A Van der Pol-Mathieu equation for the dynamics of dust grain charge in dusty plasmas

The chaotic profile of dust grain dynamics associated with dust-acoustic oscillations in a dusty plasma is considered. The collective behaviour of the dust plasma component is described via a multi-fluid model, comprising Boltzmann distributed electrons and ions, as well as an equation of continuity possessing a source term for the dust grains, the dust momentum and Poisson's equations. A Van der Pol–Mathieu-type nonlinear ordinary differential equation for the dust grain density dynamics is derived. The dynamical system is cast into an autonomous form by employing an averaging method. Critical stability boundaries for a particular trivial solution of the governing equation with varying parameters are specified. The equation is analysed to determine the resonance region, and finally numerically solved by using a fourth-order Runge–Kutta method. The presence of chaotic limit cycles is pointed out.

GPU Acceleration of Partial Differential Equation Solvers

Differential equations are often directly solvable by analytical means only in their one dimensional version. Partial differential equations are generally not solvable by analytical means in two and three dimensions, with the exception of few special cases. In all other cases, numerical approximation methods need to be utilized. One of the most popular methods is the finite element method. The main areas of focus, here, are the Poisson heat equation and the plate bending equation. The purpose of this paper is to provide a quick walkthrough of the various approaches that the authors followed in pursuit of creating optimal solvers, accelerated with the use of graphical processing units, and comparing them in terms of accuracy and time efficiency with existing or self-made non-accelerated solvers.

Analytical description of stochastic field-Line wandering in magnetic turbulence

A nonperturbative nonlinear statistical approach is presented to describe turbulent magnetic systems embedded in a uniform mean magnetic field. A general formula in the form of an ordinary differential equation for magnetic field-line wandering (random walk) is derived. By considering the solution of this equation for different limits several new results are obtained. As an example, it is demonstrated that the stochastic wandering of magnetic field-lines in a two-component turbulence model leads to superdiffusive transport, contrary to an existing diffusive picture. The validity of quasilinear theory for field-line wandering is discussed, with respect to different turbulence geometry models, and previous diffusive results are shown to be deduced in appropriate limits.

On solvability of linear differential equations in ${\Bbb R}^{\Bbb N}$

We construct $x^0$ in ${\Bbb R}^{\Bbb N}$ and a row-finite matrix $T=\{T_{i,j}(t)\}_{i,j\in\N}$ of polynomials of one real variable $t$ such that the Cauchy problem $\dot x(t)=T_tx(t)$, $x(0)=x^0$ in the Fr\'echet space $\R^\N$ has no solutions. We also construct a row-finite matrix $A=\{A_{i,j}(t)\}_{i,j\in\N}$ of $C^\infty(\R)$ functions such that the Cauchy problem $\dot x(t)=A_tx(t)$, $x(0)=x^0$ in ${\Bbb R}^{\Bbb N}$ has no solutions for any $x^0\in{\Bbb R}^{\Bbb N}\setminus\{0\}$. We provide some sufficient condition of solvability and of unique solvability for linear ordinary differential equations $\dot x(t)=T_tx(t)$ with matrix elements $T_{i,j}(t)$ analytically dependent on $t$.

Coupled consolidation of a solid, infinite cylinder using a Terzaghi formulation

Axisymmetric consolidation is a classical boundary value problem for geotechnical engineers. Under some circumstances an analysis in which the changes in pore pressure, effective stress and displacement can be uncoupled from each other is sufficient, leading to a Terzaghi formulation of the axisymmetric consolidation equation in terms of the pore pressure. However, representation of the Mandel-Cryer effect usually requires more complex, coupled, Biot formulations. A new coupled formulation for the plane strain, axisymmetric consolidation problem is presented for small, linear elastic deformations. A single, easily evaluated parameter couples changes in pore pressure to changes in effective stress, and the resulting differential equation for pore pressure dissipation is very similar to Terzaghi’s classic formulation. The governing equations are then solved using finite differences and the consolidation of a solid infinite cylinder analysed, calculating the variation with time and with radius of the excess pore pressure and the radial displacement. Comparison with a previously published semi-analytical solution indicates that the formulation successfully embodies the Mandel-Cryer effect.

