求一类非线性偏微分方程解析解的简洁构造算法

通过引入一个变换,利用齐次平衡原理和选准一个待定函数来构造求解一类非线性偏微分方程解析解的算法.作为实例,我们将该算法应用到了mKdV方程,KdV-Burgers方程和KdV-Burgers-Kuramoto方程.借助符号计算软件Mathematica获得了这些方程的解析解.不难看出,该方法不仅简洁,而且有望进一步扩展.     

基于场方法的非线性系统求解

用场方法联立多尺度法求单自由度的非线性系统的近似解．将两个状态方程的一个状态变量看作是另一个状态变量和时间的一个场函数,把原系统化为求解具有初始条件的基本方程,通过多尺度展开,逐个摄动方程求解,获得了振幅和相位的一阶近似微分方程,作为例子,求得了非线性振动系统的一阶近似解,并和数值解进行比较,两者吻合较好。     

复模态分析超临界轴向运动梁横向非线性振动

近似解析研究了简支边界条件下超临界轴向运动梁横向非线性自由振动的固有频率和模态函数.采用复模态方法处理控制方程,一个积分偏微分方程.将Galerkin截断思想用于近似处理线性化方程,一个含空间依赖系数的常微分方程.给出了不同截断项数对固有频率的影响.基于8项截断,讨论了系统参数对模态函数的影响.     

平方非线性振动方程的渐近解及其数值验证

改进的Lindstedt-Poincar啨(L-P)法在传统的L-P法的基础上,对频率的展开式作了改进;卷积分法则提供了一个求近似解的迭代格式.本文首先用这两种方法求得平方非线性振动方程的二阶渐近解,并用Picard逐步逼近法证明由卷积分法得到的渐近解在有限的时间上是一致收敛的.其次,一种数值阶验证技术证实求得的二阶渐近解对小参数都是一致有效的.最后,对这两种渐近解进行误差的数值比较,结果表明它们对大参数无效,并简明分析其失效的原因.因此,这两种方法在平方非线性振动方程中的应用受到小参数的限制.     

扁锥面单层网壳的非线性动力学特性

用拟壳法建立了正三角形网格三向扁锥面单层网壳的轴对称非线性动力学基本方程．通过分离变量函数法,用Galerkin法得到了一个含二次、三次的非线性微分方程．为了研究混沌运动,对一类非线性动力系统的自由振动方程进行了求解,给出了单层扁锥面网壳非线性自由振动微分方程的准确解．通过求Melnikov函数,给出了发生混沌运动的临界条件．数字仿真证实了混沌运动的存在．     

基于LS-SVM方法求高阶线性ODE近似解

对于线性常微分方程,解析解方便定性分析和实际应用,然而大多数微分方程没有解析解。回归的方法被应用获取近似解析解,其中最小二乘支持向量机（LS-SVM）是目前为止最好的方法。但是该方法不仅需要对核函数求高阶导数而且需要求解一个大的线性方程组。为此,把高阶线性常微分方程转化为一阶线性常微分方程组,构建含有一阶导数形式的LS-SVM回归模型。该模型利用最小化误差函数去获得合适的参数,最终通过求解三个小的线性方程组获得高精度的近似解（连续、可微）。实验结果验证了该方法的有效性。     

基于MATLAB的非线性方程组遗传解法

将非线性方程组的求解问题转化为用遗传算法求解目标函数的最小值问题,利用MATLAB的遗传算法与直接搜索工具箱（GADs）对目标函数求取最小值。计算结果表明,用该方法求得的非线性方程组近似解精度较高。     

有限元线法

一、引 言 有限元线法(Pinite Element Method of Lines,简称FEMOL)是一种新型的以常微分方程(Ordinary Differential Equation,简称ODE)求解器(Solver)为支撑软件的半解析方法.在该法中,我们首先利用有限元技术将控制微分方程半离散化为用结线函数表示的常微分方程组(ODEs),然后选用高质量的ODE求解程序直接求解(本文中采用COLSYS),得到满足用户预先指定的误差限的ODE解答,作为原问题的近似解.     

分数阶对偶Burger方程的精确解

将分数阶复变换方法和tanh函数方法相结合,得到了一种用来求解时-空分数阶非线性微分方程精确解的复变换-tanh函数方法。借助于软件Mathematica的符号计算功能,使用该方法求解了分数阶对偶Burger方程,得到了分数阶对偶Burger方程的新的精确解。     

基于δ法求解高阶时变非线性微分方程

该文基于二阶线性微分方程δ法绘制相平面原理,提出一种新颖而简单计算圆弧圆心和半径的方法实现高阶时变非线性微分方程相平面的作图,从而得到求解高阶时变非线性微分方程时域解的算法,并与龙格-库塔法等解析法相比具有计算简单、结果精度高的特点。     

An efficient computational method for a class of singularly perturbed delay parabolic partial differential equation

In this paper, a new technique is constructed skillfully in order to solve a class of singularly perturbed delay parabolic partial differential equation. The outer and inner exact solutions of the linear problem can be expressed in the form of series and the outer and inner approximate solutions of the nonlinear problem are given by the iterative formula. Compared with known investigations, the advantages of our method are that the representation of the exact solution is obtained by using a new technique in a new reproducing kernel Hilbert space and the accuracy of numerical computation is higher. Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate that it is simple and effective.     

