Дисертації з теми "Nonlocal order"

In this paper, the monotone method is extended to the initial-boundary value problems of nonlocal PDE system of first order, both quasi-monotone and non-monotone. A comparison principle is established, and a monotone scheme is given.

Dans cette thèse, nous nous intéressons à résoudre certains problèmes inverses pour des équations différentielles aux dérivées fractionnaires. Un problème inverse est généralement mal posé. Un problème mal posé est un problème qui ne répond pas à l’un des trois critères de Hadamard pour être bien posé, c’est-à-dire, soit l’existence, l’unicité ou une dépendance continue aux données n'est plus vraie, à savoir, des petits changements dans les données de mesure entraînent des changements indéfiniment importants dans la solution. La plupart des difficultés à résoudre des problèmes mal posés sont causées par l’instabilité de la solution. D’autre part, les équations différentielles fractionnaires deviennent un outil important dans la modélisation de nombreux problèmes de la vie réelle et il y a eu donc un intérêt croissant pour l’étude des problèmes inverses avec des équations différentielles fractionnaires. Le calcul fractionnaire est une branche des mathématiques qui fait référence à l’extension du concept de dérivation classique à la dérivation d’ordre non entier. Calculer une dérivée fractionnaire à un certain moment exige tous les processus précédents avec des propriétés de mémoire. C’est l’avantage principal du calcul fractionnaire d’expliquer les processus associés aux systèmes physiques complexes qui ont une mémoire à long terme et / ou des interactions spatiales à longue distance. De plus, les équations différentielles fractionnaires peuvent nous aider à réduire les erreurs découlant de paramètres négligés dans la modélisation des phénomènes physiques
In this thesis, we are interested in solving some inverse problems for fractional differential equations. An inverse problem is usually ill-posed. The concept of an ill-posed problem is not new. While there is no universal formal definition for inverse problems, Hadamard [1923] defined a problem as being ill-posed if it violates the criteria of a well-posed problem, that is, either existence, uniqueness or continuous dependence on data is no longer true, i.e., arbitrarily small changes in the measurement data lead to indefinitely large changes in the solution. Most difficulties in solving ill-posed problems are caused by solution instability. Inverse problems come into various types, for example, inverse initial problems where initial data are unknown and inverse source problems where the source term is unknown. These unknown terms are to be determined using extra boundary data. Fractional differential equations, on the other hand, become an important tool in modeling many real-life problems and hence there has been growing interest in studying inverse problems of time fractional differential equations. The Non-Integer Order Calculus, traditionally known as Fractional Calculus is the branch of mathematics that tries to interpolate the classical derivatives and integrals and generalizes them for any orders, not necessarily integer order. The advantages of fractional derivatives are that they have a greater degree of flexibility in the model and provide an excellent instrument for the description of the reality. 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, i.e., calculating timefractional derivative at some time requires all the previous processes with memory and hereditary properties

Dans cette thèse, nous nous proposons d'étudier et de traiter conjointement plusieurs problèmes phares en traitement d'images incluant le recalage d'images qui vise à apparier deux images via une transformation, la segmentation d'images dont le but est de délimiter les contours des objets présents au sein d'une image, et la décomposition d'images intimement liée au débruitage, partitionnant une image en une version plus régulière de celle-ci et sa partie complémentaire oscillante appelée texture, par des approches variationnelles locales et non locales. Les relations étroites existant entre ces différents problèmes motivent l'introduction de modèles conjoints dans lesquels chaque tâche aide les autres, surmontant ainsi certaines difficultés inhérentes au problème isolé. Le premier modèle proposé aborde la problématique de recalage d'images guidé par des résultats intermédiaires de segmentation préservant la topologie, dans un cadre variationnel. Un second modèle de segmentation et de recalage conjoint est introduit, étudié théoriquement et numériquement puis mis à l'épreuve à travers plusieurs simulations numériques. Le dernier modèle présenté tente de répondre à un besoin précis du CEREMA (Centre d'Études et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement) à savoir la détection automatique de fissures sur des images d'enrobés bitumineux. De part la complexité des images à traiter, une méthode conjointe de décomposition et de segmentation de structures fines est mise en place, puis justifiée théoriquement et numériquement, et enfin validée sur les images fournies
In this thesis, we study and jointly address several important image processing problems including registration that aims at aligning images through a deformation, image segmentation whose goal consists in finding the edges delineating the objects inside an image, and image decomposition closely related to image denoising, and attempting to partition an image into a smoother version of it named cartoon and its complementary oscillatory part called texture, with both local and nonlocal variational approaches. The first proposed model addresses the topology-preserving segmentation-guided registration problem in a variational framework. A second joint segmentation and registration model is introduced, theoretically and numerically studied, then tested on various numerical simulations. The last model presented in this work tries to answer a more specific need expressed by the CEREMA (Centre of analysis and expertise on risks, environment, mobility and planning), namely automatic crack recovery detection on bituminous surface images. Due to the image complexity, a joint fine structure decomposition and segmentation model is proposed to deal with this problem. It is then theoretically and numerically justified and validated on the provided images

