records. In the analysis of the activities, we verified that the students after using the scientific calculator, they managed to verbalize and write rules to the addition and subtraction of fractional numbers with the same denominator to a multiplication of any fractional numbers and to the division of fractional numbers that presented not only the numerators but also the multiple denominators. However, the students managed not only by using the calculator to perceive the rules for the addition and division of any fractional numbers. We understand that the calculator allowed the students to search for relations and dot not treat the fractional numbers only as two natural numbers. We also should probably have related to other didactic resources so that they could have perceived the relations between numerators and denominators for the addition and subtraction operations / O presente trabalho teve por objetivo levar um grupo de estudantes do quinto ano do ensino fundamental a construir significado para as regras operatórias fundamentais com números fracionários a partir da utilização de calculadoras científicas com representação fracionária. Com esse intuito, desenvolvemos uma sequência de ensino com quatro alunos de uma escola pública situada na periferia de Belém/PA e como aporte teórico utilizamos a Teoria das Situações Didáticas (TSD) e a Teoria de Registros de Representação Semiótica, e a Engenharia Didática como metodologia. A TSD nos auxiliou na elaboração, experimentação e análise dos resultados da sequência, enquanto a Teoria de Registros de Representação Semiótica na articulação entre registros numéricos e figurais. Na análise das atividades verificamos que os alunos conseguiram, após a utilização da calculadora, verbalizar e escrever regras para a adição e subtração de números fracionários com mesmo denominador, para a multiplicação de quaisquer números fracionários e para a divisão de números fracionários que apresentavam tanto os numeradores, quanto os denominadores múltiplos. No entanto, não conseguiram, apenas utilizando a calculadora perceber as regras para a adição e divisão de números fracionários quaisquer. Entendemos que a calculadora permitiu que os alunos buscassem relações e não tratassem os números fracionários apenas como dois números naturais, mas faltou, provavelmente, relacioná-la a outros recursos didáticos para que percebessem as relações entre numeradores e denominadores para as operações de adição e subtração

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.

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.

This thesis is concerned with the existence and spectral stability of periodic waves in the fractional Korteweg-de Vries (KdV) equation and the fractional modified Korteweg-de Vries (mKdV) equation. We study the existence of periodic travelling waves using various tools such as Green's function for fractional Laplacian operator, Petviashvili fixed point method, and a new variational characterization in which the periodic waves in fractional KdV and fractional mKdV are realized as the constrained minimizers of the quadratic part of the energy functional subject to fixed L3 and L4 norm respectively. This new variational framework allows us to identify the existence region of periodic travelling waves and to derive the criterion for spectral stability of the periodic waves with respect to perturbations of the same period. / Thesis / Doctor of Philosophy (PhD)

This thesis presents a numerical approach based on generalized fractional-order Chebyshev wavelets for solving fractional-order optimal control problems. The exact value of the Riemann– Liouville fractional integral operator of the generalized fractional-order Chebyshev wavelets is computed by applying the regularized beta function. We apply the given wavelets, the exact formula, and the collocation method to transform the studied problem into a new optimization problem. The convergence analysis of the proposed method is provided. The present method is extended for solving fractional-order, distributed-order, and variable-order optimal control problems. Illustrative examples are considered to show the advantage of this method in comparison with the existing methods in the literature.

