扰动扩散方程初值问题的近似对称约化

本文利用近似广义条件对称方法研究一类带有源项的非线性扩散方程的初值问题.给出所研究方程的分类并将偏微分方程的初值问题约化为常微分方程的初值问题,通过求解约化后的常微分方程组可得相对应偏微分方程初值问题的近似解.     

对“求一类非线性偏微分方程解析解的一种简洁方法”一文的一点注记

利用文献中所引入的变换，将一个非线性偏微分方程化为一个非线性常微分方程，再直接求解该常微分方程，从而简洁地求得了Burgers方程的几个精确解析解.所得结果与已有结果完全符合. 关键词： 非线性偏微分方程 非线性常微分方程 解析解     

非线性多点边值问题的插值矩阵法

建立了插值矩阵法的基本理论，用于解非线性混合阶常微分方程组多点边值问题，制作了常微分方程组求解器ＩＶＭＭＳ，可以支持计算力学中的有限元线法。     

场方法的改进及其在积分Riemann-Cartan空间运动方程中的应用

和Hamilton-Jacobi方法类似,Vujanovi?场方法把求解常微分方程组特解的问题转化为寻找一个一阶拟线性偏微分方程(基本偏微分方程)完全解的问题,但Vujanovi?场方法依赖于求出基本偏微分方程的完全解,而这通常是困难的,这就极大地限制了场方法的应用.本文将求解常微分方程组特解的Vujanovi?场方法改进为寻找动力学系统运动方程第一积分的场方法,并将这种方法应用于一阶线性非完整约束系统Riemann-Cartan位形空间运动方程的积分问题中.改进后的场方法指出,只要找到基本偏微分方程的包含m(m≤ n,n为基本偏微分方程中自变量的数目)个任意常数的解,就可以由此找到系统m个第一积分.特殊情况下,如果能够求出基本偏微分方程的完全解(完全解是m=n时的特例),那么就可以由此找到≤系统全部第一积分,从而完全确定系统的运动.Vujanovi?场方法等价于这种特殊情况.     

非线性偏微分方程边值问题的优化算法研究与应用

针对椭圆类非线性偏微分方程边值问题,以差分法和动态设计变量优化算法为基础,以离散网格点未知函数值为设计变量,以离散网格点的差分方程组构建为复杂程式化形式的目标函数.提出一种求解离散网格点处未知函数值的优化算法.编制了求解未知离散点函数值的通用程序.求解了具体算例.通过与解析解对比,表明了本文提出求解算法的有效性和精确性,将为更复杂工程问题分析提供良好的解决方法. 关键词： 非线性偏微分方程 边值问题 动态设计变量优化算法 程序设计     

天线振子作为初值问题求解

根据计算实践和物理考虑,指出电磁场基本方程作为一阶双曲型偏微分方程组的初值问题,可以成为有效的天线振子数值计算方法。对于隐式差分格式,建议了一种交替方向追赶和外边界插值相结合的办法。讨论了圆柱对称振子的数值结果。     

半导体瞬态问题计算方法的新进展

综述三维热传导型半导体瞬态问题计算方法的新进展.数学模型是一类由四个方程组成的非线性耦合对流-扩散偏微分方程组的初边值问题.重点研究特征分数步差分方法,修正迎风分数步差分方法,特征交替方向变网格有限元方法,区域分裂及并行计算.     

边界耦合的Marangoni对流边界层问题的近似解析解

研究了由温度梯度引起的Marangoni对流边界层问题.由于动量方程和能量方程的边界条件耦合,利用相似变换将偏微分方程组转化为常微分方程非线性边界值问题.通过巧妙引入摄动小参数对速度和温度边界层方程同时渐近展开求解,得到了问题的近似解析解,并对相应的动量、能量传递特性进行了讨论. 关键词： Marangoni对流 近似解析解 渐近展开     

计算石油地质等领域的一些新进展

主要综述应用计算数学、渗流力学的数值方法和理论研究油田勘探开发中的数值模拟、核废料污染问题的数值方法、海水入侵的预测和防治,半导体瞬态问题的数值模拟.问题的数学模型是一类非线性耦合对流 扩散偏微分方程组的初边值问题.重点讨论特征差分方法、特征有限元法、分数步数值方法及其理论分析.     

对流占优扩散问题的高精度直线法

基于常微分方程边值问题的高精度求解器SEVORD对偏微分方程作半离散,提出了求解一维对流扩散方程的高精度直线法,并采用局部一维化方法(LOD)给出了求解二维对流扩散问题的高精度交替方向直线法。     

Transient heat transfer in longitudinal fins of various profiles with temperature-dependent thermal conductivity and heat transfer coefficient

Transient heat transfer through a longitudinal fin of various profiles is studied. The thermal conductivity and heat transfer coefficients are assumed to be temperature dependent. The resulting partial differential equation is highly nonlinear. Classical Lie point symmetry methods are employed and some reductions are performed. Since the governing boundary value problem is not invariant under any Lie point symmetry, we solve the original partial differential equation numerically. The effects of realistic fin parameters such as the thermogeometric fin parameter and the exponent of the heat transfer coefficient on the temperature distribution are studied.     