Equações diferenciais de quarta ordem aparecem naturalmente na modelagem de oscilações de estruturas elásticas, como aquelas observadas em pontes pênseis. São considerados dois modelos que descrevem as oscilações no tabuleiro de uma ponte. No modelo unidimensional estudamos blow up em espaço finito de soluções de uma classe de equações diferenciais de quarta ordem. Os resultados apresentados solucionam uma conjectura apresentada em [F. Gazzola and R. Pavani. Wide oscillation finite time blow up for solutions to nonlinear fourth order differential equations. Arch. Ration. Mech. Anal., 207(2):717752, 2013] e implicam a não existência de ondas viajantes com baixa velocidade de propagação em uma viga. No modelo bidimensional analisamos uma equação não local para uma placa longa e fina, suportada nas extremidades menores, livre nas demais e sujeita a protensão. Provamos existência e unicidade de solução fraca e estudamos o seu comportamento assintótico sob amortecimento viscoso. Estudamos ainda a estabilidade de modos simples de oscilação, os quais são classificados como longitudinais ou torcionais.
Fourth order differential equations appear naturally when modeling oscillations in elastic structures such as those observed in suspension bridges. Two models describing oscillations in the roadway of a bridge are considered. In the one-dimensional model we study finite space blow up of solutions for a class of fourth order differential equations. The results answer a conjecture presented in [F. Gazzola and R. Pavani. Wide oscillation finite time blow up for solutions to nonlinear fourth order differential equations. Arch. Ration. Mech. Anal., 207(2):717752, 2013] and imply the nonexistence of beam oscillation given by traveling wave profile with low speed propagation. In the two-dimensional model we analyze a nonlocal equation for a thin narrow prestressed rectangular plate where the two short edges are hinged and the two long edges are free. We prove existence and uniqueness of weak solution and we study its asymptotic behavior under viscous damping. We also study the stability of simple modes of oscillations which are classified as longitudinal or torsional.

La motivation principale de cette thèse est l'étude du comportement en temps grand des solutions non-bornées d'équations de Hamilton-Jacobi visqueuses dans RN en présence d'un terme d'Ornstein-Uhlenbeck. Nous considérons la même question dans le cas d'une équation de Hamilton-Jacobi du premier ordre. Dans le premier cas, qui constitue le cœur de la thèse, nous généralisons les résultats de Fujita, Ishii et Loreti (2006) dans plusieurs directions. La première est de considérer des opérateurs de diffusion plus généraux en remplaçant le Laplacien par une matrice de diffusion quelconque. Nous considérons ensuite des opérateurs non-locaux intégro-différentiels de type Laplacien fractionnaire. Le second type d'extension concerne le Hamiltonien qui peut dépendre de x et est seulement supposé sous-linéaire par rapport au gradient
The main aim of this thesis is to study large time behavior of unbounded solutions of viscous Hamilton-Jacobi equations in RN in presence of an Ornstein-Uhlenbeck drift. We also consider the same issue for a first order Hamilton-Jacobi equation. In the first case, which is the core of the thesis, we generalize the results obtained by Fujita, Ishii and Loreti (2006) in several directions. The first one is to consider more general operators. We first replace the Laplacian by a general diffusion matrix and then consider a non-local integro-differential operator of fractional Laplacian type. The second kind of extension is to deal with more general Hamiltonians which are merely sublinear