[ES] El cálculo fraccionario es una extensión del cálculo clásico, donde el orden de las derivadas o integrales es un número real. Hoy en día, el cálculo fraccionario tiene numerosas aplicaciones en ciencias e ingeniería. La principal razón es el mayor grado de libertad de las herramientas del cálculo fraccionario en comparación con las herramientas del cálculo clásico. Muchos problemas reales se modelan por medio de ecuaciones diferenciales fraccionarias no lineales cuyo sistema de ecuaciones es no lineal, y por tanto, es conveniente que se adapten procedimientos iterativos para resolver problemas no lineales con el uso de derivadas fraccionarias, y observar cuál es la consecuencia en la convergencia de dicho método. En esta Tesis Doctoral diseñamos nuevos procedimientos iterativos con derivadas fraccionarias (o su aproximación) que al menos igualen a los métodos clásicos en términos de orden de convergencia, mediante la introducción de las derivadas fraccionarias de Riemann-Liouville, de Caputo y conformable (o sus aproximaciones). También, proponemos estudiar la estabilidad de estos esquemas con el uso de planos de convergencia, y planos dinámicos en algunos casos. Finalmente, pretendemos diseñar una técnica que nos permita obtener la versión fraccionaria conformable (o versión con derivada conformable o su aproximación) de cualquier procedimiento iterativo clásico para problemas no lineales. En el Capítulo 2 se exponen los conceptos previos que serán necesarios para el desarrollo de los siguientes capítulos: Se presentan los conceptos básicos relacionados con métodos de punto fijo, se muestran los esquemas clásicos que trataremos en esta memoria, y finalmente se introducen las herramientas del cálculo fraccionario que serán necesarias para el diseño de procedimientos iterativos fraccionarios. En el Capítulo 3 se diseñan métodos fraccionarios (o esquemas con derivadas fraccionarias) de tipo Newton-Raphson escalares con las derivadas de Caputo, de Riemann Liouville y la conformable. También diseñamos esquemas fraccionarios de Newton-Raphson escalares de mayor orden. Finalmente, realizamos el análisis de convergencia de dichos procedimientos y estudiamos su estabilidad. En el Capítulo 4 se diseña la versión vectorial del método de Newton-Raphson conformable visto en el Capítulo 3. Antes, es necesario definir nuevos conceptos y establecer nuevos resultados que serán necesarios para el dersarrollo de este esquema. Finalmente, realizamos el análisis de convergencia y estudiamos su estabilidad. En el Capítulo 5 se diseñan procedimientos fraccionarios de tipo Traub escalares con derivadas de Caputo y de Riemann-Liouville. También se diseña una técnica general para obtener la versión fraccionaria conformable escalar de cualquier método clásico, y se usa esta técnica para diseñar algunos esquemas conformables multipunto escalares: de tipos Traub, Chun-Kim, Ostrowski y Chun. Por último, se realiza el análisis de convergencia y se estudia la estabilidad de tales procedimientos. En el Capítulo 6 se diseñan métodos fraccionarios libres de derivadas escalares de tipos Steffensen y Secante (el cual tiene memoria), donde es necesario la aproximación de derivadas conformables. Aquí se usa la técnica general propuesta en el Capítulo 5 para obtener la versión conformable de cada esquema. Finalmente, realizamos el análisis de convergencia y se estudia la estabilidad de dichos procedimientos. En el Capítulo 7 se presentan las conclusiones y líneas futuras de investigación. / [CA] El càlcul fraccionari és una extensió del càlcul clàssic, on l'ordre de les derivades o integrals és un nombre real. Hui dia, el càlcul fraccionari té nombroses aplicacions en ciències i enginyeria. La principal raó és el major grau de llibertat de les eines del càlcul fraccionari en comparació amb les eines del càlcul clàssic. Molts problemes reals es modelen per mitjà d'equacions diferencials fraccionàries no lineals el sistema d'equacions de les quals és no lineal, i per tant, és convenient que s'adapten procediments iteratius per a resoldre problemes no lineals amb l'ús de derivades fraccionàries, i observar quina és la conseqüència en la convergència d'aquest mètode. En aquesta Tesi Doctoral dissenyem nous procediments iteratius amb derivades fraccionàries (o la seua aproximació) que almenys igualen als mètodes clàssics en termes d'ordre de convergència, mitjançant la introducció de les derivades fraccionàries de Riemann-Liouville, de Caputo i conformable (o les seues aproximacions). També, proposem estudiar l'estabilitat d'aquests esquemes amb l'ús de plans de convergència, i plans dinàmics en alguns casos. Finalment, pretenem dissenyar una tècnica que ens permeta obtindre la versió fraccionària conformable (o versió amb derivada conformable o la seua aproximació) de qualsevol procediment iteratiu clàssic per a problemes no lineals. En el Capítol 2 s'exposen els conceptes previs que seran necessaris per al desenvolupament dels següents capítols: Es presenten els conceptes bàsics relacionats amb mètodes de punt fix, es mostren els esquemes clàssics que tractarem en aquesta memòria, i finalment s'introdueixen les eines del càlcul fraccionari que seran necessàries per al disseny de procediments iteratius fraccionaris. En el Capítol 3 es dissenyen mètodes fraccionaris (o esquemes amb derivades fraccionàries) de tipus Newton-Raphson escalars amb les derivades de Caputo, de Riemann Liouville i la conformable. També dissenyem esquemes fraccionaris de Newton-Raphson escalars de major ordre. Finalment, realitzem l'anàlisi de convergència d'aquests procediments i estudiem la seua estabilitat. En el Capítol 4 es dissenya la versió vectorial del mètode de Newton-Raphson conformable vist en el Capítol 3. Abans, és necessari definir nous conceptes i establir nous resultats que seran necessaris per al dersarrollo d'aquest esquema. Finalment, realitzem l'anàlisi de convergència i estudiem la seua estabilitat. En el Capítol 5 es dissenyen procediments fraccionaris de tipus Traub escalars amb derivades de Caputo i de Riemann-Liouville. També es dissenya una tècnica general per a obtindre la versió fraccionària conformable escalar de qualsevol mètode clàssic, i s'usa aquesta tècnica per a dissenyar alguns esquemes conformables multipunt escalars: de tipus Traub, Chun-Kim, Ostrowski i Chun. Finalment, es realitza l'anàlisi de convergència i s'estudia l'estabilitat de tals procediments. En el Capítol 6 es dissenyen mètodes fraccionaris lliures de derivades escalars de tipus Steffensen i Assecant (el qual té memòria), on és necessari l'aproximació de derivades conformables. Ací s'usa la tècnica general proposta en el Capítol 5 per a obtindre la versió conformable de cada esquema. Finalment, realitzem l'anàlisi de convergència i s'estudia l'estabilitat d'aquests procediments. En el Capítol 7 es presenten les conclusions i línies futures d'investigació. / [EN] Fractional calculus is an extension of classical calculus, where the order of the derivatives or integrals is a real number. Today, fractional calculus has numerous applications in science and engineering. The main reason is the higher degree of freedom of the fractional calculus tools compared to the classical calculus tools. Many real problems are modeled by means of nonlinear fractional differential equations whose system of equations is nonlinear, and therefore it is convenient that iterative procedures are adapted to solve nonlinear problems with the use of fractional derivatives, and observe what the consequence is in the convergence of said method. In this Doctoral Thesis we design new iterative procedures with fractional derivatives (or their approximation) that are at least equal to the classical methods in terms of convergence order, by introducing the Riemann-Liouville, Caputo and conformable fractional derivatives (or their approximations). Also, we propose to study the stability of these schemes with the use of convergence planes, and dynamic planes in some cases. Finally, we intend to design a technique that allows us to obtain the conformable fractional version (or version with conformable derivative or its approximation) of any classical iterative procedure for nonlinear problems. In Chapter 2 the previous concepts that will be necessary for the development of the following chapters are exposed: The basic concepts related to fixed point methods are presented, the classic schemes that we will deal with in this memory are shown, and finally the tools of the fractional calculus that will be necessary for the design of fractional iterative procedures. In Chapter 3, scalar Newton-Raphson type fractional methods (or schemes with fractional derivatives) are designed with the Caputo, Riemann Liouville and conformable derivatives. We also design higher order scalar Newton-Raphson fractional schemes. Finally, we perform the convergence analysis of these procedures and study their stability. In Chapter 4, the vector version of the conformable Newton-Raphson method seen in Chapter 3 is designed. Before, it is necessary to define new concepts and establish new results that will be necessary for the development of this scheme. Finally, we perform the convergence analysis and study its stability. In Chapter 5, fractional procedures of the scalar Traub type with derivatives of Caputo and Riemann-Liouville are designed. A general technique is also designed to obtain the scalar conformable fractional version of any classical method, and this technique is used to design some scalar multipoint conformable schemes: of Traub, Chun-Kim, Ostrowski and Chun types. Finally, the convergence analysis is carried out and the stability of such procedures is studied. In Chapter 6 free fractional methods of scalar derivatives of Steffensen and Secant types (which has memory) are designed, where the conformable derivatives approximation is necessary. Here we use the general technique proposed in Chapter 5 to obtain the conformable version of each scheme. Finally, we carry out the convergence analysis and the stability of these procedures is studied. In Chapter 7 the conclusions and future lines of research are presented. / Candelario Villalona, GG. (2023). Métodos iterativos fraccionarios para la resolución de ecuaciones y sistemas no lineales: diseño, análisis y estabilidad [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/194270

