Multiscaling analysis of a slowly varying anaerobic digestion model

We construct the slowly varying limiting state solutions to a nonlinear dynamical system for anaerobic digestion with Monod-based kinetics involving slowly varying model parameters arising from slow environmental variation. The advantage of these approximate solutions over numerical solutions is their applicability to a wide range of parameter values. We use these limiting state solutions to develop analytic approximations to the full nonlinear system by applying a multiscaling technique. The approximate solutions are shown to compare favorably with numerical solutions.     

An approximate solution for linear boundary-value problems with slowly varying coefficients

In this paper, an approximate closed-form solution for linear boundary-value problems with slowly varying coefficient matrices is obtained. The derivation of the approximate solution is based on the freezing technique, which is commonly used in analyzing the stability of slowly varying initial-value problems as well as solving them. The error between the approximate and the exact solutions is given, and an upper bound on the norm of the error is obtained. This upper bound is proportional to the rate of change of the coefficient matrix of the boundary-value problem. The proposed approximate solution is obtained for a two-point boundary-value problem and is compared to its solution obtained numerically. Good agreement is observed between the approximate and the numerical solutions, when the rate of change of the coefficient matrix is small.     

Multiscaling analysis of a slowly varying single species population model displaying an Allee effect

We consider the growth of a single species population modelled by a logistic equation modified to accommodate an Allee effect, in which the model parameters are slowly varying functions of time. We apply a multitiming technique to construct general approximate expressions for the evolving population in the case where the population survives to a (slowly varying) finite positive limiting state, and that where the population declines to extinction. We show that these expressions give excellent agreement with the results of numerical calculations for particular instances of the changing model parameters. Copyright © 2013 John Wiley & Sons, Ltd.     

Control of error in the homotopy analysis of solutions to the Zakharov system with dissipation

We apply the method of homotopy analysis to the Zakharov system with dissipation in order to obtain analytical solutions, treating the auxiliary linear operator as a time evolution operator. Evolving the approximate solutions in time, we construct approximate solutions which depend on the convergence control parameters. In the situation where solutions are strongly coupled, there will be multiple convergence control parameters. In such cases, we will pick the convergence control parameters to minimize a sum of squared residual errors. We explain the error minimization process in detail, and then demonstrate the method explicitly on several examples of the Zakharov system held subject to specific initial data. With this, we are able to efficiently obtain approximate analytical solutions to the Zakharov system of minimal residual error using approximations with relatively few terms.     

Numerical approximation of a metastable system

Using the Becker-Döring cluster equations as an example,we highlight some of the problems that can arise in the numericalapproximation of dynamical systems with slowly varying solutions.We describe the Becker-Döring model, summarize some ofits properties and construct a numerical approximation whichallows accurate and efficient computation of solutions in thelong, slowly varying metastable phase. We use the approximationto obtain test results and discuss the clear relationship betweenthem and equilibrium solutions of the Becker-Döring equations.     

On the solitary wave dynamics,under slowly varying medium,for nonlinear Schrödinger equations

We consider the problem of a solitary wave propagation, in a slowly varying medium, for a variable-coefficients nonlinear Schrödinger equation. We prove global existence and uniqueness of solitary wave solutions for a large class of slowly varying media. Moreover, we describe for all time the behavior of these solutions, which include refracted and reflected solitary waves, depending on the initial energy.     

Analysis of the power law logistic population model with slowly varying coefficients

We apply a multiscale method to construct general analytic approximations for the solution of a power law logistic model, where the model parameters vary slowly in time. Such approximations are a useful alternative to numerical solutions and are applicable to a range of parameter values. We consider two situations—positive growth rates, when the population tends to a slowly varying limiting state; and negative growth rates, where the population tends to zero in infinite time. The behavior of the population when a transition between these situations occurs is also considered. These approximations are shown to give excellent agreement with the numerical solutions of test cases. Copyright © 2012 John Wiley & Sons, Ltd.     

缓变深度分层流体中的准周期波和准孤立波

本文讨论具缓变深度二流体系统中的非线性波,该系统由一不规则底部与一水平固壁间的两层常密度无粘流体所组成.文中用约化摄动法导出了所考虑模型的变系数Korteweg-de Vries方程,并用多重尺度法求出了该方程的近似解,发现底部固壁的不规则变化将产生所谓准周期波和准孤立波.它们的周期、波速和波形将发生缓慢变化,文中给出了准周期波的周期随深度的变化关系式以及准孤立波波幅、波速随深度的变化关系式,底部水平情形和单层流体情形可看成是本文的特例.     

Bifurcations of autoresonant modes in oscillating systems with combined excitation

A mathematical model describing the capture of nonlinear systems into the autoresonance by a combined parametric and external periodic slowly varying perturbation is considered. The autoresonance phenomenon is associated with solutions having an unboundedly growing amplitude and a limited phase mismatch. The paper investigates the behavior of such solutions when the parameters of the excitation take bifurcation values. In particular, the stability of different autoresonant modes is analyzed and the asymptotic approximations of autoresonant solutions on asymptotically long time intervals are proposed by a modified averaging method with using the constructed Lyapunov functions.     

