Numerical Approximation of a Fractional-In-Space Diffusion Equation, I

2000 Mathematics Subject Classification: 26A33 (primary), 35S15 (secondary)

Anti-Periodic Boundary Value Problem for Impulsive Fractional Integro Differential Equations

MSC 2010: 34A37, 34B15, 26A33, 34C25, 34K37

Crack Theory with Singularities at the Boundary

2002 Mathematics Subject Classification: 35S15, 35J70, 35J40, 38J40

Two-phase flow analogy as an effective boundary condition for modelling liquids at atomistic resolution

A hybrid Molecular Dynamics/Fluctuating Hydrodynamics framework based on the analogy with two-phase hydrodynamics has been extended to dynamically tracking the feature of interest at all-atom resolution. In the model, the hydrodynamics description is used as an effective boundary condition to close the molecular dynamics solution without resorting to standard periodic boundary conditions. The approach is implemented in a popular Molecular Dynamics package GROMACS and results for two biomolecular systems are reported. A small peptide dialanine and a complete capsid of a virus porcine circovirus 2 in water are considered and shown to reproduce the structural and dynamic properties compared to those obtained in theory, purely atomistic simulations, and experiment.

Buraco negros, correspondência AdS/BCFT e fluido/gravidade

Einstein’s equations with negative cosmological constant possess the so-called anti de Sitter space, AdSd+1, as one of its solutions. We will later refer to this space as to the "bulk". The holographic principle states that quantum gravity in the AdSd+1 space can be encoded by a d−dimensional quantum field theory on the boundary of AdSd+1 space, invariant under conformal transformations, a CFTd. In the most famous example, the precise statement is the duality of the type IIB string theory in the space AdS5 × S 5 and the 4−dimensional N = 4 supersymmetric Yang-Mills theory. Another example is provided by a relation between Einstein’s equations in the bulk and hydrodynamic equations describing the effective theory on the boundary, the so-called fluid/gravity correspondence. An extension of the "AdS/CFT duality"for the CFT’s with boundary was proposed by Takayanagi, which was dubbed the AdS/BCFT correspondence. The boundary of a CFT extends to the bulk and restricts a region of the AdSd+1. Neumann conditions imposed on the extension of the boundary yield a dynamic equation that determines the shape of the extension. From the perspective of fluid/gravity correspondence, the shape of the Neumann boundary, and the geometry of the bulk is sourced by the energy-momentum tensor Tµν of a fluid residing on this boundary. Clarifying the relation of the Takayanagi’s proposal to the fluid/gravity correspondence, we will study the consistence of the AdS/BCFT with finite temperature CFT’s, or equivalently black hole geometries in the bulk.

Advanced thermal analysis of microelectronics using spreading resistance models

Thermal analysis of electronic devices is one of the most important steps for designing of modern devices. Precise thermal analysis is essential for designing an effective thermal management system of modern electronic devices such as batteries, LEDs, microelectronics, ICs, circuit boards, semiconductors and heat spreaders. For having a precise thermal analysis, the temperature profile and thermal spreading resistance of the device should be calculated by considering the geometry, property and boundary conditions. Thermal spreading resistance occurs when heat enters through a portion of a surface and flows by conduction. It is the primary source of thermal resistance when heat flows from a tiny heat source to a thin and wide heat spreader. In this thesis, analytical models for modeling the temperature behavior and thermal resistance in some common geometries of microelectronic devices such as heat channels and heat tubes are investigated. Different boundary conditions for the system are considered. Along the source plane, a combination of discretely specified heat flux, specified temperatures and adiabatic condition are studied. Along the walls of the system, adiabatic or convective cooling boundary conditions are assumed. Along the sink plane, convective cooling with constant or variable heat transfer coefficient are considered. Also, the effect of orthotropic properties is discussed. This thesis contains nine chapters. Chapter one is the introduction and shows the concepts of thermal spreading resistance besides the originality and importance of the work. Chapter two reviews the literatures on the thermal spreading resistance in the past fifty years with a focus on the recent advances. In chapters three and four, thermal resistance of a twodimensional flux channel with non-uniform convection coefficient in the heat sink plane is studied. The non-uniform convection is modeled by using two functions than can simulate a wide variety of different heat sink configurations. In chapter five, a non-symmetrical flux channel with different heat transfer coefficient along the right and left edges and sink plane is analytically modeled. Due to the edge cooling and non-symmetry, the eigenvalues of the system are defined using the heat transfer coefficient on both edges and for satisfying the orthogonality condition, a normalized function is calculated. In chapter six, thermal behavior of two-dimensional rectangular flux channel with arbitrary boundary conditions on the source plane is presented. The boundary condition along the source plane can be a combination of the first kind boundary condition (Dirichlet or prescribed temperature) and the second kind boundary condition (Neumann or prescribed heat flux). The proposed solution can be used for modeling the flux channels with numerous different source plane boundary conditions without any limitations in the number and position of heat sources. In chapter seven, temperature profile of a circular flux tube with discretely specified boundary conditions along the source plane is presented. Also, the effect of orthotropic properties are discussed. In chapter 8, a three-dimensional rectangular flux channel with a non-uniform heat convection along the heat sink plane is analytically modeled. In chapter nine, a summary of the achievements is presented and some systems are proposed for the future studies. It is worth mentioning that all the models and case studies in the thesis are compared with the Finite Element Method (FEM).

