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Academic literature on the topic 'Numerical calculu'
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Dissertations / Theses on the topic "Numerical calculu"
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Abdelsheed, Ismail Gad Ameen. "Fractional calculus: numerical methods and SIR models." Doctoral thesis, Università degli studi di Padova, 2016. http://hdl.handle.net/11577/3422267.
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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.<br>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.
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Baladron, Pezoa Javier. "Exploring the neural codes using parallel hardware." Phd thesis, Université Nice Sophia Antipolis, 2013. http://tel.archives-ouvertes.fr/tel-00847333.
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The aim of this thesis is to understand the dynamics of large interconnected populations of neurons. The method we use to reach this objective is a mixture of mesoscopic modeling and high performance computing. The rst allows us to reduce the complexity of the network and the second to perform large scale simulations. In the rst part of this thesis a new mean eld approach for conductance based neurons is used to study numerically the eects of noise on extremely large ensembles of neurons. Also, the same approach is used to create a model of one hypercolumn from the primary visual cortex where the basic computational units are large populations of neurons instead of simple cells. All of these simulations are done by solving a set of partial dierential equations that describe the evolution of the probability density function of the network. In the second part of this thesis a numerical study of two neural eld models of the primary visual cortex is presented. The main focus in both cases is to determine how edge selection and continuation can be computed in the primary visual cortex. The dierence between the two models is in how they represent the orientation preference of neurons, in one this is a feature of the equations and the connectivity depends on it, while in the other there is an underlying map which denes an input function. All the simulations are performed on a Graphic Processing Unit cluster. Thethesis proposes a set of techniques to simulate the models fast enough on this kind of hardware. The speedup obtained is equivalent to that of a huge standard cluster.
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Simpson, Arthur Charles. "Numerical methods for the solution of fractional differential equations." Thesis, University of Liverpool, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.250281.
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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.
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Souza, João Artur de. "Calculo numerico da exponencial de uma matriz." reponame:Repositório Institucional da UFSC, 1993. http://repositorio.ufsc.br/xmlui/handle/123456789/75933.
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Dissertação (mestrado) - Universidade Federal de Santa Catarina. Centro de Ciencias Fisicas e Matematicas<br>Made available in DSpace on 2012-10-16T05:50:28Z (GMT). No. of bitstreams: 0Bitstream added on 2016-01-08T18:29:54Z : No. of bitstreams: 1
93362.pdf: 1944049 bytes, checksum: df7a8432cb7ee0c3cc00c1fd481b723f (MD5)<br>A importância em resolver a Equação Diferencial x'(t) = Ax(t)
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Farah, Jad. "Amélioration des mesures anthroporadiamétriques personnalisées assistées par calcul Monte Carlo : optimisation des temps de calculs et méthodologie de mesure pour l’établissement de la répartition d’activité." Thesis, Paris 11, 2011. http://www.theses.fr/2011PA112183/document.
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Afin d’optimiser la surveillance des travailleuses du nucléaire par anthroporadiamétrie, il est nécessaire de corriger les coefficients d’étalonnage obtenus à l’aide du fantôme physique masculin Livermore. Pour ce faire, des étalonnages numériques basés sur l’utilisation des calculs Monte Carlo associés à des fantômes numériques ont été utilisés. De tels étalonnages nécessitent d’une part le développement de fantômes représentatifs des tailles et des morphologies les plus communes et d’autre part des simulations Monte Carlo rapides et fiables. Une bibliothèque de fantômes thoraciques féminins a ainsi été développée en ajustant la masse des organes internes et de la poitrine suivant la taille et les recommandations de la chirurgie plastique. Par la suite, la bibliothèque a été utilisée pour étalonner le système de comptage du Secteur d’Analyses Médicales d’AREVA NC La Hague. De plus, une équation décrivant la variation de l’efficacité de comptage en fonction de l’énergie et de la morphologie a été développée. Enfin, des recommandations ont été données pour corriger les coefficients d’étalonnage du personnel féminin en fonction de la taille et de la poitrine. Enfin, pour accélérer les simulations, des méthodes de réduction de variance ainsi que des opérations de simplification de la géométrie ont été considérées.Par ailleurs, pour l’étude des cas de contamination complexes, il est proposé de remonter à la cartographie d’activité en associant aux mesures anthroporadiamétriques le calcul Monte Carlo. La méthode développée consiste à réaliser plusieurs mesures spectrométriques avec différents positionnements des détecteurs. Ensuite, il s’agit de séparer la contribution de chaque organe contaminé au comptage grâce au calcul Monte Carlo. L’ensemble des mesures réalisées au LEDI, au CIEMAT et au KIT ont démontré l’intérêt de cette méthode et l’apport des simulations Monte Carlo pour une analyse plus précise des mesures in vivo, permettant ainsi de déterminer la répartition de l’activité à la suite d’une contamination interne<br>To optimize the monitoring of female workers using in vivo spectrometry measurements, it is necessary to correct the typical calibration coefficients obtained with the Livermore male physical phantom. To do so, numerical calibrations based on the use of Monte Carlo simulations combined with anthropomorphic 3D phantoms were used. Such computational calibrations require on the one hand the development of representative female phantoms of different size and morphologies and on the other hand rapid and reliable Monte Carlo calculations. A library of female torso models was hence developed by fitting the weight of internal organs and breasts according to the body height and to relevant plastic surgery recommendations. This library was next used to realize a numerical calibration of the AREVA NC La Hague in vivo counting installation. Moreover, the morphology-induced counting efficiency variations with energy were put into equation and recommendations were given to correct the typical calibration coefficients for any monitored female worker as a function of body height and breast size. Meanwhile, variance reduction techniques and geometry simplification operations were considered to accelerate simulations.Furthermore, to determine the activity mapping in the case of complex contaminations, a method that combines Monte Carlo simulations with in vivo measurements was developed. This method consists of realizing several spectrometry measurements with different detector positioning. Next, the contribution of each contaminated organ to the count is assessed from Monte Carlo calculations. The in vivo measurements realized at LEDI, CIEMAT and KIT have demonstrated the effectiveness of the method and highlighted the valuable contribution of Monte Carlo simulations for a more detailed analysis of spectrometry measurements. Thus, a more precise estimate of the activity distribution is given in the case of an internal contamination
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Kimeu, Joseph M. "Fractional Calculus: Definitions and Applications." TopSCHOLAR®, 2009. http://digitalcommons.wku.edu/theses/115.
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Tavares, Dina dos Santos. "Fractional calculus of variations." Doctoral thesis, Universidade de Aveiro, 2017. http://hdl.handle.net/10773/22184.
Full textAbstract:
Doutoramento em Matemática e Aplicações<br>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.<br>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.
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Banks, Nicola E. "Insights from the parallel implementation of efficient algorithms for the fractional calculus." Thesis, University of Chester, 2015. http://hdl.handle.net/10034/613841.
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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.
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Cairo, Giuseppe. "Curve di Bezier e Calcolo Numerico." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2012. http://amslaurea.unibo.it/4463/.
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La, Monaca Liliana. "Calcolo numerico ed esplorazioni con geogebra." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/8771/.
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Il lavoro di tesi svolto propone diversi argomenti di geometria la cui costruzione è stata fatta con il software GeoGebra. Propone anche alcuni metodi di integrazione numerica realizzati con esso e anche un modo di approssimare la superficie di rotazione di una funzione sfruttando tali metodi. Gli argomenti trattati spaziano da quelli classici della geometria euclidea a temi affrontati più recentemente esaminando sia oggetti rappresentabili sul piano sia nello spazio tridimensionale.