厚环壳的渐近求解方程和作用弯矩M0的解

本文从三维弹性力学基本方程出发，利用几何小参数ａ＝ｒ＿０／Ｒ＿摄动展开，得到了任意载荷下，厚环壳的各级渐近求解方程。它可以分成两组类似平面应变问题和扭转问题的独立方程组。用此方程求得了厚环壳受弯矩Ｍ＿０作用的两级渐近解。     

关于耦合型对流-扩散方程组的奇异摄动边值问题

本文考虑下述耦合型对流-扩散方程组的奇异摄动边值问题:本文提出两种方法:一种是初值化解法,用这种方法,原始问题转化成一系列没有扰动的一阶微分方程或方程组的初值问题,从而得到一个渐近展开式;第二种是边值化解法,用这种方法,原始问题转化成一组没有边界层现象的边值问题,从而可以得到精确解和使用经典的数值方法去得到具有关于摄动参数ε一致的高精度数值解.     

A General Approach to Obtain Series Solutions of Nonlinear Differential Equations

Based on homotopy, which is a basic concept in topology, a general analytic method (namely the homotopy analysis method) is proposed to obtain series solutions of nonlinear differential equations. Different from perturbation techniques, this approach is independent of small/large physical parameters. Besides, different from all previous analytic methods, it provides us with a simple way to adjust and control the convergence of solution series. Especially, it provides us with great freedom to replace a nonlinear differential equation of order n into an infinite number of linear differential equations of order k , where the order k is even unnecessary to be equal to the order n . In this paper, a nonlinear oscillation problem is used as example to describe the basic ideas of the homotopy analysis method. We illustrate that the second-order nonlinear oscillation equation can be replaced by an infinite number of (2κ)th-order linear differential equations, where κ≥ 1 can be any a positive integer. Then, the homotopy analysis method is further applied to solve a high-dimensional nonlinear differential equation with strong nonlinearity, i.e., the Gelfand equation. We illustrate that the second-order two or three-dimensional nonlinear Gelfand equation can be replaced by an infinite number of the fourth or sixth-order linear differential equations, respectively. In this way, it might be greatly simplified to solve some nonlinear problems, as illustrated in this paper. All of our series solutions agree well with numerical results. This paper illustrates that we might have much larger freedom and flexibility to solve nonlinear problems than we thought traditionally. It may keep us an open mind when solving nonlinear problems, and might bring forward some new and interesting mathematical problems to study.     

Simplified renormalization group method for ordinary differential equations

The renormalization group (RG) method for differential equations is one of the perturbation methods which allows one to obtain invariant manifolds of a given ordinary differential equation together with approximate solutions to it. This article investigates higher order RG equations which serve to refine an error estimate of approximate solutions obtained by the first order RG equations. It is shown that the higher order RG equation maintains the similar theorems to those provided by the first order RG equation, which are theorems on well-definedness of approximate vector fields, and on inheritance of invariant manifolds from those for the RG equation to those for the original equation, for example. Since the higher order RG equation is defined by using indefinite integrals and is not unique for the reason of the undetermined integral constants, the simplest form of RG equation is available by choosing suitable integral constants. It is shown that this simplified RG equation is sufficient to determine whether the trivial solution to time-dependent linear equations is hyperbolically stable or not, and thereby a synchronous solution of a coupled oscillators is shown to be stable.     

Modified poisson series to implement the method of multiple scales

The method of multiple scales is a global perturbation technique that has resulted to be very useful in perturbed ordinary differential equations characterized by disparate time scales. The general principle behind the method is that the solution to the differential equation is uniformly expanded in terms of two or more independent variables, referred to as time scales. In this article, we present a mathematical object based on a Poisson series to apply the method of multiple scales via specific symbolic computation. Copyright © 2016 John Wiley & Sons, Ltd.     

