一种基于积分方程的数值微分方法

本文研究了目前一些求解数值微分的方法无法求出端点导数或是求出的端点附近导数不可用的问题.利用构造一类积分方程的方法,将数值微分问题转化为这类积分方程的求解,并用一种加速的迭代正则化方法来求解积分方程. 数值实验结果表明该算法可以有效求出端点的导数,且具有数值稳定、计算简单等优点.     

首次积分法求时空-分数阶MkdV-ZK方程新的精确解

为了给物理学中的动力学行为研究提供依据,更好解释一些物理现象.首先使用分数阶复变换将时空-分数阶MKdV-ZK方程转换为非线性常微分方程组,其次使用除法定理寻求常微分方程组的首次积分,最后使用首次积分求解出原方程的许多精确解,得到了时空-分数阶MKdV-ZK方程的新精确解.数值结果表明首次积分法是有效的,该方法具有简单便捷等优点.     

基于周期变换求解带尖角区域的声波散射问题

利用周期变换和位势理论将声波散射问题转化为第二类边界积分方程,再利用Nystrom方法来求解该边界积分方程.给出二维空间的数值例子,结果表明该方法简单,可行并且具有较好的精度.     

波形松弛算法及其在计算流体力学中的应用

本文介绍求解线性常系数微分代数方程组的波形松弛算法, 基于Laplace积分变换得到该算法新的收敛理论. 进一步将波形松弛算法应用于求解非定常Stokes方程, 介绍并讨论了连续时间波形松弛算法CABSOR算法和离散时间波形松弛算法DABSOR算法.     

具任意次非线性项的Camassa-Holm方程的双孤子新解

给出辅助方程、函数变换与变量分离解相结合的方法,构造了具任意次非线性项的Camassa-Holm方程的双孤子和双周期新解.首先,通过两个辅助方程、函数变换与变量分离解,将具任意次非线性项的Camassa-Holm方程的求解问题转化为非线性代数方程的求解问题.然后,借助符号计算系统Mathematica求出该方程组的解,并用辅助方程的相关结论,构造了双周期解和双孤子新解.     

广义Camassa-Holm方程的几种新结论

给出一种辅助方程的几种新结论, 构造了广义 Camassa-Holm 方程的多种无穷序列新解. 首先, 利用首次积分与函数变换, 给出了一种辅助方程的新解、B¨acklund 变换和解的非线性叠加公式. 然后, 通过函数变换, 把广义Camassa-Holm 方程的求解问题转化为非线性常微分方程的求解问题. 最后, 借助符号计算系统 Mathematica, 构造了广义Camassa-Holm方程的多种无穷序列新解.     

基于不动点方法求解非线性Falkner-Skan流动方程

Falkner-Skan流动方程描述绕楔面的流动,该方程具有很强的非线性.首先通过引入变换式,将原半无限大区域上的流动问题转化为有限区间上的两点边值问题.接着基于泛函分析中的不动点理论,采用不动点方法求解两点边值问题从而得到Falkner Skan流动方程的解.最后将不动点方法给出的结果和文献中的数值结果相比较,发现不动点方法得到的结果具有很高的精度,并且解的精度很容易通过迭代而不断得到提高.表明不动点方法是一种求解非线性微分方程行之有效的方法.     

广义Camassa-Holm方方程的几种新结论

给出一种辅助方程的几种新结论,构造了广义Camassa-Holm方程的多种无穷序列新解.首先,利用首次积分与函数变换,给出了一种辅助方程的新解、B¨acklund变换和解的非线性叠加公式.然后,通过函数变换,把广义Camassa-Holm方程的求解问题转化为非线性常微分方程的求解问题.最后,借助符号计算系统Mathematica,构造了广义Camassa-Holm方程的多种无穷序列新解.     

基于Riccati-Bernoulli辅助常微分方程的Davey-Stewartson方程的行波解

Riccati-Bernoulli辅助常微分方程方法可以用来构造非线性偏微分方程的行波解.利用行波变换,将非线性偏微分方程化为非线性常微分方程, 再利用Riccati-Bernoulli方程将非线性常微分方程化为非线性代数方程组, 求解非线性代数方程组就能直接得到非线性偏微分方程的行波解.对Davey-Stewartson方程应用这种方法, 得到了该方程的精确行波解.同时也得到了该方程的一个Backlund变换.所得结果与首次积分法的结果作了比较.Riccati-Bernoulli辅助常微分方程方法是一种简单、有效地求解非线性偏微分方程精确解的方法.     

一些高振荡积分、高振荡积分方程的高性能计算

电磁、声波散射问题的研究涉及一类数学物理问题, 此类问题具有深刻的理论价值和重要的应用背景, 亟待解决. 高振荡微分、积分方程是刻画这些问题的重要的数学模型, 其数值计算存在许多挑战性研究课题. 本文从积分方程解法角度出发, 综述了求解这类高振荡问题的一些最新进展, 特别是针对广义Fourier 变换、Bessel 变换的高效算法、高振荡核Volterra 积分方程的数值解法作了详细介绍. 这些数值方法共有特点是振荡频率越高算法精度愈高, 且可望为电磁计算的研究提供一些新的高效算法.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

Solution of a laminar boundary layer flow via a numerical method

In this paper, the numerical solution of the Blasius problem is obtained using the collocation method based on rational Chebyshev functions. The Blasius equation is a nonlinear ordinary differential equation which arises in the boundary layer flow. The method reduces solving the equation to solving a system of nonlinear algebraic equations. The results presented here demonstrate reliability and efficiency of the method.     

