Numerical methods for strong solutions of stochastic differential equations: an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

Non-linear analysis of the thermo-electro-mechanical behaviour of shear deformable FGM plates with piezoelectric actuators

This paper investigates the non-linear bending behaviour of functionally graded plates that are bonded with piezoelectric actuator layers and subjected to transverse loads and a temperature gradient based on Reddy's higher-order shear deformation plate theory. The von Karman-type geometric non-linearity, piezoelectric and thermal effects are included in mathematical formulations. The temperature change is due to a steady-state heat conduction through the plate thickness. The material properties are assumed to be graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The plate is clamped at two opposite edges, while the remaining edges can be free, simply supported or clamped. Differential quadrature approximation in the X-axis is employed to convert the partial differential governing equations and the associated boundary conditions into a set of ordinary differential equations. By choosing the appropriate functions as the displacement and stress functions on each nodal line and then applying the Galerkin procedure, a system of non-linear algebraic equations is obtained, from which the non-linear bending response of the plate is determined through a Picard iteration scheme. Numerical results for zirconia/aluminium rectangular plates are given in dimensionless graphical form. The effects of the applied actuator voltage, the volume fraction exponent, the temperature gradient, as well as the characteristics of the boundary conditions are also studied in detail. Copyright (C) 2004 John Wiley Sons, Ltd.

Partial regularity of weak solutions of the liquid crystal equilibrium system

For a parameter, we consider the modified relaxed energy of the liquid crystal system. Each minimizer of the modified relaxed energy is a weak solution to the liquid crystal equilibrium system. We prove the partial regularity of minimizers of the modified relaxed energy. We also prove the existence of infinitely many weak solutions for the special boundary value x.

Analytical solutions of linear finite- and small-strain one-dimensional consolidation

Analytical solutions are presented for linear finite-strain one-dimensional consolidation of initially unconsolidated soil layers with surcharge loading for both one- and two-way drainage. These solutions complement earlier solutions for initially unconsolidated soil layers without surcharge and initially normally consolidated soil layers with surcharge. Small-strain solutions for the consolidation of initially unconsolidated soil layers with surcharge loading are also presented, and the relationship between the earlier solutions for initially unconsolidated soil without surcharge and the corresponding small-strain solutions, which was not addressed in the earlier work, is clarified. The new solutions for initially unconsolidated soil with surcharge loading can be applied to the analysis of low stress consolidation tests and to the partial validation of numerical solutions of non-linear finite-strain consolidation. They also clarify a formerly perplexing aspect of finite-strain solution charts first noted in numerical solutions. Copyright (C) 2004 John Wiley Sons, Ltd.

On multiple solutions of the exterior neumann problem involving critical sobolev exponent

In this paper we consider the exterior Neumann problem involving a critical Sobolev exponent. We establish the existence of two solutions having a prescribed limit at infinity.

Global behavior of positive solutions of nonlinear three-point boundary value problems

We investigate the structure of the positive solution set for nonlinear three-point boundary value problems of the form u('') + h(t) f(u) = 0, u(0) = 0, u(1) = lambdau(eta), where eta epsilon (0, 1) is given lambda epsilon (0, 1/n) is a parameter, f epsilon C ([0, infinity), [0, infinity)) satisfies f (s) > 0 for s > 0, and h epsilon C([0, 1], [0, infinity)) is not identically zero on any subinterval of [0, 1]. Our main results demonstrate the existence of continua of positive solutions of the above problem. (C) 2004 Elsevier Ltd. All rights reserved.

Numerical solutions of the sediment conservation law: A review and improved formulation for coastal morphological modelling

Numerical solutions of the sediment conservation law are reviewed in terms of their application to bed update schemes in coastal morphological models. It is demonstrated that inadequately formulated numerical techniques lead to the introduction of diffusion, dispersion and the bed elevation oscillations previously reported in the literature. Four different bed update schemes are then reviewed and tested against benchmark analytical solutions. These include a first order upwind scheme, two Lax-Wendroff schemes and a non-oscillating centred scheme (NOCS) recently applied to morphological modelling by Saint-Cast [Saint-Cast, F., 2002. Modelisation de la morphodynamique des corps sableux en milieu littoral (Modelling of coastal sand banks morphodynamics), University Bordeaux 1, Bordeaux, 245 pp.]. It is shown that NOCS limits and controls numerical errors while including all the sediment flux gradients that control morphological change. Further, no post solution filtering is required, which avoids difficulties with selecting filter strength. Finally, NOCS is compared to a recent Lax-Wendroff scheme with post-solution filtering for a longer term simulation of the morphological evolution around a trained river entrance. (C) 2006 Elsevier B.V. All rights reserved.

Positive solutions of a Neumann problem with competing critical nonlinearities

We present existence results for a Neumann problem involving critical Sobolev nonlinearities both on the right hand side of the equation and at the boundary condition.. Positive solutions are obtained through constrained minimization on the Nehari manifold. Our approach is based on the concentration 'compactness principle of P. L. Lions and M. Struwe.

