Moment decay rates of solutions of stochastic differential equations

The objective of this paper is to investigate the p-ίh moment asymptotic stability decay rates for certain finite-dimensional Itό stochastic differential equations. Motivated by some practical examples, the point of our analysis is a special consideration of general decay speeds, which contain as a special case the usual exponential or polynomial type one, to meet various situations. Sufficient conditions for stochastic differential equations (with variable delays or not) are obtained to ensure their asymptotic properties. Several examples are studied to illustrate our theory.

An improved result of exponential stability of stochastic semilinear evolution equations

Sufficient conditions for the exponential stability of a class ofnonlinear, non-autonomous stochastic differential equations in infinitedimensions are studied. The analysis consists of introducing a suitableapproximating solution systems and using a limiting argument to pass onstability of strong solutions to mild ones. As a consequence, the classicalcriteriaof stability in A. Ichikawa [8] are improved and extended to cover a class ofnon-autonomous stochastic evolution equations.Two examples are investigated to illustrate our theory.

Multiscale numerical modelling of crack propagation in two-dimensional metal plate

A novel multiscale model of brittle crack propagation in an Ag plate with macroscopic dimensions has been developed. The model represents crack propagation as stochastic drift-diffusion motion of the crack tip atom through the material, and couples the dynamics across three different length scales. It integrates the nanomechanics of bond rupture at the crack tip with the displacement and stress field equations of continuum based fracture theories. The finite element method is employed to obtain the continuum based displacement and stress fields over the macroscopic plate, and these are then used to drive the crack tip forward at the atomic level using the molecular dynamics simulation method based on many-body interatomic potentials. The linkage from the nanoscopic scale back to the macroscopic scale is established via the Ito stochastic calculus, the stochastic differential equation of which advances the tip to a new position on the macroscopic scale using the crack velocity and diffusion constant obtained on the nanoscale. Well known crack characteristics, such as the roughening transitions of the crack surfaces, crack velocity oscillations, as well as the macroscopic crack trajectories, are obtained.

Modelling meso-scale diffusion processes in stochastic fluid bio-membranes

The space–time dynamics of rigid inhomogeneities (inclusions) free to move in a randomly fluctuating fluid bio-membrane is derived and numerically simulated as a function of the membrane shape changes. Both vertically placed (embedded) inclusions and horizontally placed (surface) inclusions are considered. The energetics of the membrane, as a two-dimensional (2D) meso-scale continuum sheet, is described by the Canham–Helfrich Hamiltonian, with the membrane height function treated as a stochastic process. The diffusion parameter of this process acts as the link coupling the membrane shape fluctuations to the kinematics of the inclusions. The latter is described via Ito stochastic differential equation. In addition to stochastic forces, the inclusions also experience membrane-induced deterministic forces. Our aim is to simulate the diffusion-driven aggregation of inclusions and show how the external inclusions arrive at the sites of the embedded inclusions. The model has potential use in such emerging fields as designing a targeted drug delivery system.

A multi-scale numerical modelling of crack propagation in a 2D metallic plate

A new multi-scale model of brittle fracture growth in an Ag plate with macroscopic dimensions is proposed in which the crack propagation is identified with the stochastic drift-diffusion motion of the crack-tip atom through the material. The model couples molecular dynamics simulations, based on many-body interatomic potentials, with the continuum-based theories of fracture mechanics. The Ito stochastic differential equation is used to advance the tip position on a macroscopic scale before each nano-scale simulation is performed. Well-known crack characteristics, such as the roughening transitions of the crack surfaces, as well as the macroscopic crack trajectories are obtained.

Conservative semi-Lagrangian finite volume schemes

Semi-Lagrangian finite volume schemes for the numerical approximation of linear advection equations are presented. These schemes are constructed so that the conservation properties are preserved by the numerical approximation. This is achieved using an interpolation procedure based on area-weighting. Numerical results are presented illustrating some of the features of these schemes.

