Age-dependent Population Growth

This paper considers a non-linear deterministic model, in whichthe death rate rises as the population grows. The model is inthe form of a semi-linear first-order partial differential equation,with respect to time and age. It is an age-dependent versionof a simple equation (the logistic equation) often used to describepopulations whose growth is controlled by limited resources.The problem reduces to the solution of a non-linear integralequation; it has a constant solution, which is proved to beglobally asymptotically stable. This implies that there areno steady oscillations, and that in the long run the populationsize and age-structure become fixed, independent of the initialconditions. Further details of the solution are discussed, includingnumerical results.     

EXACT EXPLICIT SOLUTIONS OF THE NONLINEAR SCHR(O)DINGER EQUATION COUPLED TO THE BOUSSINESQ EQUATION

A system comprised of the nonlinear Schrodinger equation coupled to theBoussinesq equation (S-B equations) which dealing with the stationary propagation of cou-pled non-linear upper-hybrid and magnetosonic waves in magnetized plasma is proposed.To examine its solitary wave solutions, a reduced set of ordinary differential equations areconsidered by a simple traveling wave transformation. It is then shown that several newsolutions (either functional or parametrical) can be obtained systematically, in addition torederiving all known ones by means of our simple and direct algebra method with the helpof the computer algebra system Maple.     

An approach for relating social structure to cognitive structure

There are three different reasons why non-linear functions between social macro-variables (aggregates) may arise. They can be related to three basic steps of a specific model of explanation of social phenomena, defined by Coleman: the logic of situation, the logic of selection and the logic of transition. Starting with a model of one linear difference equation, a change to three different non-linear system equations can generate stable cycles, bifurcations, and chaos. These non-linear system equations can be deduced from simple assumptions about individual or institutional social attributes. It is shown that a) non-linear individual reactions, b) different selection rules for actors having different social attributes and c) institutional constraints resulting in different transition processes are possible causes for non-linearity at the system level. Furthermore it is demonstrated that the assumption of non-linear but homogeneous reactions of all persons have a similar effect on non-linearity, like it is the case for different selection rules. However, despite of being able to show mathematically the possibility of chaos, it has to be said that chaos as a durable state of social systems is very improbable.     

On a kinetic model for shallow water waves

The system of shallow water waves is one of the classical examples for non-linear, two-dimensional conservation laws. The paper investigates a simple kinetic equation depending on a parameter ? which leads for ? → 0 to the system of shallow water waves. The corresponding ‘equilibrium’ distribution function has a compact support which depends on the eigenvalues of the hyperbolic system. It is shown that this kind of kinetic approach is restricted to a special class of non-linear conservation laws. The kinetic model is used to develop a simple particle method for the numerical solution of shallow water waves. The particle method can be implemented in a straightforward way and produces in test examples sufficiently accurate results.     

EXACT EXPLICIT SOLUTIONS OF THE NONLINEAR SCHRDINGER EQUATION COUPLED TO THE BOUSSINESQ EQUATION

A system comprised of the nonlinear Schrodinger equation coupled to the Boussinesq equation (S-B equations) which dealing with the stationary propagation of coupled non-linear upper-hybrid and magnetosonic waves in magnetized plasma is proposed. To examine its solitary wave solutions, a reduced set of ordinary differential equations are considered by a simple traveling wave transformation. It is then shown that several new     

An improved pegasus method for root finding

A method of Regula Falsi type for finding a simple root of a non-linear equation is presented. It is similar to the Pegasus procedure, but exhibits a higher asymptotic convergence efficiency. This is accomplished by eliminating one or two of the slower Pegasus substeps.Work performed under the auspices of the United States Atomic Energy Commission.     

一个三阶牛顿变形方法

基于反函数建立的积分方程,结合Simpson公式,给出了一个非线性方程求根的新方法,即为牛顿变形方法.证明了它至少三次收敛到单根,与牛顿法相比,提高了收敛阶和效率指数.文末给出数值试验,且与牛顿法和同类型牛顿变形法做了比较.结果表明方法具有较好的优越性,它丰富了非线性方程求根的方法.     

用图像研究非线性迭代模型

用蛛网图、迭代函数图、周期分叉图以及分布直方图等几何图像研究了L og istic模型的迭代轨道,研究了非线性迭代轨道进入混沌状态的条件.     

一类三阶牛顿变形方法

A class of third-order convergence methods of solving roots for non-linear equation,which are variant Newton's method,are given.Their convergence properties are proved.They are at least third order convergence near simple root and one order convergence near multiple roots.In the end,numerical tests are given and compared with other known Newton's methods.The results show that the proposed methods have some more advantages than others.They enrich the methods to find the roots of non-linear equations and they are important in both theory and application.     

