Periodic Plane Waves for the Oregonator

The Oregonator is a set of differential equations proposed by R. J. Field and R. M. Noyes as a model for the oscillating chemical reaction first studied by B. P. Belousov and A. M. Zhabotinskii. In this paper it is shown that the associated diffusion equations have periodic plane waves for parameter values not covered in earlier work. This amounts to studying a singularly perturbed system when nothing is known about the stability of periodic solutions for the reduced system.     

Block-iterative methods for consistent and inconsistent linear equations

Summary We shall in this paper consider the problem of computing a generalized solution of a given linear system of equations. The matrix will be partitioned by blocks of rows or blocks of columns. The generalized inverses of the blocks are then used as data to Jacobi- and SOR-types of iterative schemes. It is shown that the methods based on partitioning by rows converge towards the minimum norm solution of a consistent linear system. The column methods converge towards a least squares solution of a given system. For the case with two blocks explicit expressions for the optimal values of the iteration parameters are obtained. Finally an application is given to the linear system that arises from reconstruction of a two-dimensional object by its one-dimensional projections.     

Positive steady-state solutions of the Noyes–Field model for Belousov–Zhabotinskii reaction

In the paper, we investigate the Noyes–Field model for Belousov–Zhabotinskii reaction and study positive steady-state solutions of this model with the homogeneous Neumann boundary condition. We obtain the existence and non-existence of non-constant positive steady-state solutions.     

The similarity forms and invariant solutions of two-layer shallow-water equations

In this study symmetry group properties and general similarity forms of the two-layer shallow-water equations are discussed by Lie group theory. We represent that Lie group theory can be used as an effective approach for investigation of the self-similar solutions for the shallow-water equations with variable inflow as the generalization of dimensional analysis that was used so far for a regular approach in the literature. We also represent that the results obtained by dimensional analysis are just a special case of the results obtained by Lie group theory and it is possible to obtain the new similarity forms and the new variable inflow functions for the study of gravity currents in two-layer flow under shallow-water approximations based on Lie group theory. The symmetry groups of the system of nonlinear partial differential equations are found and the corresponding similarity and reduced forms are obtained. Some similarity solutions of the reduced equations are investigated. It is shown that reduced equations and similarity forms of the system depend on the group parameters. We show that an analytic similarity solution for the system of equations can be found for some special values of them. For other values of the group parameters, the similarity solutions of the two-layer shallow-water equations representing the gravity currents with a variable inflow are found by the numeric integration.     

构造非线性微分差分方程精确解的一种方法

把最近提出的G′/G展开法推广到了非线性微分差分方程,利用该方法成功构造了一种修正的Volterra链和Toda链的双曲函数、三角函数以及有理函数三类涉及任意参数的行波解,当这些参数取特殊值时,可得这两个方程的扭状孤立波解、奇异行波解以及三角函数状的周期波解等.研究结果表明,该算法探讨非线性微分差分方程精确解十分有效、简洁.     

Chaos in a bienzymatic cyclic model with two autocatalytic loops

A general bienzymatic cyclic system including two autocatalytic loops is studied and used as a basic design principle for modelling extracellular matrix turnover. Using classical enzyme kinetic rates, the model is described by a set of four ordinary differential equations and numerically studied by bifurcation diagrams and Poincaré sections. We observe limit-cycle oscillations and chaotic behaviors arising from period-doubling cascades or intermittency. Chaotic oscillations originate from distinct strange attractors that undergo boundary and internal crisis. For some parameter values, the system presents several bistable areas, where a limit cycle coexists with another one or with a strange attractor. The dynamics are qualitatively modified when the weight of the autocatalytic loops on the system varies, resulting in the change in the number of attractors.     

On Analyses of the Solution to a Linear Programming Problem

Many complex problem situations in various contexts have been represented in recent years by the linear programming model. The simplex method can then be used to give the optimal values of the variables corresponding to a given set of values of the parameters. However, in many situations it is useful to have the solution to many other related problems which differ from the original problem only in the values of some of the parameters. This paper presents procedures by which the solutions to the changed problems can be derived from the simplex solution tableau corresponding to the original problem. The method will be illustrated by means of an example problem, and it will be shown how quantitative information obtained from such analyses can aid management in decision making.     

On Computing Travelling Wave Solutions in a Model for the Belousov Zhabotinskii Reaction

The Belousov Zhabotinskii reaction is a visually dramatic oscillatingphenomenon for which a chemical model has been developed byField & Noyes (1974). This model results in a pair of non-linearparabolic partial differential equations for which we shallcompute a difference solution. Each approximate solution consistsof a steady state waveform and the speed with which this wavepropagates.     

Stability sets of multiparameter Hamiltonian systems

A real linear Hamiltonian system with constant coefficients that depend on several real parameters is considered. A method is proposed for calculating the sets of all values of the parameters for which the stationary solution of this system is stable for fixed values of the parameters (that is, the stability sets). The application of the method is demonstrated for a gyroscopic problem described by a Hamiltonian system with four degrees of freedom and three parameters. Computer algebra, in particular, a Gröbner basis and a Power Geometry are used. It is shown that the four-parameter generalization of this problem does not contain fundamentally new difficulties.     

