scholarly journals Examination of Particle Tails

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Tim Blackwell ◽  
Dan Bratton
Keyword(s):  
Power Law ◽  
Difference Equations ◽  
Particle Swarm ◽  
Second Order ◽  
Particle Swarm Optimisation ◽  
Position Distribution ◽  
Length Scales ◽  
Stochastic Difference Equations ◽  
First Order ◽  
Second Order Equations

The tail of the particle swarm optimisation (PSO) position distribution at stagnation is shown to be describable by a power law. This tail fattening is attributed to particle bursting on all length scales. The origin of the power law is concluded to lie in multiplicative randomness, previously encountered in the study of first-order stochastic difference equations, and generalised here to second-order equations. It is argued that recombinant PSO, a competitive PSO variant without multiplicative randomness, does not experience tail fattening at stagnation.

scholarly journals Weierstrass elliptic difference equations

1987 ◽  
Vol 35 (1) ◽  
pp. 43-48 ◽  
Author(s):  
Renfrey B. Potts
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Elliptic Function ◽  
Order Differential Equation ◽  
Second Order ◽  
Elliptic Difference ◽  
First Order ◽  
Second Order Differential Equation ◽  
Weierstrass Elliptic Function

The Weierstrass elliptic function satisfies a nonlinear first order and a nonlinear second order differential equation. It is shown that these differential equations can be discretized in such a way that the solutions of the resulting difference equations exactly coincide with the corresponding values of the elliptic function.

Linear Stochastic Difference Equations

2013 ◽  
Author(s):  
Lars Peter Hansen ◽  
Thomas J. Sargent
Keyword(s):  
Time Series ◽  
Difference Equation ◽  
Difference Equations ◽  
Martingale Difference ◽  
Stochastic Difference Equation ◽  
Stochastic Difference Equations ◽  
Order Linear ◽  
Economic Agents ◽  
First Order ◽  
Competitive Equilibria

This chapter describes the vector first-order linear stochastic difference equation. It is first used to represent information flowing to economic agents, then again to represent competitive equilibria. The vector first-order linear stochastic difference equation is associated with a tidy theory of prediction and a host of procedures for econometric application. Ease of analysis has prompted the adoption of economic specifications that cause competitive equilibria to have representations as vector first-order linear stochastic difference equations. Because it expresses next period's vector of state variables as a linear function of this period's state vector and a vector of random disturbances, a vector first-order vector stochastic difference equation is recursive. Disturbances that form a “martingale difference sequence” are basic building blocks used to construct time series. Martingale difference sequences are easy to forecast, a fact that delivers convenient recursive formulas for optimal predictions of time series.

Diffusion approximations to linear stochastic difference equations with stationary coefficients

1977 ◽  
Vol 14 (1) ◽  
pp. 58-74 ◽  
Author(s):  
Harry A. Guess ◽  
John H. Gillespie
Keyword(s):  
Difference Equations ◽  
Time Scale ◽  
Population Biology ◽  
Correction Term ◽  
Random Environments ◽  
Diffusion Approximations ◽  
Uhlenbeck Process ◽  
Stochastic Difference Equations ◽  
First Order ◽  
Approach Zero

It is shown that solutions to linear first-order stochastic difference equations with stationary autocorrelated coefficients converge weakly in D[0,1] to an Ito stochastic integral plus a correction term when the time scale is shifted so that the means, variances, and covariances of the coefficients all approach zero at the same rate. Other limit theorems applicable to different time scale shifts are also given. These results yield two different continuous time limits to a recent model of Roughgarden (1975) for population growth in stationary random environments. One limit, an Ornstein-Uhlenbeck process, is applicable in the presence of rapidly fluctuating autocorrelated environments; the other limit, which is not a diffusion process, applies to the case of slowly varying, highly autocorrelated environments. Other applications in population biology and genetics are discussed.

Acoustofluidic Simulations: Ultrasound Handling of Liquids and Particles in Microfluidic Systems

2008 ◽  
Author(s):  
Peder Skafte-Pedersen ◽  
Henrik Bruus
Keyword(s):  
Acoustic Radiation ◽  
Acoustic Streaming ◽  
Radiation Force ◽  
Second Order ◽  
Order Effects ◽  
Perturbation Approach ◽  
First Order ◽  
At Resonance ◽  
Liquid Flows ◽  
Second Order Equations

Within the field of lab-on-a-chip systems large efforts are devoted to the development of onchip tools for particle handling and mixing in viscosity-dominated liquid flows on the sub-mm scale. One technology involves ultrasound with frequencies in the MHz range, which leads to wavelengths of the order of 0.1–1 mm suitable for mm-sized microchambers. Due to the nonlinearity of the governing acoustofluidic equations, second-order effects will induce steady forces on fluids and suspended particles through the effects known as acoustic streaming and acoustic radiation force. We extend the basic perturbation approach for treating these effects in systems at resonance in various geometries. The first-order eigenmodes are used as source terms for the time-averaged viscous second-order equations. The theory is applied to explain experimental results on aqueous microbead solutions in silicon-glass microchips.