Des mesures de magnétorésistances locales et non locales dans des monocouches de graphène obtenues par sublimation sur la face silicium du carbure de silicium (SiC) sont présentées dans cette thèse. L’objectif est d’étudier les phénomènes physiques qui apparaissent proche de point de neutralité de charge (dopage faible en trous) dans ces monocouches. Or, celles-ci sont généralement fortement dopées en électrons à cause de l’interaction avec la couche d’interface et le substrat. Des dispositifs en forme de barre de Hall encapsulés par une résine sont utilisés. Le contrôle du niveau de Fermi dans ces dispositifs est réalisé en utilisant la méthode de décharge corona. L’amplitude du désordre est évaluée dans ces monocouches de graphène en : (i) ajustant la courbe de résistivité en fonction du coefficient de Hall obtenue à température ambiante ; (ii) ajustant les courbes de dépendance en température de la densité de Hall pour les échantillons proche de point de neutralité de charge. Toutes ces analyses donnent une amplitude du désordre de l’ordre de (20 ±10) meV. Les échantillons préparés avec un faible dopage en trous sont ensuite étudiés en régime d'effet Hall quantique. Les mesures de magnétorésistances montrent que la résistance de Hall présente un comportement ambipolaire en fonction du champ magnétique. Ce comportement coïncide avec l’apparition d’un maximum local dans la résistance longitudinale. Ces résultats sont expliqués via un modèle de transfert de charge entre régions de différents dopages dans le graphène. Néanmoins, l’origine microscopique de ces régions est mal connue. Finalement, des mesures non locales sont effectuées sur ces mêmes échantillons et montrent l’apparition des résistances non locales importantes dont la valeur peut, dans certains cas, dépasser les résistances longitudinales correspondantes. L'analyse de ces résultats montre que la contribution du courant de spin et des effets thermiques dans l’apparition de ces tensions non locales est négligeable. Cependant, les données expérimentales sont raisonnablement reproduites par un modèle de conduction basé sur des états de bord rétrodiffusés par le "bulk" isolant
Local and nonlocal magnetoresistances measurements on monolayer graphene grown on the silicon face of silicon carbide (SiC) are reported. The purpose of this work is to understand the physical phenomena appearing close to the charge neutrality point in these monolayers. The first issue to overcome was that graphene is generally strongly doped with electrons due to the interaction with the substrate. The control of the Fermi level has been realised using the corona discharge method. The disorder amplitude has been evaluated in these structures by : (i) fitting the resistivity dependence curve of the Hall coefficient obtained at room temperature; (ii) fitting the temperature dependence of the Hall density for samples that were prepared near the charge neutrality point. All these analyses gave a disorder strength equal to (20 ± 10) meV. It is then shown that for samples with low hole doping, the Hall resistance exhibits an ambipolar behavior as a function of the magnetic field. This behavior is accompanied by the appearance of a local maximum in the longitudinal resistance.This behavior is been explained by a charge transfer model between regions of different doping in graphene. Nevertheless, the microscopic origin of these regions is poorly known. Finally, nonlocal measurements carried out on these samples showed the appearance of important nonlocal resistances which in some cases exceed the corresponding longitudinal resistances. The analysis of these results shows that the contribution of spin current and thermal effects on the occurrence of these nonlocal voltages is neglegible. In contrast, the experimental data are reproduced quite well by a model based on counter-propagating edge states backscattered by the bulk

This dissertation investigates whether higher-statistical-order nonlocal turbulence-closure can be utilized to describe some of the complex features of atmospheric turbulent flows. Transilient turbulence theory, a nonlocal forcing and mixing model, was chosen as a framework. This theory describes turbulent transport as a two-dimensional 'transilient' matrix of exchange coefficients between different layers in the fluid. A new higher-statistical-order nonlocal parameterization of the transilient matrix is proposed. The studies of former parameterizations of transilient matrices have indicated the following main deficiencies: lack of asymmetry of upward and downward motions in convective boundary layers, improper vertical distribution of turbulent mixing, ad hoc definition of diagonal elements of transilient matrices, and unrealistically large ranges of turbulent transport. This dissertation proposes a parameterization that addresses these deficiencies. The following concepts are utilized in the new parameterization: a virtual turbulent-transport eddy, convective potential-energy and shear potential-energy of mean-flow instabilities, decomposition of forcing into symmetric and asymmetric components, prognostic equations for the nonlocal turbulence kinetic energy budget and for the 3rd statistical moment of the vertical velocity, a limited turbulent-transport range, and a transilient tendency-matrix as a precursor of the transilient matrix. The parameterization utilizes eight parameters. A testbed atmospheric boundary-layer model is created to investigate, evaluate, and compare the new closure with three other turbulence-closure models and with two large-eddy-simulation data sets. The evaluation of testbed model led to the following conclusions. The parameterization of asymmetry qualitatively predicts the overall asymmetry of turbulent transport. An improper vertical distribution of mixing in convective boundary layers is evident, caused by improper height and scale-wise structure of buoyancy forcing. The utilization of the new transilient tendencymatrix successfully removed the necessity to parameterize the diagonal elements of transilient matrices. The limited transport-range approximation causes sensitivity discretization. Finally, the parameterization is computationally expensive, mainly due to the cost of solving the prognostic equations. In summary, the new parameterization has demonstrated the potential to improve turbulence parameterization, but more work needs to be done.
Science, Faculty of
Earth, Ocean and Atmospheric Sciences, Department of
Graduate

The study of second order ordinary differential equations is vital given their proliferation in mechanics. The group theoretic approach devised by Lie is one of the most successful techniques available for solving these equations. However, many second order equations cannot be reduced to quadratures due to the lack of a sufficient number of point symmetries. We observe that increasing the order will result in a third order differential equation which, when reduced via an alternate symmetry, may result in a solvable second order equation. Thus the original second order equation can be solved. In this dissertation we will attempt to classify second order differential equations that can be solved in this manner. We also provide the nonlocal transformations between the original second order equations and the new solvable second order equations. Our starting point is third order differential equations. Here we concentrate on those invariant under two- and three-dimensional Lie algebras.
Thesis (M.Sc.)-University of Natal, Durban, 2000.

碩士
國立臺北教育大學
數學暨資訊教育學系(含數學教育碩士班)
96
We consider the existence of positive solutions of a nonlinear n-th order boundary value problem. In particular, we establish the existence of at least one positive solution if f is “superlinear” by applying the fixed point theorem in cones due to Krasnoselkiˇı and Guo.