Exact results for a tunnel-coupled pair of trapped Bose-Einstein condensates

A model describing coherent quantum tunnelling between two trapped Bose-Einstein condensates is discussed. It is not well known that the model admits an exact solution, obtained some time ago, with the energy spectrum derived through the algebraic Bethe ansatz. An asymptotic analysis of the Bethe ansatz equations leads us to explicit expressions for the energies of the ground and the first excited states in the limit of weak tunnelling and all energies for strong tunnelling. The results are used to extract the asymptotic limits of the quantum fluctuations of the boson number difference between the two Bose-Einstein condensates and to characterize the degree of coherence in the system.

Selecting methods to solve multi-well master equations

There are several competing methods commonly used to solve energy grained master equations describing gas-phase reactive systems. When it comes to selecting an appropriate method for any particular problem, there is little guidance in the literature. In this paper we directly compare several variants of spectral and numerical integration methods from the point of view of computer time required to calculate the solution and the range of temperature and pressure conditions under which the methods are successful. The test case used in the comparison is an important reaction in combustion chemistry and incorporates reversible and irreversible bimolecular reaction steps as well as isomerizations between multiple unimolecular species. While the numerical integration of the ODE with a stiff ODE integrator is not the fastest method overall, it is the fastest method applicable to all conditions.

An approach to verifying and debugging simulation models governed by ordinary differential equations: Part 1. Methodology for residual generation

For dynamic simulations to be credible, verification of the computer code must be an integral part of the modelling process. This two-part paper describes a novel approach to verification through program testing and debugging. In Part 1, a methodology is presented for detecting and isolating coding errors using back-to-back testing. Residuals are generated by comparing the output of two independent implementations, in response to identical inputs. The key feature of the methodology is that a specially modified observer is created using one of the implementations, so as to impose an error-dependent structure on these residuals. Each error can be associated with a fixed and known subspace, permitting errors to be isolated to specific equations in the code. It is shown that the geometric properties extend to multiple errors in either one of the two implementations. Copyright (C) 2003 John Wiley Sons, Ltd.

An approach to verifying and debugging simulation models governed by ordinary differential equations: Part 2. Residuals analysis and a case study

In Part 1 of this paper a methodology for back-to-back testing of simulation software was described. Residuals with error-dependent geometric properties were generated. A set of potential coding errors was enumerated, along with a corresponding set of feature matrices, which describe the geometric properties imposed on the residuals by each of the errors. In this part of the paper, an algorithm is developed to isolate the coding errors present by analysing the residuals. A set of errors is isolated when the subspace spanned by their combined feature matrices corresponds to that of the residuals. Individual feature matrices are compared to the residuals and classified as 'definite', 'possible' or 'impossible'. The status of 'possible' errors is resolved using a dynamic subset testing algorithm. To demonstrate and validate the testing methodology presented in Part 1 and the isolation algorithm presented in Part 2, a case study is presented using a model for biological wastewater treatment. Both single and simultaneous errors that are deliberately introduced into the simulation code are correctly detected and isolated. Copyright (C) 2003 John Wiley Sons, Ltd.

A new wavelet-based adaptive method for solving population balance equations

A new wavelet-based adaptive framework for solving population balance equations (PBEs) is proposed in this work. The technique is general, powerful and efficient without the need for prior assumptions about the characteristics of the processes. Because there are steeply varying number densities across a size range, a new strategy is developed to select the optimal order of resolution and the collocation points based on an interpolating wavelet transform (IWT). The proposed technique has been tested for size-independent agglomeration, agglomeration with a linear summation kernel and agglomeration with a nonlinear kernel. In all cases, the predicted and analytical particle size distributions (PSDs) are in excellent agreement. Further work on the solution of the general population balance equations with nucleation, growth and agglomeration and the solution of steady-state population balance equations will be presented in this framework. (C) 2002 Elsevier Science B.V. All rights reserved.