Modelling the mechanical response of the Reed-mouthpiece-lip system of a clarinet. Part I. A one-dimensional distributed model

The motion of a clarinet reed that is clamped to a mouthpiece and supported by a lip is simulated in the time-domain using finite difference methods. The reed is modelled as a bar with non-uniform cross section, and is described using a one-dimensional, fourth-order partial differential equation. The interactions with the mouthpiece Jay and the player's lip are taken into account by incorporating conditional contact forces in the bar equation. The model is completed by clamped-free boundary conditions for the reed. An implicit finite difference method is used for discretising the system, and values for the physical parameters are chosen both from laboratory measurements and by accurate tuning of the numerical simulations. The accuracy of the numerical system is assessed through analysis of frequency warping effects and of resonance estimation. Finally, the mechanical properties of the system are studied by analysing its response to external driving forces. In particular, the effects of reed curling are investigated.

Some results on solvability of ordinary linear differential equations in locally convex spaces

Let $\Gamma$ be the class of sequentially complete locally convex spaces such that an existence theorem holds for the linear Cauchy problem $\dot x = Ax$, $x(0) = x_0$ with respect to functions $x: R\to E$. It is proved that if $E\in \Gamma$, then $E\times R^A$ is-an-element-of $\Gamma$ for an arbitrary set $A$. It is also proved that a topological product of infinitely many infinite-dimensional Frechet spaces, each not isomorphic to $\omega$, does not belong to $\Gamma$.

Interactions between invasive and native crustaceans: differential functional responses of intraguild predators towards juvenile hetero-specifics

Intraguild predation (IGP) between invasive and native species can lead to species exclusions or co-existence, dependent on the direction and strength of the interaction. Recently, derivation of 'functional responses' has been identified as a means of comparing the community impacts of invasive and native species. Here, we employ a novel use of this functional response methodology to evaluate any IGP asymmetries between the invasive Ponto-Caspian amphipod Echinogammarus ischnus and the North American native Gammarus fasciatus. The direction and magnitude of intraguild predation of adult males on hetero-specific adult females has previously been shown to reverse across a water conductivity gradient. This partially explains field patterns, but does not predict the co-existence of the two species observed in many habitats and locations. Here, we compared intraguild predation by both species on each other's juveniles in high- and low- conductivity water. G. fasciatus has a higher type II functional response towards E. ischnus juveniles compared to the reciprocal interaction. Conductivity did not influence the predation rate on juveniles of either E. ischnus or G. fasciatus. Thus, the male/female IGP advantage to the native G. fasciatus in low conductivity water is compounded by a juvenile IGP asymmetry, which also counteracts the male/female IGP advantage to E. ischnus in high conductivity waters, helping to explain field patterns of exclusion and co-existence. Thus, complex asymmetries in mutual IGP associated with inherent species differences, environmental modulation, and life-history effects can help us understand and predict the population and community level outcomes of species invasions.

Differential ecological impacts of an invader and a native under environmental change are revealed by comparative functional responses.

Predicting the ecological impacts of damaging invasive species under relevant environmental contexts is a major challenge, for which comparative functional responses (the relationship between resource availability and consumer uptake rate) have great potential. Here, the functional responses of Gammarus pulex, an ecologically damaging invader in freshwaters in Ireland and other islands, were compared with those of a native trophic equivalent Gammarus duebeni celticus. Experiments were conducted at two dissolved oxygen concentrations (80 and 50 % saturation), representative of anthropogenic water quality changes, using two larval prey, blackfly (Simuliidae spp.) and mayfly (Baetis rhodani). Overall, G. pulex had higher Type II functional responses and hence predatory impacts than G. d. celticus and the functional responses of both predators were reduced by lowered oxygen concentration. However, this reduction was of lower magnitude for the invader as compared to the native. Further, the invader functional response at low oxygen was comparable to that of the native at high oxygen. Attack rates of the two predators were similar, with low oxygen reducing these attack rates, but this effect occurred more strongly for blackfly than mayfly prey. Handling times were significantly lower for the invader compared with the native, and significantly higher at low oxygen, however, the effect of lowered oxygen on handling times was minimal for the invader and pronounced for the native. Maximum feeding rates were significantly greater for the invader compared with the native, and significantly reduced at low oxygen, with this effect again lesser for the invader as compared to the native. The greater functional responses of the invader corroborate with its impacts on recipient macroinvertebrate communities when it replaces the native. Further, our experiments predict that the impact of the invader will be less affected than the native under altered oxygen regimes driven by anthropogenic influences.