Homogenization of the p-Laplacian with nonlinear boundary condition on critical size particles: Identifying the strange term for the some non smooth and multivalued operators

We extend previous papers in the literature concerning the homogenization of Robin type boundary conditions for quasilinear equations, in the case of microscopic obstacles of critical size: here we consider nonlinear boundary conditions involving some maximal monotone graphs which may correspond to discontinuous or non-Lipschitz functions arising in some catalysis problems.

Second-order elliptic PDE with discontinuous boundary data

We shall consider the weak formulation of a linear elliptic model problem with discontinuous Dirichlet boundary conditions. Since such problems are typically not well-defined in the standard H^1-H^1 setting, we will introduce a suitable saddle point formulation in terms of weighted Sobolev spaces. Furthermore, we will discuss the numerical solution of such problems. Specifically, we employ an hp-discontinuous Galerkin method and derive an L^2-norm a posteriori error estimate. Numerical experiments demonstrate the effectiveness of the proposed error indicator in both the h- and hp-version setting. Indeed, in the latter case exponential convergence of the error is attained as the mesh is adaptively refined.

Migração lateral da desembocadura do Rio Itapocú, SC, Brasil: evolução morfológica e condicionantes físicas

Desembocaduras são ambientes bastante dinâmicos e sujeitos à complexa interação entre fatores estabilizadores e desestabilizadores. Dependendo dessa interação, desembocaduras podem apresentar a tendência de migração ao longo de barreiras arenosas. Um dos mecanismos mais eficientes de transporte de sedimento paralelo à costa, e consequentemente migração de canais, são as correntes longitudinais geradas pelas ondas se aproximando obliquamente à costa. A motivação do presente trabalho é entender o comportamento morfodinâmico do sistema de desembocadura do rio Itapocú, localizado no centro-norte de Santa Catarina (SC), frente aos processos forçantes que atuam na sua migração ao longo da linha de costa. A morfologia dos pontais arenosos foi obtida a partir de levantamentos morfológicos com o uso de DGPS. Para analisar a refração de ondas foi utilizado o modelo numérico MIKE 21 SW, sendo considerados como condições de contorno os dados de ondas referentes ao ano de 2002 e os dados de ondas previstos referentes ao período de coleta. Os dados de saída do modelo foram utilizados para estimar a deriva litorânea potencial na região. Os resultados morfológicos obtidos demonstraram uma migração da desembocadura para o norte durante o período analisado, sendo mais intenso durante o inverno e o verão. Ondas incidentes do quadrante sul sofreram mais o fenômeno da refração e as ondas de leste apresentaram menor variação angular ao se aproximarem à costa. A deriva litorânea potencial anual para os dados de ondas de 2002 apresentou sentido norte-sul, com inversão de sentido durante o outono. Utilizando os dados de ondas previstas para o período dos levantamentos, a deriva litorânea potencial estimada apresentou sentido sul-norte, concordando com a migração observada. Na região próxima a desembocadura, nos pontais arenosos, a deriva potencial apresentou direção para o norte durante todas as estações. Os dados de descarga fluvial não apresentaram influência na migração do canal, porém apresentaram uma relação com a largura do mesmo sazonalmente.Os dados de morfologia juntamente com os dados de deriva litorânea referentes às ondas de 2004/2005 mostraram claramente a migração do canal para o norte sendo a deriva a principal contribuinte para a migração da desembocadura.

Brazilian offshore wave climate based on NWW3 reanalysis

This paper provides a description of the wave climate off the Brazilian coast based on an eleven-year time series (Jan/1997-Dec/2007) obtained from the NWW3 operational model hindcast reanalysis. Information about wave climate in Brazilian waters is very scarce and mainly based on occasional short-term observations, the present analysis being the first covering such temporal and spatial scales. To define the wave climate, six sectors were defined and analyzed along the Brazilian shelf-break: South (W1), Southeast (W2), Central (W3), East (W4), Northeast (W5) and North (W6). W1, W2 and W3 wave regimes are determined by the South Atlantic High (SAH) and the passage of synoptic cold fronts; W4, W5 and W6 are controlled by the Intertropical Convergence Zone (ITCZ) and its meridional oscillation. The most energetic waves are from the S, generated by the strong winds associated to the passage of cold fronts, which mainly affect the southern region. Wave power presents a decrease in energy levels from south to north, with its annual variation showing that the winter months are the most energetic in W1 to W4, while in W5 and W6 the most energetic conditions occur during the austral summer. The information presented here provides boundary conditions for studies related to coastal processes, fundamental for a better understanding of the Brazilian coastal zone.

