1,654 research outputs found
Mean-square convergence and stability of the backward Euler method for stochastic differential delay equations with highly nonlinear growing coefficients
Over the last few decades, the numerical methods for stochastic differential
delay equations (SDDEs) have been investigated and developed by many scholars.
Nevertheless, there is still little work to be completed. By virtue of the
novel technique, this paper focuses on the mean-square convergence and
stability of the backward Euler method (BEM) for SDDEs whose drift and
diffusion coefficients can both grow polynomially. The upper mean-square error
bounds of BEM are obtained. Then the convergence rate, which is one-half, is
revealed without using the moment boundedness of numerical solutions.
Furthermore, under fairly general conditions, the novel technique is applied to
prove that the BEM can inherit the exponential mean-square stability with a
simple proof. At last, two numerical experiments are implemented to illustrate
the reliability of the theories
Finite Temperature Models of Bose-Einstein Condensation
The theoretical description of trapped weakly-interacting Bose-Einstein
condensates is characterized by a large number of seemingly very different
approaches which have been developed over the course of time by researchers
with very distinct backgrounds. Newcomers to this field, experimentalists and
young researchers all face a considerable challenge in navigating through the
`maze' of abundant theoretical models, and simple correspondences between
existing approaches are not always very transparent. This Tutorial provides a
generic introduction to such theories, in an attempt to single out common
features and deficiencies of certain `classes of approaches' identified by
their physical content, rather than their particular mathematical
implementation.
This Tutorial is structured in a manner accessible to a non-specialist with a
good working knowledge of quantum mechanics. Although some familiarity with
concepts of quantum field theory would be an advantage, key notions such as the
occupation number representation of second quantization are nonetheless briefly
reviewed. Following a general introduction, the complexity of models is
gradually built up, starting from the basic zero-temperature formalism of the
Gross-Pitaevskii equation. This structure enables readers to probe different
levels of theoretical developments (mean-field, number-conserving and
stochastic) according to their particular needs. In addition to its `training
element', we hope that this Tutorial will prove useful to active researchers in
this field, both in terms of the correspondences made between different
theoretical models, and as a source of reference for existing and developing
finite-temperature theoretical models.Comment: Detailed Review Article on finite temperature theoretical techniques
for studying weakly-interacting atomic Bose-Einstein condensates written at
an elementary level suitable for non-experts in this area (e.g. starting PhD
students). Now includes table of content
Models of Delay Differential Equations
This book gathers a number of selected contributions aimed at providing a balanced picture of the main research lines in the realm of delay differential equations and their applications to mathematical modelling. The contributions have been carefully selected so that they cover interesting theoretical and practical analysis performed in the deterministic and the stochastic settings. The reader will find a complete overview of recent advances in ordinary and partial delay differential equations with applications in other multidisciplinary areas such as Finance, Epidemiology or Engineerin
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