A non-linear parabolic control problem with non-homogeneous boundary condition—convergence of Galerkin approximation

The paper is concerned with optimal control problem for a non-linear parabolic equation with non-homogenous boundary condition and quadratic cost. The control is acting in a nonlinear equation. We derive some results on the existence of optimal controls. Then we treat optimal control problem by Galerkin method and we prove the convergence of optimal values for approximated control problems to the one for the original problem. Finally, we apply the results to give a simple example. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

A family of ellipse methods for solving non-linear equations

This note presents a method for the numerical approximation of simple zeros of a non-linear equation in one variable. In order to do so, the method uses an ellipse rather than a tangent approach. The main advantage of our method is that it does not fail even if the derivative of the function is either zero or very small in the vicinity of the required root.     

Ricci-flat metrics on the complexification of a compact rank one symmetric space

We construct a complete Ricci-flat Kähler metric on the complexification of a compact rank one symmetric space. Our method is to look for a Kähler potential of the form ψ = ?(u), whereu satisfies the homogeneous Monge-Ampère equation. We use the high degree of symmetry present to reduce the non-linear partial differential equation governing the Ricci curvature to a simple second-order ordinary differential equation for the functionf. To prove that the resulting metric is complete requires some techniques from symplectic geometry.     

Meshless local Petrov-Galerkin (MLPG) approximation to the two dimensional sine-Gordon equation

During the past few years, the idea of using meshless methods for numerical solution of partial differential equations (PDEs) has received much attention throughout the scientific community, and remarkable progress has been achieved on meshless methods. The meshless local Petrov-Galerkin (MLPG) method is one of the “truly meshless” methods since it does not require any background integration cells. The integrations are carried out locally over small sub-domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. In this paper the MLPG method for numerically solving the non-linear two-dimensional sine-Gordon (SG) equation is developed. A time-stepping method is employed to deal with the time derivative and a simple predictor-corrector scheme is performed to eliminate the non-linearity. A brief discussion is outlined for numerical integrations in the proposed algorithm. Some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped SG equation is investigated. The final numerical results confirm the ability of proposed method to deal with the unsteady non-linear problems in large domains.     

The inverse electromagnetic scattering problem by a penetrable cylinder at oblique incidence

In this work, we consider the method of non-linear boundary integral equation for solving numerically the inverse scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinder in three dimensions. We consider the indirect method and simple representations for the electric and the magnetic fields in order to derive a system of five integral equations, four on the boundary of the cylinder and one on the unit circle where we measure the far-field pattern of the scattered wave. We solve the system iteratively by linearizing only the far-field equation. Numerical results illustrate the feasibility of the proposed scheme.     

Non-linear expectations in spaces of Colombeau generalized functions

We introduce the notion of a non-linear expectation in spaces of Colombeau generalized functions and provide its characterization in terms of the upper expectation over a set of probability measures. We then study a fully non-linear backward stochastic differential equation in the Colombeau setting via its connection with the corresponding fully non-linear partial differential equation.     

One dimensional stochastic partial differential equations and the branching measure diffusion

Summary An existence theorem in the spirit of Keisler [Ke], is proved for the simple one-dimensional diffusion equation driven by white noise modulated by a non-linear function of the solution. This is used to obtain a density and a Stochastic Partial Differential Equation in one dimension for the critical branching diffusion studied by Dawson et al.This research was carried out while the author was at the Department of Mathematics, University of British Columbia     

求解非线性算子方程的加速迭代格式

研究非线性算子方程的近似求解方法.首先对通常的求解非线性方程加速迭代格式进行推广,得到高阶收敛速度的加速迭代格式,最后把这种加速迭代格式推广到非线性算子方程的求解中去,利用非线性算子的渐进展开,证明了这种加速格式具有三阶的收敛速度.     

Equivalence group of a fourth-order evolution equation unifying various non-linear models

A fourth-order non-linear evolutionary partial differential equation containing several arbitrary functions of the dependent variable is considered. This equation arises as a generalization of various non-linear models describing a non-linear heat diffusion, the dynamics of thin liquid films, etc. Equivalence transformations give more flexibility to the unified model. We determine the generators of the equivalence group and use them for specifying certain types of arbitrary functions when the model equation has additional symmetries, and hence admits non-trivial group invariant solutions.     

Runge-Kutta方法用于非线性方程求根

将Runge-Kutta方法用于非线性方程求根问题,给出二阶,三阶和四阶对应的三个新的方程求根公式,证明了它们至少三次收敛到单根,线性收敛到重根.文末给出数值试验,且与其它已知求根公式做了比较.结果表明此方法具有较好的优越性,它们丰富了非线性方程求根的方法,在理论上和应用上都有一定的价值.