一类时滞Logistic方程正周期解的存在性

In the present paper,we consider the existence of positive periodic solu-tions for a kind of delay Logistic equations.By using a fixed point theorem in cones,we give some new existence results of single and multiple positive periodic solutionsfor a kind of delay Logistic equations.Some biomathematical models are presentedto illustrate our results.     

Approximation to a parabolic system modeling the thermoelastic contacts of two rods

In this article, we study a sequence of finite difference approximate solutions to a parabolic system, which models two dissimilar rods that may come into contact as a result of thermoelastic expansion. We construct the approximate solutions based on a set of finite difference schemes to the system, and we will prove that the approximate solutions converge strongly to the exact solutions. Moreover, we obtain and prove rigorously the error bound, which measures the difference between the exact solutions and approximate solutions in a reasonable norm. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 1–25, 1998     

Adiabatic approximations for Landau-Lifshitz equations

The asymptotics with respect to a small parameter for solutions of a system of Landau-Lifshitz equations with slowly varying coefficients and small dissipative terms is investigated. These equations are a mathematical model of a uniaxial ferromagnet in a time-dependent magnetic field. The asymptotics constructed make it possible to describe the magnetization reversal effect and to reveal the influence of the parameters of the external magnetic field and dissipation on the stability of this process.     

Transitions in a logistic population model incorporating an Allee effect

In some species, the population may decline to zero; that is, the species becomes extinct if the population falls below a given threshold. This phenomenon is well known as an Allee effect. In most Allee models, the model parameters are constants, and the population tends either to a nonzero limiting state (survival) or to zero (extinction). However, when environmental changes occur, these parameters may be slowly varying functions of time. Then, application of multitiming techniques allows us to construct approximations to the evolving population in cases where the population survives to a slowly varying surviving state and those where the population declines to zero. Here, we investigate the solution of a logistic population model exhibiting an Allee effect, when the carrying capacity and the limiting density interchange roles, via a transition point. We combine multiscaling analysis with local asymptotic analysis at the transition point to obtain an overall expression for the evolution of the population. We show that this shows excellent agreement with the results of numerical computations. Copyright © 2015 John Wiley & Sons, Ltd.     

Approximate periodic solutions of a nonlinear parabolic system and an identification problem

This work continues our study in [L. Lei, Identification of parameters through the approximate periodic solutions of a linear parabolic system, preprint, 2005] on the identification problem for the coefficients for the lower order terms in a parabolic system, through its approximate periodic solutions. Different from the work in [L. Lei, Identification of parameters through the approximate periodic solutions of a linear parabolic system, preprint, 2005], our system now is nonlinear and the coefficients to be detected are from the first order term. From the application point of view, we now try to determine the diffusion coefficients for the system by the observation over a subregion of the physical domain. The existence and uniqueness problem of the approximate periodic solutions is studied in the first part of the paper.     

Cosmological inflation models in the kinetic approximation

We consider the construction of cosmological inflation models with an approximate linear dependence of the kinetic energy of the scalar field on the state parameter. We compare the obtained solutions with known cosmological models and calculate the main parameters of cosmological perturbations.     

Survival to extinction in a slowly varying harvested logistic population model

This work considers a harvested logistic population for which birth rate, carrying capacity and harvesting rate all vary slowly with time. Asymptotic results from earlier work, obtained using a multiscaling technique, are combined to construct approximate expressions for the evolving population for the situation where the population initially survives to a slowly varying limiting state, but then, due to increasing harvesting, is reduced to extinction in finite time. These results are shown to give very good agreement with those obtained from numerical computation.     

On one class of solutions of a countable quasilinear system of differential equations with slowly varying parameters

For a countable quasilinear differential system whose coefficients are represented as Fourier series with slowly varying coefficients and frequency, we present conditions under which solutions of this system have analogous structure. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1121–1128, August, 1998.     

Approximate symmetries in viscoelasticity

In the framework of approximate symmetries, we investigate a perturbed system of partial differential equations for viscoelastic media with nonlinear dissipation. We completely classify the approximate symmetries and prove a theorem on the relation between the symmetries of two related models. In some physical cases, we find approximate solutions using the generator of the group of transformations taken in the first-order approximation.     

Classical solvability of a parabolic system of two nonlinear equations

We study the complete regularity of solutions of a nondiagonal parabolic system of quasilinear second-order differential equations in divergence form assuming that the coefficients are sufficiently slowly varying functions of their arguments and the off-diagonal terms in the coefficient matrix are sufficiently small. To this end, we use a method based on the successive approximation to the solution by smooth functions with the use of Schauder estimates at each step.