A seminumeric approach for solution of the Eikonal partial differential equation and its applications

In this work, a partial differential equation, which has several important applications, is investigated, and some techniques based on semianalytic (or quasi‐numerical) approaches are developed to find its solution. In this article, the homotopy perturbation method (HPM), Adomian decomposition method, and the modified homotopy perturbation method are proposed to solve the Eikonal equation. HPM yields solution in convergent series form with easily computable terms, and in some case, yields exact solutions in one iteration. In other hand, in Adomian decomposition method, the approximate solution is considered as an infinite series usually converges to the accurate solution. Moreover, these methods do not require any discretization, linearization, or small perturbation, and therefore reduce the numerical computation a lot. Several test problems are given and results are compared with the variational iteration method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Large fluctuations and growth rates of linear Volterra summation equations

This paper concerns the asymptotic behaviour of solutions of a linear convolution Volterra summation equation with an unbounded forcing term. In particular, we suppose the kernel is summable and ascribe growth bounds to the exogenous perturbation. If the forcing term grows at a geometric rate asymptotically, or is bounded by a geometric sequence, then the solution (appropriately scaled) omits a convenient asymptotic representation. Moreover, this representation is used to show that additional growth properties of the perturbation are preserved in the solution. If the forcing term fluctuates asymptotically, we prove that fluctuations of the same magnitude will be present in the solution and we also connect the finiteness of time averages of the solution with those of the perturbation. Our results, and corollaries thereof, apply to stochastic as well as deterministic equations, and we demonstrate this by studying some representative classes of examples.     

An iterative method based on equation decomposition for the fourth‐order singular perturbation problem

In this article, we propose an iterative method based on the equation decomposition technique ( 1 ) for the numerical solution of a singular perturbation problem of fourth‐order elliptic equation. At each step of the given method, we only need to solve a boundary value problem of second‐order elliptic equation and a second‐order singular perturbation problem. We prove that our approximate solution converges to the exact solution when the domain is a disc. Our numerical examples show the efficiency and accuracy of our method. Our iterative method works very well for singular perturbation problems, that is, the case of 0 < ε ? 1, and the convergence rate is very fast. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Solitary wave solution of higher-order Korteweg–de Vries equation

An attempt has been made to obtain exact analytical traveling wave solution or simple wave solution of higher-order Korteweg–de Vries (KdV) equation by using tanh-method or hyperbolic method. The higher-order equation can be derived for magnetized plasmas by using the reductive perturbation technique. It is found that the exact solitary wave solution of higher-order KdV equation is obtained by tanh-method. Using this method, different kinds of nonlinear wave equations can be evaluated. The higher-order nonlinearity and higher-order dispersive effect can be observed from the solutions of the equations. The method is applicable for other nonlinear wave equations.     

A unified approach for solving nonlinear regular perturbation problems

This aritcle describes a simple alternative unified method of solving nonlinear regular perturbation problems. The procedure is based upon the manipulation of Taylor's approximation for the expansion of the nonlinear term in the perturbed equation.

An essential feature of this technique is the relative simplicity used and the associated unified computational procedure that is employed. As such it should be of interest to teachers of applied mathematics courses, particularly those courses which include perturbation methods.

One of the merits of this approach is that it leads on naturally to a scheme based on Taylor's expansions and, consequently, allows the regular perturbation method to be introduced into a course much earlier than is currently common.

The method is illustrated by implementing it to four perturbation problems, including two algebraic equations and two initial-value problems. The new approach is compared and contrasted with the traditional perturbation scheme in order to demonstrate its relative merit.     

一个两流体系统中mKdV孤立波的迎撞*

本文从文[2]的基本方程出发,采用约化摄动方法和PLK方法,讨论了三阶非线性和色散效应相平衡的修正的KdV(mKdV)孤立波迎撞问题.这些波在流体密度比等于流体深度比平方的两流体系统界面上传播.我们求得了二阶摄动解,发现在不考虑非均匀相移的情况下,碰撞后孤立波保持原有的形状,这与Fornberg和whitham[6]的追撞数值分析结果一致,但当考虑波的非均匀相移后,碰撞后波形将变化.     

A hybrid method for pricing European options based on multiple assets with transaction costs

The problem of pricing European options based on multiple assets with transaction costs is considered. These options include, for example, quality options and options on the minimum of two or more risky assets. The value of these options is the solution of a nonlinear parabolic partial differential equation subject to a final condition given by the payoff function associated with the option. A computationally efficient method to solve this final-value problem is proposed. This method is based on an asymptotic expansion of the required solution with respect to the parameters related to the transaction costs followed by the numerical solution of the linear partial differential equations obtained at each order in perturbation theory. The numerical solution of these linear problems involves an implicit finite-difference scheme for the parabolic equation and the use of the fast Fourier sine transform to solve the resulting elliptic problems. Numerical results obtained on test problems with the method proposed here are shown and discussed.     

