NUMERICAL SOLUTION OF NONLINEAR ORDINARY DIFFERENTIAL EQUATION FOR A TURNING POINT PROBLEM

By using the method in [3], several useful estimations of the derivatives of the solution of the boundary value problem for a nonlinear ordinary differential equation with a turning point are obtained. With the help of the technique in [4], the uniform convergence on the small parameter e for a difference scheme is proved. At the end of this paper, a numerical example is given. The numerical result coincides with theoretical analysis.

Second-order difference scheme for a nonlinear model of wood drying process

A numerical simulation for a model of wood drying process is considered. The model is given by a couple of nonlinear differential equations. One is a nonlinear parabolic equation and the other one is a nonlinear ordinary equation. A difference scheme is derived by the method of reduction of order. First, a new variable is introduced and the original problem is rewritten into a system of the first-order differential equations. Secondly, a difference scheme is constructed for the later problem. The solvability, stability and convergence of the difference scheme are proved by the energy method. The convergence order of the difference scheme is secondorder both in time and in space. A prior error estimate is put forward. The new variable is put aside to reduce the computational cost. A numerical example testifies the theoretical result.

Some new solutions derived from the nonlinear (2+1)-dimensional Toda equation-an efficient method of creating solutions

This paper presents a new and efficient approach for constructing exact solutions to nonlinear differential-difference equations （NLDDEs） and lattice equation. By using this method via symbolic computation system MAPLE, we obtained abundant soliton-like and/or period-form solutions to the （2＋1）-dimensional Toda equation. It seems that solitary wave solutions are merely special cases in one family. Furthermore, the method can also be applied to other nonlinear differential-difference equations.

ECONOMICAL DIFFERENCE SCHEME FOR ONE MULTI-DIMENSIONAL NONLINEAR SYSTEM

The multi-dimensional system of nonlinear partial differential equations is considered. In two-dimensional case, this system describes process of vein formation in higher plants. Variable directions finite difference scheme is constructed. The stability and convergence of that scheme are studied. Numerical experiments are carried out. The appropriate graphical illustrations and tables are given.

非标准有限差分法求解Kuramoto-Sivashinsky方程

给出一种非线性四阶Kuramoto-Sivashinsky(KS)方程的非标准有限差分格式,证明该格式的稳定性,并针对具体算例进行数值试验,比较了数值解与真实解,验证该方法的有效性。

ON LINEARIZED FINITE DIFFERENCE SIMULATION FOR THE MODEL OF NUCLEAR REACTOR DYNAMICS

The model of nuclear reactor dynamics is an initial-boundary value problems of a cou- pled nonlinear integrodifferential equation system of one ordinary differential equation and one par-- tial differential equation. In this this,paper,a linearized difference scheme is derived by the method of reduction of order.It is proved that the scheme is uniquely solvable and unconditionally convergent with the convergence rate of order two both in discrete H1norm and in discrete maxinum narm,and one needs only to solve a tridiagonal system of linear algebraic equations at each time lev- el.The method of reduction of order is an indirect constructing-difference-scheme method,which aim is for the analysis of solvablity and convergence of the constructed difference scheme.

一类非线性微分-差分方程的整函数解

考虑非线性微分-差分方程f(z)+q(z)e^(Q(z))f^((k))(z+c)=p_(1)e^(α1z)+p2^(eα2z)的超越整函数解,其中k≠0是整数,q(z)是非零多项式,Q(z)是非常数多项式,c,p_(1),p_(2),α_(1),α_(2)为非零常数,α_(1)≠α_(2).所得结果改进并推广了已有成果.

非线性差分-微分方程的显示精确解

将广义投影Riccati方程法应用于求解非线性差分-微分方程,并在符号计算机系统Maple帮助下得到了离散(2+1)维Toda lattice方程和离散mKdV lattice方程一些新的精确解,其中包括双曲函数解和三角函数解.

广义Riccati方程有理展开法在非线性差分-微分方程中的应用

将广义Riccati方程有理展开法应用于求解非线性差分-微分方程.并在符号计算机系统Maple的帮助下,以离散的非线性mKdV lattice方程和离散的非线性(2+1)维Toda lattice方程为例,得到了一些新的精确解,其中包括双曲函数解和三角函数解.

三Riccati方程的新展开法及其在差分-微分方程中的应用(英文)

将三Riccati方程的新展开法应用于求解非线性差分-微分方程,借助符号计算系统Maple,得到了离散KdV方程和离散mKdVlattice方程的一些新的精确解,并具体给出了双曲函数解.