Modelling air quality in street canyons: a review

High pollution levels have been often observed in urban street canyons due to the increased traffic emissions and reduced natural ventilation. Microscale dispersion models with different levels of complexity may be used to assess urban air qualityand support decision-making for pollution control strategies and traffic planning. Mathematical models calculate pollutant concentrations by solving either analytically a simplified set of parametric equations or numerically a set of differential equations that describe in detail wind flow and pollutant dispersion. Street canyon models, which might also include simplified photochemistry and particle deposition–resuspension algorithms, are often nested within larger-scale urban dispersion codes. Reduced-scale physical models in wind tunnels may also be used for investigating atmospheric processes within urban canyons and validating mathematical models. A range of monitoring techniques is used to measure pollutant concentrations in urban streets. Point measurement methods (continuous monitoring, passive and active pre-concentration sampling, grab sampling) are available for gaseous pollutants. A number of sampling techniques (mainlybased on filtration and impaction) can be used to obtain mass concentration, size distribution and chemical composition of particles. A combination of different sampling/monitoring techniques is often adopted in experimental studies. Relativelysimple mathematical models have usually been used in association with field measurements to obtain and interpret time series of pollutant concentrations at a limited number of receptor locations in street canyons. On the other hand, advanced numerical codes have often been applied in combination with wind tunnel and/or field data to simulate small-scale dispersion within the urban canopy.

A computational model of coupled heat and moisture transfer with phase change in granular sugar during varying environmental conditions

As part of a comprehensive effort to predict the development of caking in granular materials, a mathematical model is introduced to model simultaneous heat and moisture transfer with phase change in porous media when undergoing temperature oscillations/cycling. The resulting model partial differential equations were solved using finite-volume procedures in the context of the PHYSICA framework and then applied to the analysis of sugar in storage. The influence of temperature on absorption/desorption and diffusion coefficients is coupled into the transport equations. The temperature profile, the depth of penetration of the temperature oscillation into the bulk solid, and the solids moisture content distribution were first calculated, and these proved to be in good agreement with experimental data. Then, the influence of temperature oscillation on absolute humidity, moisture concentration, and moisture migration for different parameters and boundary conditions was examined. As expected, the results show that moisture near boundary regions responds faster than farther away from them with surface temperature changes. The moisture absorption and desorption in materials occurs mainly near boundary regions (where interactions with the environment are more pronounced). Small amounts of solids moisture content, driven by both temperature and vapour concentration gradients, migrate between boundary and center with oscillating temperature.

Multi-physics modelling for microelectronics and microsystems - current capabilities and future challenges

At present the vast majority of Computer-Aided- Engineering (CAE) analysis calculations for microelectronic and microsystems technologies are undertaken using software tools that focus on single aspects of the physics taking place. For example, the design engineer may use one code to predict the airflow and thermal behavior of an electronic package, then another code to predict the stress in solder joints, and then yet another code to predict electromagnetic radiation throughout the system. The reason for this focus of mesh-based codes on separate parts of the governing physics is essentially due to the numerical technologies used to solve the partial differential equations, combined with the subsequent heritage structure in the software codes. Using different software tools, that each requires model build and meshing, leads to a large investment in time, and hence cost, to undertake each of the simulations. During the last ten years there has been significant developments in the modelling community around multi- physics analysis. These developments are being followed by many of the code vendors who are now providing multi-physics capabilities in their software tools. This paper illustrates current capabilities of multi-physics technology and highlights some of the future challenges

Application of modelling for predicting the behaviour of polymers and adhesives in microelectronic and photonic devices

With the growth in computing power, and advances in numerical methods for the solution of partial differential equations, modeling technologies based around computational fluid dynamics, finite element analysis and optimisation are now being widely used by researchers and industry. Polymer and adhesive materials are now being widely used in electronic and photonic devices. This paper will illustrate the use of modeling tools to predict the behaviour of these materials from product assembly to its performance and reliability.

Concise introduction to image processing using C++

A Concise Intro to Image Processing using C++ presents state-of-the-art image processing methodology, including current industrial practices for image compression, image de-noising methods based on partial differential equations, and new image compression methods such as fractal image compression and wavelet compression. It includes elementary concepts of image processing and related fundamental tools with coding examples as well as exercises. With a particular emphasis on illustrating fractal and wavelet compression algorithms, the text covers image segmentation, object recognition, and morphology. An accompanying CD-ROM contains code for all algorithms.