This thesis scrutinizes the condensed matter physics phenomenon known as the fractional quantum Hall effect (FQHE), in particular fractional quantum Hall effects occurring in multicomponent systems. Broadly speaking, the FQHE can be defined as a many-electron quantum phenomenon, driven by strong interactions, that occurs in two-dimensional electron gasses in the presence of a perpendicular external magnetic field (and it is also predicted to occur for any two-dimensional particles, such as confined cold atoms, in an external gauge field). Multicomponent systems are systems where the constituent particles (such as electrons or cold atoms) possess internal degrees of freedom, for instance a spin or valley index. These internal degrees of freedom are often overlooked when modeling the FQHE. Taking into account the multicomponent degree of freedom yields an abundance of possibilities for the intellection of new types of so-called “topological phases of matter”, which are ubiquitously associated with the FQHE. In this thesis several different cases are considered. The first topic discussed herein is a study of phase transitions that can take place between FQHE phases with different net values of their multicomponent degrees of freedom. Examples are phase transitions between phases of different uniform net spin polarization, tunable as a function of certain system parameters. Some significant technical refinements are made to a previous model and comparisons are made with a variety of different experiments. The results are relevant for multicomponent FQHEs occurring in GaAs,AlAs and SiGe semiconductor systems where the electronic structure is confined to two dimensions, as well as in two-dimensional materials such as graphene. The second topic discussed herein is the introduction of the multiparticle multicomponent pseudopotential formalism. This methodology is oriented towards considerably expanding an existing framework for the construction of exactly solvable FQHE models by parameterizing multicomponent interactions. The final topic is the first example application of this new formalism to the construction of an exactly solvable FQHE model.

This dissertation investigates the consequences of fractional dynamics for political modeling. Using Monte Carlo analyses, Chapters II and III investigate the threats to statistical inference posed by including fractionally integrated variables in bivariate and multivariate regressions. Fractional differencing is the most appropriate tool to guard against spurious regressions and other threats to inference. Using fractional differencing, multivariate models of British politics are developed in Chapter IV to compare competing theories regarding which subjective measure of economic evaluations best predicts support levels for the governing party; egocentric measures outperform sociotropic measures. The concept of fractional cointegration is discussed and the value of fractionally integrated error correction mechanisms are both discussed and demonstrated in models of Conservative party support. In Chapter V models of presidential approval in the United States are reconfigured in light of the possibilities of fractionally integrated variables. In both the British and American case accounting for the fractional character of all variables allows the development of more accurate multivariate models.