scholarly journals Applying the method of normal forms to second-order nonlinear vibration problems

2010 ◽  
Vol 467 (2128) ◽  
pp. 1141-1163 ◽  
Author(s):  
Simon A. Neild ◽  
David J. Wagg
Keyword(s):  
Normal Form ◽  
Nonlinear Vibration ◽  
Normal Forms ◽  
Equations Of Motion ◽  
Second Order ◽  
Form Analysis ◽  
First Order ◽  
Invariance Properties ◽  
Vibration Problems ◽  
Second Order Equations

Vibration problems are naturally formulated with second-order equations of motion. When the vibration problem is nonlinear in nature, using normal form analysis currently requires that the second-order equations of motion be put into first-order form. In this paper, we demonstrate that normal form analysis can be carried out on the second-order equations of motion. In addition, for forced, damped, nonlinear vibration problems, we show that the invariance properties of the first- and second-order transforms differ. As a result, using the second-order approach leads to a simplified formulation for forced, damped, nonlinear vibration problems.

scholarly journals Lie symmetry analysis and similarity solutions for the Jimbo – Miwa equation and generalisations

2020 ◽  
Vol 21 (7-8) ◽  
pp. 767-779
Author(s):  
Amlan K. Halder ◽  
Andronikos Paliathanasis ◽  
Rajeswari Seshadri ◽  
Peter G. L. Leach
Keyword(s):  
Singularity Analysis ◽  
Travelling Wave ◽  
Similarity Solutions ◽  
Second Order ◽  
Lie Symmetry Analysis ◽  
Group Approach ◽  
First Order ◽  
Reduced Equations ◽  
Second Order Equations ◽  
Hypergeometric Solutions

AbstractWe study the Jimbo – Miwa equation and two of its extended forms, as proposed by Wazwaz et al., using Lie’s group approach. Interestingly, the travelling – wave solutions for all the three equations are similar. Moreover, we obtain certain new reductions which are completely different for each of the three equations. For example, for one of the extended forms of the Jimbo – Miwa equation, the subsequent reductions leads to a second – order equation with Hypergeometric solutions. In certain reductions, we obtain simpler first – order and linearisable second – order equations, which helps us to construct the analytic solution as a closed – form function. The variation in the nonzero Lie brackets for each of the different forms of the Jimbo – Miwa also presents a different perspective. Finally, singularity analysis is applied in order to determine the integrability of the reduced equations and of the different forms of the Jimbo – Miwa equation.

Rheological Behaviour and Time Dependent Characterisation of Gilaboru Juice (Viburnum opulus L.)

2005 ◽  
Vol 11 (2) ◽  
pp. 129-137 ◽  
Author(s):  
A. Altan ◽  
S. Kus ◽  
A. Kaya
Keyword(s):  
Power Law ◽  
Apparent Viscosity ◽  
Rheological Behaviour ◽  
Single Equation ◽  
Second Order ◽  
Time Dependent ◽  
Viburnum Opulus ◽  
Flow Curves ◽  
First Order ◽  
Order Stress

The flow curves and time-dependent rheological behaviour of gilaboru ( Viburnum opulus L.), a traditional drink in the middle Anatolia region of Turkey, with different solid concentrations (59.7, 56.3, 53.1, 43 and 35°Brix) were studied at different temperatures (5-60°C) using a controlled stress rheometer. Gilaboru samples exhibited thixotropic behaviour for all concentrations, both in the forward and backward measurements were characterised by the power law. A single equation was proposed for the apparent viscosity 1. Temperature played a major role in determining the magnitudes of the apparent viscosity 1. The completely destructed gilaboru samples flow curves were also measured after subjecting the samples to a high shear rate for 2h. After eliminating thixotropy by shearing, samples showed shearthinning properties that fitted well to the power law model. Three models were used to describe the time-dependent behaviour, namely, the second-order structural kinetic, Weltman and first-order stress decay models. Among the models, the first-order stress decay model fitted well compared to the second- order structural kinetic and Weltman models.

scholarly journals Second Order Equations in Functional Spaces: Qualitative and Discrete Well-Posedness

2015 ◽  
Vol 2015 ◽  
pp. 1-63 ◽  
Author(s):  
A. Ashyralyev ◽  
J. Pastor ◽  
S. Piskarev ◽  
H. A. Yurtsever
Keyword(s):  
Approximate Solution ◽  
Difference Equations ◽  
Second Order ◽  
Difference Schemes ◽  
Well Posedness ◽  
Present Survey ◽  
The Stability ◽  
Well Posed ◽  
Second Order Equations

The present survey contains the recent results on the local and nonlocal well-posed problems for second order differential and difference equations. Results on the stability of differential problems for second order equations and of difference schemes for approximate solution of the second order problems are presented.

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