Turing pattern outside of the Turing domain

There are two simple solutions to reaction-diffusion systems with limit-cycle reaction kinetics, producing oscillatory behaviour. The reaction parameter mu gives rise to a 'space-invariant' solution, and mu versus the ratio of the diffusion coefficients gives rise to a 'time-invariant' solution. We consider the case where both solution types may be possible. This leads to a refinement of the Turing model of pattern formation. We add convection to the system and investigate its effect. More complex solutions arise that appear to combine the two simple solutions. The convective system sheds light on the underlying behaviour of the diffusive system.     

Parametric analysis of the oscillatory solutions to stochastic differential equations with the Wiener and Poisson components by the Monte Carlo method

Using the Monte Carlo method, we address the influence of the Wiener and Poisson random noises on the behavior of oscillatory solutions to systems of stochastic differential equations (SDEs). For the linear and Van der Pol oscillators, we study the accuracy of estimates of the functionals of numerical solutions to SDEs obtained by the generalized explicit Euler method. For a linear oscillator, we obtain the exact analytical expressions for the mathematical expectation and the variance of the SDE solution. These expressions allow us to investigate the dependence of the accuracy of estimates of the solution moments on the values of SDE parameters, the size of meshsize, and the ensemble of simulated trajectories of the solution. For the Van der Pol oscillator, we study the dependence of the frequency and the damping rate of the oscillations of the mathematical expectation of SDE solution on the values of parameters of the Poisson component. The results of the numerical experiments are presented.     

Phase transitions for Belousov–Zhabotinsky reactions

The main objective of this article is to study the dynamic phase transitions associated with the spatial–temporal oscillations of the BZ reactions, given by Field, Körös and Noyes, also referred as the Oregonator. Two criteria are derived to determine (1) the existence of either multiple equilibria or spatiotemporal oscillations and (2) the types of transitions. These criteria gives a complete characterization of the dynamic transitions of the BZ systems from the homogeneous states. The analysis is carried out using a dynamic transition theory developed recently by the authors, which has been successfully applied to a number of problems in science. Copyright © 2011 John Wiley & Sons, Ltd.     

The first fundamental problem of the theory of elasticity for a rectangular parallelepiped

In 1852 Lame [1] formulated the first fundamental problem of the theory of elasticity for a rectangular parallelepiped. An approximate solution to this problem was given by Filonenko-Borodich [2 and 3] who used Castigliano's variational principle. Later Mishonov [4] obtained an approximate solution to Lamé's problem in the form of divergent triple Fourier series. These series contain constants which are found from infinite systems of linear equations. Teodorescu [5] has considered a particular case of Lame's problem. Using his own method the author solves the problem in the form of double series analogous to those used in [6 to 8] and by Baida in [9 and 10] in solving problems on the equilibrium of a rectangular parallelepiped. The solution of the problem reduces to three infinite system of linear equations and the author asserts that these infinite systems are regular. It is shown in Section 5 that the infinite systems obtained by Teodorescu, on the other hand, will not be regular.

In the references mentioned above which investigate Lamé's problem the authors confine their attention either to obtaining a solution by an approximate method, or to reducing the solution process to one of obtaining infinite systems, leaving these uninvestigated. It must be emphasized that the main difficulty in solving this problem lies in investigating the infinite systems obtained which are significantly different from the infinite systems of the corresponding plane problem.

In this paper a solution is given to the first fundamental problem of the theory of elasticity for a rectangular parallelepiped with prescribed external stresses on the surface (Sections 2, 3 and 4). For the solution of this problem the author has used a form of the general solution of the homogeneous Lamé equations which contains five arbitrary harmonic functions and which constitutes a generalization of the familiar Papkovich-Neuber solution (Section 1). The solution is expressed in the form of double series containing four series of unknown constants which can be found from four infinite systems of linear algebraic equations. The infinite systems of linear equations obtained is studied for values of Poisson's ratio within the range 0 < σ ≤ 0.18. It is shown that for these values of Poisson's ratio the infinite systems are quasi-fully regular.     

The free-parameter solution of the convection equations in a vertical cylinder with a volume heat source

An exact solution of the free-convection equations is constructed in the Oberbeck–Boussinesq approximation, describing the flow of a viscous heat-conducting fluid in a vertical cylinder of large radius when heated by radiation. The initial problem is reduced to an operator equation with an extremely non-linear operator, satisfying Schauder's theorem in C[0,1]. An iteration procedure is proposed for determining the independent parameter, that occurs in the solution, which enables three different values to be obtained and, correspondingly, three classes of solution of the initial problem. The linear stability of all the solutions obtained is investigated and it is shown that, for chosen values of the problem parameters, the most dangerous one is the plane wave mode and two instability mechanisms are present. The flow structure and the type of instability depend considerably on the values of the free parameter.     