Operator-splitting methods for the 2D convective Cahn–Hilliard equation

In this work, we present operator-splitting methods for the two-dimensional nonlinear fourth-order convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The full problem is split into hyperbolic, nonlinear diffusion and linear fourth-order problems. We prove that the semi-discrete approximate solution obtained from the operator-splitting method converges to the weak solution. Numerical methods are then constructed to solve each sub equations sequentially. The hyperbolic conservation law is solved by efficient finite volume methods and dimensional splitting method, while the one-dimensional hyperbolic conservation laws are solved using front tracking algorithm. The front tracking method is based on the exact solution and hence has no stability restriction on the size of the time step. The nonlinear diffusion problem is solved by a linearized implicit finite volume method, which is unconditionally stable. The linear fourth-order equation is solved using a pseudo-spectral method, which is based on an exact solution. Finally, some numerical experiments are carried out to test the performance of the proposed numerical methods.     

非线性受控系统状态方程的任意阶近似解

针对工作在理想状态附近的受控系统,通过对其非线性状态方程进行Taylor展开,使之变为无穷级数形式的常微分方程组;然后在线性状态方程组解的基础上采用常数变异法,使之变换成积分方程;最后采用逐次逼近法求得非线性状态方程的任意阶近似解,并进一步讨论了系统状态的方均包络矩阵的转移规律.     

The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure

In this article, we propose a new approach for solving an initial–boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional procedures, the series-based technique of the Adomian decomposition technique is shown to evaluate solutions accurately and efficiently. The method is very reliable and effective that provides the solution in terms of rapid convergent series. Several examples are tested to support our study.     

Causes of instabilities in numerial integration techniques

This paper presents the causes of instabilities which arise during the numerical solution of ordinary differential equations. Using the numerical integration routines presently available, one actually approximates the differential equation by a difference equation. If the difference equation is of higher order than the original differential equation, the approximate solution contains extraneous solutions which are not at all related to the true solution. It is the behavior of these extraneous solutions that one is concerned with in a stability analysis.     

Vanishing Moment Method and Moment Solutions for Fully Nonlinear Second Order Partial Differential Equations

This paper concerns with numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for fully nonlinear second order PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called the vanishing moment method, and hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods. The main idea of the proposed vanishing moment method is to approximate a fully nonlinear second order PDE by a higher order, in particular, a quasilinear fourth order PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist. This work was partially supported by the NSF grants DMS-0410266 and DMS-0710831.     

Error analysis of solutions of second-order boundary value problems obtained with residual correction method

Based on the maximum principle of differential equations and with the aid of asymptotic iteration technique, this paper tries to establish monotonic relation of second-order obstacle boundary value problems with their approximate solutions to eventually obtain the upper and lower approximate solutions of the exact solution. To obtain numerical solutions, the cubic spline approximation method is applied to discretize equations, and then according to the ‘residual correction method’ proposed in this paper, residual correction values are added into discretized grid points to translate once complex inequalities’ constraint mathematical programming problems into simple equational iteration problems. The numerical results also show that such method has the characteristic of correcting residual values to symmetrical values for such problems, as a result, the mean approximate solutions obtained even with a considerably small quantity of grid points still quite approximate the exact solution. Furthermore, the error range of approximate solutions can be identified very easily by using the obtained upper and lower approximate solutions, even if the exact solution is unknown.     

Intelligent computing to solve fifth-order boundary value problem arising in induction motor models

In this study, biologically inspired intelligent computing approached based on artificial neural networks (ANN) models optimized with efficient local search methods like sequential quadratic programming (SQP), interior point technique (IPT) and active set technique (AST) is designed to solve the higher order nonlinear boundary value problems arise in studies of induction motor. The mathematical modeling of the problem is formulated in an unsupervised manner with ANNs by using transfer function based on log-sigmoid, and the learning of parameters of ANNs is carried out with SQP, IPT and ASTs. The solutions obtained by proposed methods are compared with the reference state-of-the-art numerical results. Simulation studies show that the proposed methods are useful and effective for solving higher order stiff problem with boundary conditions. The strong motivation of this research work is to find the reliable approximate solution of fifth-order differential equation problems which are validated through strong statistical analysis.

     

Oscillators with nonlinear elastic and damping forces

Generalization to the previous oscillators is done by introducing the nonlinear elastic and damping forces. The mathematical model of the system is a second order differential equation with nonlinear elastic and damping terms whose order is integer and/or noninteger. Cveticanin’s solving procedure is extended for solving such a strong nonlinear differential equation. The approximate solution obtained is a function of initial amplitude and initial phase. A damping coefficient and order of damping interaction with an elastic coefficient and order of elasticity for the generalized oscillators are also determined. Special attention is paid to obtain the relation between initial amplitude and phase, on the one hand, and initial displacement and velocity on the other hand. Correction to the frequency of vibration for the linear oscillators with nonlinear damping and for the pure nonlinear oscillators with linear damping is obtained and analyzed. Analytical results given in this paper are compared with numerically obtained ones and show a good agreement.     

A new stochastic approach for solution of Riccati differential equation of fractional order

In this article, a stochastic technique has been developed for the solution of nonlinear Riccati differential equation of fractional order. Feed-forward artificial neural network is employed for accurate mathematical modeling and learning of its weights is made with heuristic computational algorithm based on swarm intelligence. In this scheme, particle swarm optimization is used as a tool for the rapid global search method, and simulating annealing for efficient local search. The scheme is equally capable of solving the integer order or fractional order Riccati differential equations. Comparison of results was made with standard approximate analytic, as well as, stochastic numerical solvers and exact solutions.