Modelagem da dispersão de odores em ambientes urbanos

O presente trabalho investigou o problema da modelagem da dispersão de compostos odorantes em presença de obstáculos (cúbicos e com forma complexa) sob condição de estabilidade atmosférica neutra. Foi empregada modelagem numérica baseada nas equações de transporte (CFD1) bem como em modelos algébricos baseados na pluma Gausseana (AERMOD2, CALPUFF3 e FPM4). Para a validação dos resultados dos modelos e a avaliação do seu desempenho foram empregados dados de experimentos em túnel de vento e em campo. A fim de incluir os efeitos da turbulência atmosférica na dispersão, dois diferentes modelos de sub-malha associados à Simulação das Grandes Escalas (LES5) foram investigados (Smagorinsky dinâmico e WALE6) e, para a inclusão dos efeitos de obstáculos na dispersão nos modelos Gausseanos, foi empregado o modelo PRIME7. O uso do PRIME também foi proposto para o FPM como uma inovação. De forma geral, os resultados indicam que o uso de CFD/LES é uma ferramenta útil para a investigação da dispersão e o impacto de compostos odorantes em presença de obstáculos e também para desenvolvimento dos modelos Gausseanos. Os resultados também indicam que o modelo FPM proposto, com a inclusão dos efeitos do obstáculo baseado no PRIME também é uma ferramenta muito útil em modelagem da dispersão de odores devido à sua simplicidade e fácil configuração quando comparado a modelos mais complexos como CFD e mesmo os modelos regulatórios AERMOD e CALPUFF. A grande vantagem do FPM é a possibilidade de estimar-se o fator de intermitência e a relação pico-média (P/M), parâmetros úteis para a avaliação do impacto de odores. Os resultados obtidos no presente trabalho indicam que a determinação dos parâmetros de dispersão para os segmentos de pluma, bem como os parâmetros de tempo longo nas proximidades da fonte e do obstáculo no modelo FPM pode ser melhorada e simulações CFD podem ser usadas como uma ferramenta de desenvolvimento para este propósito. Palavras chave: controle de odor, dispersão, fluidodinâmica computacional, modelagem matemática, modelagem gaussiana de pluma flutuante, simulação de grandes vórtices (LES).

Contrasts in the basins of attraction of structurally identical iterative root finding methods

A numerical comparison is performed between three methods of third order with the same structure, namely BSC, Halley’s and Euler–Chebyshev’s methods. As the behavior of an iterative method applied to a nonlinear equation can be highly sensitive to the starting points, the numerical comparison is carried out, allowing for complex starting points and for complex roots, on the basins of attraction in the complex plane. Several examples of algebraic and transcendental equations are presented.

A class of singular first order differential equations with applications in reaction-diffusion

Agências Financiadoras: FCT e MIUR

Systems of Navier-Stokes equations on cantor sets

We present systems of Navier-Stokes equations on Cantor sets, which are described by the local fractional vector calculus. It is shown that the results for Navier-Stokes equations in a fractal bounded domain are efficient and accurate for describing fluid flow in fractal media.

Controllability results for impulsive mixed type functional integro-differential evolution equations with nonlocal conditions

In this paper, we establish the controllability for a class of abstract impulsive mixed-type functional integro-differential equations with finite delay in a Banach space. Some sufficient conditions for controllability are obtained by using the Mönch fixed point theorem via measures of noncompactness and semigroup theory. Particularly, we do not assume the compactness of the evolution system. An example is given to illustrate the effectiveness of our results.

Combined mutation differential evolution to solve systems of nonlinear equations

This paper presents a differential evolution heuristic to compute a solution of a system of nonlinear equations through the global optimization of an appropriate merit function. Three different mutation strategies are combined to generate mutant points. Preliminary numerical results show the effectiveness of the presented heuristic.

Self-adaptive combination of global tabu search and local search for nonlinear equations

Solving systems of nonlinear equations is a very important task since the problems emerge mostly through the mathematical modelling of real problems that arise naturally in many branches of engineering and in the physical sciences. The problem can be naturally reformulated as a global optimization problem. In this paper, we show that a self-adaptive combination of a metaheuristic with a classical local search method is able to converge to some difficult problems that are not solved by Newton-type methods.

Algebraic structure for the crossing of balanced and stair nested designs

Stair nesting allows us to work with fewer observations than the most usual form of nesting, the balanced nesting. In the case of stair nesting the amount of information for the different factors is more evenly distributed. This new design leads to greater economy, because we can work with fewer observations. In this work we present the algebraic structure of the cross of balanced nested and stair nested designs, using binary operations on commutative Jordan algebras. This new cross requires fewer observations than the usual cross balanced nested designs and it is easy to carry out inference.

Solving nonlinear equations by a tabu search strategy

Solving systems of nonlinear equations is a problem of particular importance since they emerge through the mathematical modeling of real problems that arise naturally in many branches of engineering and in the physical sciences. The problem can be naturally reformulated as a global optimization problem. In this paper, we show that a metaheuristic, called Directed Tabu Search (DTS) [16], is able to converge to the solutions of a set of problems for which the fsolve function of MATLAB® failed to converge. We also show the effect of the dimension of the problem in the performance of the DTS.