关于钱氏摄动法的高阶解的计算机求解和收敛性的研究

本文借助于中心受集中载荷圆板小挠度问题的积分方程,获得了摄动参数为中心挠度的任意n阶摄动解的解析式.于是,任意次摄动解的所有待定系数能用计算机求解.因此,获得了相当高阶的摄动解.在此基础上,讨论了钱氏摄动法的渐近性和适用区.     

一类非线性离散系统的指数二分性及其在数值分析和计算中的应用

本文中我们考虑一类二阶非线性常微分方程的边值问题的迎风差分格式.我们运用奇异摄动方法构造了该迎风差分方程解的渐近近似,并利用指数二分性理论证明了有一个低阶方程其解是该迎风方程式的在边界外的一个良好近似.我们还构造了校正项,使校正项与低阶方程的解之和是一个渐近近似.最后一些数值例子用于显示本文方法的应用.     

Adomian’s decomposition method and homotopy perturbation method in solving nonlinear equations

The Adomian’s decomposition method and the homotopy perturbation method are two powerful methods which consider the approximate solution of a nonlinear equation as an infinite series usually converging to the accurate solution. By theoretical analysis of the two methods, we show, in the present paper, that the two methods are equivalent in solving nonlinear equations.     

A Volterra integral formulation for determining the periodic solutions of some autonomous, nonlinear, third-order ordinary differential equations

A Volterra integral formulation based on the introduction of a term proportional to the velocity times the square of the (unknown) frequency of oscillation, a new independent variable equal to the original one times the (unknown) frequency of oscillation, the method of variation of parameters and series expansions of both the solution and the frequency of oscillation, is used to determine the periodic solutions to three nonlinear, autonomous, third-order, ordinary differential equations. It is shown that the first term of the series expansion of the frequency of oscillation coincides with that determined from a first-order harmonic balance procedure, whereas the two-term approximation to the frequency of oscillation is shown to be more accurate than that of a parameter perturbation procedure for the second equation considered in this paper. For the third equation, it is shown that the two-term approximation presented in this paper is more accurate than the corresponding one for one of the parameter perturbation methods, and for initial velocities less than one, for the other parameter perturbation approach.     

Automated derivation of optimal regulators for nonlinear systems by symbolic manipulation of Poisson series

Concepts of programmability and compact programmability are defined relative to a class of modified Poisson series. Lie-series-based canonical perturbation methods from astrodynamics are applied to the Hamiltonian system boundary-value problem, and more usual methods are applied to the perturbed Hamilton-Jacobi-Bellman partial differential equation, in order to obtain a complete set of equations for the perturbed optimal feedback control law for both infinite-time and finite-time regulator problems. The relative advantages of each approach are evaluated. A major aim of the paper is to determine the largest class of perturbations, within the set of Poisson series, for which the equations can be derived on a computer by symbolic manipulation. The more general the class, the more accurate the perturbation solution can be, for a given order. The solutions developed are complete; all that remains is to program them in order to have computerized derivations of the optimal nonlinear feedback control laws.This research was supported by NSF Grant No. ENG-78-10232. This paper was presented at the 1982 Conference on Information Sciences and Systems, Princeton, New Jersey, and appears in the Proceedings.     

Symmetries and Large Time Asymptotics of Compressible Euler Flows with Damping

Large time asymptotics of compressible Euler equations for a polytropic gas with and without the porous media equation are constructed in which the Barenblatt solution is embedded. Invariance analysis for these governing equations are carried out using the classical and the direct methods. A new second order nonlinear partial differential equation is derived and is shown to reduce to an Euler–Painlevé equation. A regular perturbation solution of a reduced ordinary differential equation is determined. And an exact closed form solution of a system of ordinary differential equations is derived using the invariance analysis.     

An analytical technique for solving a class of strongly nonlinear conservative systems

Recently, an analytical technique has been developed to determine approximate solutions of strongly nonlinear differential equations containing higher order harmonic terms. Usually, a set of nonlinear algebraic equations is solved in this method. However, analytical solutions of these algebraic equations are not always possible, especially in the case of a large oscillation. Previously such algebraic equations for the Duffing equation were solved in powers of a small parameter; but the solutions measure desired results when the amplitude is an order of 1. In this article different parameters of the same nonlinear problems are found, for which the power series produces desired results even for the large oscillation. Moreover, two or three terms of this power series solution measure a good result when the amplitude is an order of 1. Besides these, a suitable truncation formula is found in which the solution measures better results than existing solutions. The method is mainly illustrated by the Duffing oscillator but it is also useful for many other nonlinear problems.