Non-ideal monitoring of a qubit state using a quantum tunnelling device

We propose a model for non-ideal monitoring of the state of a coupled quantum dot qubit by a quantum tunnelling device. The non-ideality is modelled using an equivalent measurement circuit. This allows realistically available measurement results to be related to the state of the quantum system (qubit). We present a quantum trajectory that describes the stochastic evolution of the qubit state conditioned by tunnelling events (i.e. current) through the device. We calculate and compare the noise power spectra of the current in an ideal and a non-ideal measurement. The results show that when the two qubit dots are strongly coupled the non-ideal measurement cannot detect the qubit state precisely. The limitation of the ideal model for describing a realistic system maybe estimated from the noise spectra.

A unique behavior of sub-critical hydrocarbon permeability in activated carbon at low pressures

In our study on sub-critical hydrocarbon permeation in activated carbon, a minimum in the total permeability (B-T) at low pressure has been observed for only long-chain hydrocarbons such as n-hexane and n-heptane. Such an observation suggests that the minimum appearance depends on the properties of permeating vapors as well as the porous medium. In this paper a permeation model is presented to explain the minimum behavior with the allowance of the collision-reflection factor in the Knudsen diffusivity to be a function of surface loading. Surface diffusion was found to be very significant compared to other transport mechanisms such as Knudsen diffusion and gaseous viscous flow at low pressures. Since the gaseous viscous flow contributes negligibly to the B, at low pressures, the minimum appearance in the B, is mainly attributed to the interplay between Knudsen diffusion and surface diffusion. Also, the molecular structure of adsorbates plays an important role in the minimum appearance.

Highly organized structure in the non-coding region of the psbA minicircle from clade C Symbiodinium

The chloroplast genes of dinoflagellates are distributed among small, circular dsDNA molecules termed minicircles. In this paper, we describe the structure of the non-coding region of the psbA minicircle from Symbiodinium. DNA sequence was obtained from five Symbiodinium strains obtained from four different coral host species (Goniopora tenuidens, Heliofungia actiniformis, Leptastrea purpurea and Pocillopora damicornis), which had previously been determined to be closely related using LSU rDNA region D1/D2 sequence analysis. Eight distinct sequence blocks, consisting of four conserved cores interspersed with two metastable regions and flanked by two variable regions, occurred at similar positions in all strains. Inverted repeats (IRs) occurred in tandem or 'twin' formation within two of the four cores. The metastable regions also consisted of twin IRs and had modular behaviour, being either fully present or completely absent in the different strains. These twin IRs are similar in sequence to double-hairpin elements (DHEs) found in the mitochondrial genomes of some fungi, and may be mobile elements or may serve a functional role in recombination or replication. Within the central unit (consisting of the cores plus the metastable regions), all IRs contained perfect sequence inverses, implying they are highly evolved. IRs were also present outside the central unit but these were imperfect and possessed by individual strains only. A central adenine-rich sequence most closely resembled one in the centre of the non-coding part of Amphidinium operculatum minicircles, and is a potential origin of replication. Sequence polymorphism was extremely high in the variable regions, suggesting that these regions may be useful for distinguishing strains that cannot be differentiated using molecular markers currently available for Symbiodinium.

Spider ballooning in soybean and non-crop areas of southeast Queensland

Ballooning is a form of aerial movement practiced by most miniature and some adult spiders. Very few studies have investigated the composition and rate of spider ballooning in Australian agroecosystems. Water traps were used to compare ballooning rates in irrigated soybean crops and nearby non-crop areas in southeast Queensland over two summer seasons. The highest ballooning rate (14.8 spiders/m(2) per day) was recorded in a soybean field, non-crop areas (7.0 spiders/m(2) per day) and a dry land mungbean field (6.8 spiders/m(2) per day) having similar rates. Spider ballooning in soybean increased throughout the season and showed three peaks and intervening troughs. A similar pattern in ballooning peaks was observed in non-crop areas however the numbers were lower. Peaks in ballooning activity where synchronised across habitat types and some spider groups. Composition of the ballooning fauna was different from that of the ground-dwelling fauna, some families being present in both. Ballooning is an important behaviour in terms of population dynamics for a number of spider groups in soybean and the implications for pest control are discussed. (C) 2004 Elsevier BN. All rights reserved.

Dynamic interfacial tension of aqueous solutions of PVAAs and its role in liquid-liquid dispersion stabilization

In liquid-liquid dispersion systems, the dynamic change of the interfacial properties between the two immiscible liquids plays an important role in both the emulsification process and emulsion stabilization. In this paper, experimentally measured dynamic interfacial tensions of 1-chlorobutane in the aqueous solutions of various random copolymers of polyvinyl acetate and polyvinyl alcohol (PVAA) are presented. Theoretical analyses on these results suggest that the adsorption of the polymer molecules is controlled neither by the bulk diffusion process nor the activation energy barrier for the adsorption but the conformation of polymer molecules. Based on the concept of critical concentration of condensation for polymer adsorption, as well as the observation that the rate at which the dynamic interfacial tension changes does not correlate to the PVAA's ability to stabilize a single drop, it is postulated that the main stabilization mechanism for the PVAAs is by steric hindrance, not the Gibbs-Marangoni effect offered by the small molecule surfactants.