A new correlation method for the aerodynamic service loads determination of a rigid wing based on CFD analysis

A new correlation method for the aerodynamic service loads determination of a rigid wing based on CFD analysis is presented. All flight conditions can be handled by the proposed method. The derived correlation equations are achieved by considering a training fighter aircraft as a prototype. Each wing of aircraft is divided into thirty three parts in the span wise direction. Extensive numerical solutions have been attempted by varying a number of parameters that directly affect the wings aerodynamic loads, such as Mach numbers, angle of attack, control surfaces deflections and etc. For each set of input parameters, the corresponding aerodynamic loads applied to different wing parts are calculated. The resulted loads and the corresponding input parameters are incorporated into a linear regression method in order to develop the appropriate correlation equations. The outputs of the developed equations are the aerodynamic loads at each part of the wing based on the independent variables, which are the above mentioned input parameters. The validity of the developed equations is shown by comparing the loads obtained from the latter equations with the corresponding ones calculated through numerical analysis for different flight conditions. The correlation equations can now be used to calculate the aerodynamic loads at each part for any set of arbitrary values assigned to the input parameters.     

线性定常系统特征模型的特征参量辨识

说明线性定常系统特征模型的特征参量是一组由高阶线性定常系统的相关信息压缩而成,于是不能简单的作为与状态无关的慢时变参数来处理. 基于特征建模思想,建立了线性定常系统特征模型的特征参量与子空间方法之间的联系,给出了一种该特征模型的特征参量 的合成辨识算法.同时证明了在用于子空间辨识的样本量充分大和用于状态估计的时间充分长的情况下, 特征参量的估计值与真值之间的误差达到充分小. 最后,对于一个六阶的单输入单输出线性定常系统的仿真例子,对投影的带遗忘因子最小二乘算法和合成辨识算法进行了比较,验证了合成辨识算法的有效性.     

Exact solutions of potentiostatic current transients for a corrosion reaction under mixed charge transfer and diffusion control

An analytical solution has been obtained for the current response transients to a potentiostatic step for a corrosion reaction where a planar metal electrode is immersed in an acidic electrolyte. This solution can be used to provide values of important kinetic parameters of the reaction such as corrosion current and Tafel slopes. The model is derived for conditions of mixed charge transfer and diffusion control and includes the metal ion deposition partial reaction. A Nernstian diffusion model is used, where anodic and cathodic species diffuse in a finite diffusive layer. A numerical solution has previously been published [1], but it is believed that this is the first time an analytical solution has been presented. Transients obtained by including both the metal dissolution and metal-ion deposition (back) reactions, are compared to those obtained by neglecting the metalion deposition reaction. It is shown that neglecting the back reaction can cause significant errors especially when large cathodic potential steps are applied and as the corrosion current approaches the limiting current density of the cathodic reaction.     

圆薄板大挠度问题的摄动参数

本文研究了在用正规摄动法求解均布载荷下的圆薄板大挠度问题时,与载荷,挠度,转角,内力等有关的各种摄动参数,并对一般摄动参数情形用变分原理求得了解答本文从实验的角度阐明了各参数的适用范围,结果表明,相应的解答与用中心挠度为参数的解有较好的一致性;对均布载荷的情形,中心挠度仍可看做是较为合适的摄动参数;本文推荐的摄动参数及用变分原理确定摄动解的方法,具有普遍的适用性,可以用来处理载荷联合作用等更为复杂的情形.     

Thermal analysis of a longitudinal trapezoidal fin with temperature-dependent thermal conductivity and heat transfer coefficient

A homotopy analysis method (HAM) is used to develop analytical solution for the thermal performance of a straight fin of trapezoidal profile when both the thermal conductivity and the heat transfer coefficient are temperature dependent. Results are presented for the temperature distribution, heat transfer rate, and fin efficiency for a range of values of parameters appearing in the mathematical model. Since the HAM algorithm contains a parameter that controls the convergence and accuracy of the solution, its results can be verified internally by calculating the residual error. The HAM results were also found to be accurate to at least three places of decimal compared with the direct numerical solution of the mathematical model generated using a fourth–fifth-order Runge–Kutta–Fehlberg method. The HAM solution appears in terms of algebraic expressions which are not only easy to compute but also give highly accurate results covering a wide range of values of the parameters rather than the small values dictated by the perturbation solution.     

Inversion of seismic data: sensitivity and stability via SVD analysis

The problem of recovery of seismic parameters of the media via wavefield produced by array of sources and recorded by array of receivers is considered in this work. In order to invert these data and recover elastic parameters one can use the optimization technique based on the gradient–like or Newton–like methods. In seismic applications this approach is known as “Full waveform inversion”. According to it we search for solution which minimizes mean–square deviation of the observed wavefield from the computed one for current values of elastic parameters. Surely convergence is governed by the properties of the Frechet derivative of nonlinear operator that maps medium parameters to the observed data. Thus studying these properties is an important step before development the numerical methods and algorithms of this inversion. For a simple case it can be shown that this derivative is a compact operator so implementation of any Newton–like approach is connected with necessity to resolve ill–posed problem of resolution of the first–order linear integral operational equation. In order to study the main peculiarities of this operator Singular Value Decomposition is applied. Two acquisition systems are dealt with – the offset vertical seismic profiling and cross–well tomography. Numerical results for realistic media are presented and the main differences of inverse problems for these two acquisition are shown. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)