Homotopy Continuation Approach to Electrostatic Boundary-Value Problems of Nonlinear Media

The homotopy continuation method is employed to solve electrostatic boundaryvalue problems of nonlinear media. The difficulty associated with matching the inherently nonlinear boundary conditions on the interface is overcome by the mode expansion method, by which the nonlinear partial differential equations of the original problem are transformed into an infinite set of nonlinear ordinary differential equations. In this regard, the homotopy method has to be modified to handle the nonlinear boundary conditions. As an illustration, we study two cases:(a) nonlinear inclusion in linear host and (b) linear inclusion-in nonlinear host, both in two dimensions. The homotopy method is validated by comparing the results with the exact solution of case (a) and the results derived by perturbation method in case (b).     

蒙特卡罗方法在解微分方程边值问题中的应用

介绍了蒙特卡罗方法的基本原理以及随机数的产生方法。基于蒙特卡罗方法的思想,结合有限差分方法,建立了求解微分方程边值问题的随机概率模型,并以第一类边界条件的拉普拉斯方程和一个给定初值及边界条件的非稳态热传导方程为数值算例,研究了蒙特卡罗方法在求解微分方程边值问题中的应用。结果表明:利用蒙特卡罗方法,不仅可以有效解决给定边界条件的微分方程,对于给定初值条件的微分方程,也可以从时域有限差分方程出发,采用蒙特卡罗方法进行求解。数值模拟和对误差的理论分析均表明,增加蒙特卡罗试验中的模拟粒子点数,可以提高计算结果的精度。     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

An Alternative Algorithm for the Symmetry Classification of Ordinary Differential Equations

This is the first paper on symmetry classification for ordinary differential equations (ODEs) based on Wu’s method. We carry out symmetry classification of two ODEs, named the generalizations of the Kummer-Schwarz equations which involving arbitrary function. First, Lie algorithm is used to give the determining equations of symmetry for the given equations, which involving arbitrary functions. Next, differential form Wu’s method is used to decompose determining equations into a union of a series of zero sets of differential characteristic sets, which are easy to be solved relatively. Each branch of the decomposition yields a class of symmetries and associated parameters. The algorithm makes the classification become direct and systematic. Yuri Dimitrov Bozhkov, and Pammela Ramos da Conceição have used the Lie algorithm to give the symmetry classifications of the equations talked in this paper in 2020. From this paper, we can find that the differential form Wu’s method for symmetry classification of ODEs with arbitrary function (parameter) is effective, and is an alternative method.     

Conservation Laws of a Class of Combined Equations

In this paper, we investigate conservation laws of a class of partial differential equations, which combines the nonlinear telegraph equations and the nonlinear diffusion-convection equations. Moreover, some special conservation laws of the combined equations are obtained by means of symmetry classifications of wave equations uxx = H （x）utt.     

Numerical derivation of the generalized Ginzburg-Landau equations of wavy vortex flow

Starting from the Navier-Stokes equations we treat the onset of the wavy vortex flow of the Taylor problem in a fully nonlinear manner. To this end we first transform the original partial differential equations into a set of ordinary differential equations by means of a Galerkin method. By a specific choice of the basic functions the Galerkin coefficients acquire a very concise form. Since these functions fulfil the boundary conditions individually, there are no additional constraints on these coefficients, which usually stem from the boundary conditions. We then first calculate the Taylor vortex flow and perform a linear stability analysis of it. Finally we calculate the coefficients of the generalized Ginzburg-Landau equations in the vicinity of the second threshold and solve these equations.     

Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach

In this paper an efficient computational method based on extending the sensitivity approach(SA) is proposed to find an analytic exact solution of nonlinear differential difference equations.In this manner we avoid solving the nonlinear problem directly.By extension of sensitivity approach for differential difference equations(DDEs),the nonlinear original problem is transformed into infinite linear differential difference equations,which should be solved in a recursive manner.Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained.Numerical examples are employed to show the effectiveness of the proposed approach.     

A singularly perturbed reaction diffusion problem forthe nonlinear boundary condition with two parameters

A class of singularly perturbed initial boundary value problems of reaction diffusion equations for the nonlinear boundary condition with two parameters is considered. Under suitable conditions, by using the theory of differential inequalities, the existence and the asymptotic behaviour of the solution for the initial boundary value problem are studied. The obtained solution indicates that there are initial and boundary layers and the thickness of the boundary layer is less than the thickness of the initial layer.