The solution of high-order nonlinear ordinary differential equations by Chebyshev Series

By the use of the Chebyshev series, a direct computational method for solving the higher order nonlinear differential equations has been developed in this paper. This method transforms the nonlinear differential equation into the matrix equation, which corresponds to a system of nonlinear algebraic equations with unknown Chebyshev coefficients, via Chebyshev collocation points. The solution of this system yields the Chebyshev coefficients of the solution function. An algorithm for this nonlinear system is also proposed in this paper. The method is valid for both initial-value and boundary-value problems. Several examples are presented to illustrate the accuracy and effectiveness of the method.     

Quasi-Newton Methods for Discretized Non-linear Boundary Problems

The discretization of non-linear boundary problems generallyleads to a finite system of non-linear algebraic equations,and it is to be expected that this latter has special structurearising both from the boundary problem and the method of discretizationused. The numerical solution of the algebraic system representsa serious numerical problem, and it is the point of this paperto indicate that, in certain important cases, special purposequasi-Newton methods can be constructed. We illustrate by consideringa single nonlinear differential equation discretized by collocationand present experimental results which indicate that an improvementin performance can be expected from the special methods.     

Integrability and beyond

An algebraic approach to solving nonlinear functional equations in the Riemann theta functions is stated. By the inverse scattering method and some general methods of the theory of partial differential equations, the solution of the initial boundary value problem for the nonlinear Schrödinger equation is presented. Bibliography:17 titles.     

The Sinc-collocation method for solving the Thomas–Fermi equation

A numerical technique for solving nonlinear ordinary differential equations on a semi-infinite interval is presented. We solve the Thomas–Fermi equation by the Sinc-Collocation method that converges to the solution at an exponential rate. This method is utilized to reduce the nonlinear ordinary differential equation to some algebraic equations. This method is easy to implement and yields very accurate results.     

Numerical approximations for population growth model by rational Chebyshev and Hermite functions collocation approach: A comparison

This paper aims to compare rational Chebyshev (RC) and Hermite functions (HF) collocation approach to solve Volterra's model for population growth of a species within a closed system. This model is a nonlinear integro‐differential equation where the integral term represents the effect of toxin. This approach is based on orthogonal functions, which will be defined. The collocation method reduces the solution of this problem to the solution of a system of algebraic equations. We also compare these methods with some other numerical results and show that the present approach is applicable for solving nonlinear integro‐differential equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Finite-difference methods for solving the reaction-diffusion equations of a simple isothermal chemical system

Numerical methods are proposed for the numerical solution of a system of reaction-diffusion equations, which model chemical wave propagation. The reaction terms in this system of partial differential equations contain nonlinear expressions. Nevertheless, it is seen that the numerical solution is obtained by solving a linear algebraic system at each time step, as opposed to solving a nonlinear algebraic system, which is often required when integrating nonlinear partial differential equations. The development of each numerical method is made in the light of experience gained in solving the system of ordinary differential equations, which model the well-stirred analogue of the chemical system. The first-order numerical methods proposed for the solution of this initialvalue problem are characterized to be implicit. However, in each case it is seen that the numerical solution is obtained explicitly. In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it is seen that the proposed methods have superior stability properties to those of the well-known, first-order, Euler method to which they are compared. Incorporating the proposed methods into the numerical solution of the partial differential equations is seen to lead to two economical and reliable methods, one sequential and one parallel, for solving the travelling-wave problem. © 1994 John Wiley & Sons, Inc.     

An approximate solution of the MHD Falkner–Skan flow by Hermite functions pseudospectral method

Based on a new approximation method, namely pseudospectral method, a solution for the three order nonlinear ordinary differential laminar boundary layer Falkner–Skan equation has been obtained on the semi-infinite domain. The proposed approach is equipped by the orthogonal Hermite functions that have perfect properties to achieve this goal. This method solves the problem on the semi-infinite domain without truncating it to a finite domain and transforming domain of the problem to a finite domain. In addition, this method reduces solution of the problem to solution of a system of algebraic equations. We also present the comparison of this work with numerical results and show that the present method is applicable.     

Be careful with the Exp-function method – Additional remarks

It is shown that the solution produced by the Exp-function method may not hold for all initial conditions. Riccati and Maccari nonlinear differential equations are used to illustrate that fact. Conditions of existence for the produced solution in the space of initial conditions and in the space of system’s parameters are derived using the operator method based on the generalized operator of differentiation. The concept of the expansion of an ordinary differential equation is introduced and it is shown that the algebraic–analytical solution of Maccari equation can be produced by solving Riccati equation.