PARTITIONED ANALYSIS FOR DIMENSIONALLY-HETEROGENEOUS HYDRAULIC NETWORKS

In this work an iterative strategy is developed to tackle the problem of coupling dimensionally-heterogeneous models in the context of fluid mechanics. The procedure proposed here makes use of a reinterpretation of the original problem as a nonlinear interface problem for which classical nonlinear solvers can be applied. Strong coupling of the partitions is achieved while dealing with different codes for each partition, each code in black-box mode. The main application for which this procedure is envisaged arises when modeling hydraulic networks in which complex and simple subsystems are treated using detailed and simplified models, correspondingly. The potentialities and the performance of the strategy are assessed through several examples involving transient flows and complex network configurations.

Hidden vorticity in binary Bose-Einstein condensates

We consider a binary Bose-Einstein condensate (BEC) described by a system of two-dimensional (2D) Gross-Pitaevskii equations with the harmonic-oscillator trapping potential. The intraspecies interactions are attractive, while the interaction between the species may have either sign. The same model applies to the copropagation of bimodal beams in photonic-crystal fibers. We consider a family of trapped hidden-vorticity (HV) modes in the form of bound states of two components with opposite vorticities S(1,2) = +/- 1, the total angular momentum being zero. A challenging problem is the stability of the HV modes. By means of a linear-stability analysis and direct simulations, stability domains are identified in a relevant parameter plane. In direct simulations, stable HV modes feature robustness against large perturbations, while unstable ones split into fragments whose number is identical to the azimuthal index of the fastest growing perturbation eigenmode. Conditions allowing for the creation of the HV modes in the experiment are discussed too. For comparison, a similar but simpler problem is studied in an analytical form, viz., the modulational instability of an HV state in a one-dimensional (1D) system with periodic boundary conditions (this system models a counterflow in a binary BEC mixture loaded into a toroidal trap or a bimodal optical beam coupled into a cylindrical shell). We demonstrate that the stabilization of the 1D HV modes is impossible, which stresses the significance of the stabilization of the HV modes in the 2D setting.

Transport properties of single vacancies in nanotubes

We present density of states and electronic transport calculations of single vacancies in carbon nanotubes. We confirm that the defect reconstructs into a pentagon and a nonagon, following the removal of a single carbon atom. This leads to the formation of a dangling bond. Finally, we demonstrate that care must be taken when calculating the density of states of impurities in one-dimensional systems in general. Traditional treatments of these systems using periodic boundary conditions leads to the formation of minigaps even in the limit of large unit cells.

Classical noncommutative electrodynamics with external source

In a U(1)(*)-noncommutative gauge field theory we extend the Seiberg-Witten map to include the (gauge-invariance-violating) external current and formulate-to the first order in the noncommutative parameter-gauge-covariant classical field equations. We find solutions to these equations in the vacuum and in an external magnetic field, when the 4-current is a static electric charge of a finite size a, restricted from below by the elementary length. We impose extra boundary conditions, which we use to rule out all singularities, 1/r included, from the solutions. The static charge proves to be a magnetic dipole, with its magnetic moment being inversely proportional to its size a. The external magnetic field modifies the long-range Coulomb field and some electromagnetic form factors. We also analyze the ambiguity in the Seiberg-Witten map and show that at least to the order studied here it is equivalent to the ambiguity of adding a homogeneous solution to the current-conservation equation.

Renyi entropy and parity oscillations of anisotropic spin-s Heisenberg chains in a magnetic field

Using the density matrix renormalization group, we investigate the Renyi entropy of the anisotropic spin-s Heisenberg chains in a z-magnetic field. We considered the half-odd-integer spin-s chains, with s = 1/2, 3/2, and 5/2, and periodic and open boundary conditions. In the case of the spin-1/2 chain we were able to obtain accurate estimates of the new parity exponents p(alpha)((p)) and p(alpha)((o)) that gives the power-law decay of the oscillations of the alpha-Renyi entropy for periodic and open boundary conditions, respectively. We confirm the relations of these exponents with the Luttinger parameter K, as proposed by Calabrese et al. [Phys. Rev. Lett. 104, 095701 (2010)]. Moreover, the predicted periodicity of the oscillating term was also observed for some nonzero values of the magnetization m. We show that for s > 1/2 the amplitudes of the oscillations are quite small and get accurate estimates of p(alpha)((p)) and p(alpha)((o)) become a challenge. Although our estimates of the new universal exponents p(alpha)((p)) and p(alpha)((o)) for the spin-3/2 chain are not so accurate, they are consistent with the theoretical predictions.