推广的Riccati方程法构造非线性差分-微分方程的精确解

将推广的Riccati方程法应用于求解非线性差分-微分方程求解领域.并在符号计算机系统Maple的帮助下,以离散的非线性(2+1)-维Toda lattice方程为应用实例,构造了该方程的一些新精确解,其中包括有理形式的双曲函数解和有理形式的三角函数周期解.

非线性微分-差分二元Volterra晶格方程的精确解

把(G′/G)展开法推广应用于研究非线性微分-差分二元Volterra晶格方程的精确解问题,借助数学软件计算得到该方程的双曲函数和三角函数等形式的精确解;当参数取特定的值时,应用该方法又得到一些特殊形式的扭结型孤立波解及奇异行波解。比较发现,该方法比用双曲正切法能得到更多类型的精确解,从而证实了该方法研究非线性微分-差分方程精确解问题的有效性。

Stability and convergence of the variable directions difference scheme for one nonlinear two-dimensional model

The system of two-dimensional nonlinear partial differential equations is considered. This system describes the vein formation in meristematic tissues of young leaves. Variable directions difference scheme is constructed and investigated. Absolute stability regarding space and time steps of scheme is shown. The convergence statement for the constructed scheme is proved. Rate of convergence is given. Various numerical experiments are carried out and results of some of them are considered in this paper. Comparison of numerical experiments with the results of the theoretical investigation is given too.

一类非线性复微分差分方程超越整函数解的性质

本文主要运用亚纯函数值分布理论及其差分模拟和Hadamard分解定理,研究了一类非线性复微分差分方程超越整函数解的增长级和零点收敛指数问题,并分析了该类方程的超越整函数解是指数单项式形式解时的情形。

一般格子方程的Jacobi椭圆函数精确解

将推广的投影Riccati方程法应用到非线性差分-微分方程求解领域,并以一般格子方程为例,在符号计算系统Maple的帮助下,得到该方程一些新的Jacobi椭圆函数精确解.当m→1和m→0,所得的解将分别退化为双曲函数解和三角函数解.

一类非线性微分差分方程亚纯解的性质

利用值分布理论,研究了几类非线性差分方程是否有有限级超越亚纯解的问题,还考虑了"微分差分方程fn(z)+M(z,f)=h(z)是否存在有限级超越整函数解"的问题,其中n(≥3)是整数,h(z)是非零的有理函数,M(z,f)是关于f的线性微分差分多项式.

关于非线性复方程整函数解的几个结果

经典的Nevanlinna理论是研究复微分方程解的性质的有效工具,而Halburd和Korhonen,Chiang和Feng分别给出了差分的对数导数引理的不同版本,在此基础上建立起来的差分的Nevanlinna理论为研究复差分方程,复差分多项式理论提供了理论的基础。主要利用差分的Nevanlinna理论,研究各种类型的非线性的复差分方程的超越整函数解的存在性问题,得到关于几个不同方程解的存在性质的结果,从而把一些复微分方程中的结论推广到了复差分方程中。

基于指数函数展开法构造非线性差分微分方程新的精确解

以双曲正切函数展开法、Jacobi椭圆函数展开法和试探函数法为基础,给出指数函数展开法,借助符号计算系统Mathematica,构造了一般格子方程和(2+1)维Toda格子方程等非线性差分微分方程新的精确解,其中包括精确孤立波解.该方法在构造非线性差分微分方程精确解领域具有普遍意义.

四阶非线性偏微分方程的有限差分法研究

研究了四阶非线性偏微分方程的有限差分法,分别对时间、空间导数项用差商进行近似,得到相应的有限差分格式.并以BCF模型方程为例,针对两个具体算例进行数值实验,比较和分析了数值解与真实解之间的误差.实验结果验证了该方法的有效性,高精度以及易于编程上机实现.

一类分数阶q型差分边值问题中的混合单调方法

为了研究一类非线性分数阶q型差分方程边值问题非平凡解的存在唯一性。首先,在一个新的集合上定义一个新概念,再利用正规锥的定义,建立了2个混合单调算子唯一不动点的存在性,获得了线性分数阶q型边值问题的Green函数,并且对Green函数的上下界进行了估计,由此可得到特解的表达形式。其次,运用抽象定理,讨论了符合定理条件的非线性项,建立了上述问题的唯一解的存在性,并获得逼近唯一解的迭代序列,进而证明了分数阶q型差分方程边值问题非平凡解的存在唯一性。最后,通过列举一个例子来说明主要定理和结果的有效性。研究结果表明,定理条件得证且方程组边值问题非平凡解满足存在唯一性。研究方法在理论证明和边值问题方面都得到了良好的结果,对探究其他边值问题具有一定的借鉴意义。