奇异杂交边界点方法求解泊松方程

As a boundary-type meshless method,the singular hybrid boundary node method(SHBNM)is based on the modified variational principle and the moving least square(MLS)approximation,so it has the advantages of both boundary element method(BEM)and meshless method.In this paper,the dual reciprocity method(DRM)is combined with SHBNM to solve Poisson equation in which the solution is divided into particular solution and general solution.The general solution is achieved by means of SHBNM,and the particular solution is approximated by using the radial basis function(RBF).Only randomly distributed nodes on the bounding surface of the domain are required and it doesn't need extra equations to compute internal parameters in the domain.The postprocess is very simple.Numerical examples for the solution of Poisson equation show that high convergence rates and high accuracy with a small node number are achievable.     

Dual reciprocity hybrid radial boundary node method for the analysis of Kirchhoff plates

A meshless method of dual reciprocity hybrid radial boundary node method (DHRBNM) for the analysis of arbitrary Kirchhoff plates is presented, which combines the advantageous properties of meshless method, radial point interpolation method (RPIM) and BEM. The solution in present method comprises two parts, i.e., the complementary solution and the particular solution. The complementary solution is solved by hybrid radial boundary node method (HRBNM), in which a three-field interpolation scheme is employed, and the boundary variables are approximated by RPIM, which is applied instead of moving least square (MLS) and obtains the Kronecker’s delta property where the traditional HBNM does not satisfy. The internal variables are interpolated by two groups of symmetric fundamental solutions. Based on those, a hybrid displacement variational principle for Kirchhoff plates is developed, and a meshless method of HRBNM for solving biharmonic problems is obtained, by which the complementary solution can be solved.     

The Tikhonov regularization method in elastoplasticity

The numerical simulation of the mechanical behavior of industrial materials is widely used for viability verification, improvement and optimization of designs. Elastoplastic models have been used to forecast the mechanical behavior of different materials. The numerical solution of most elastoplastic models comes across problems of ill-condition matrices. A complete representation of the nonlinear behavior of such structures involves the nonlinear equilibrium path of the body and handling of singular (limit) points and/or bifurcation points. Several techniques to solve numerical problems associated to these points have been disposed in the specialized literature. Two examples are the load-controlled Newton–Raphson method and displacement controlled techniques. However, most of these methods fail due to convergence problems (ill-conditioning) in the neighborhood of limit points, specially when the structure presents snap-through or snap-back equilibrium paths. This study presents the main ideas and formalities of the Tikhonov regularization method and shows how this method can be used in the analysis of dynamic elastoplasticity problems. The study presents a rigorous mathematical demonstration of existence and uniqueness of the solution of well-posed dynamic elastoplasticity problems. The numerical solution of dynamic elastoplasticity problems using Tikhonov regularization is presented in this paper. The Galerkin method is used in this formulation. Effectiveness of Tikhonov’s approach in the regularization of the solution of elastoplasticity problems is demonstrated by means of some simple numerical examples.     

A meshless method based on boundary integral equations and radial basis functions for biharmonic-type problems

This paper presents a meshless method, which replaces the inhomogeneous biharmonic equation by two Poisson equations in terms of an intermediate function. The solution of the Poisson equation with the intermediate function as the right-hand term may be written as a sum of a particular solution and a homogeneous solution of a Laplace equation. The intermediate function is approximated by a series of radial basis functions. Then the particular solution is obtained via employing Kansa’s method, while the homogeneous solution is approximated by using the boundary radial point interpolation method by means of boundary integral equations. Besides, the proposed meshless method, in conjunction with the analog equation method, is further developed for solving generalized biharmonic-type problems. Some numerical tests illustrate the efficiency of the method proposed.     

A convergent criss-cross method

Our paper presents a new Criss-Cross method for solving linear programming problems. Starting from a neither primal nor dual feasible solution, we reach an optimal solution in finite number of steps if it exists. If there is no optimal solution, then we show that there is not primal feasible or dual feasible solution, We prove the finiteness of this procedure. Our procedure is not the same as the primal or dual simplex method if we have a primal or dual feasible solution, so we have constructed a quite new procedure for solving linear programming problems.     

A non-overlapping domain decomposition dual reciprocity method for solving the forward–backward heat equation in two-dimension

We apply a boundary element dual reciprocity method (DRBEM) to the numerical solution of the forward–backward heat equation in a two-dimensional case. The method is employed for the spatial variable via the fundamental solution of the Laplace equation and the Crank–Nicolson finite difference scheme is utilized to treat the time variable. The physical domain is divided into two non-overlapping subdomains resulting in two standard forward and backward parabolic equations. The subproblems are then treated by the underlying method assuming a virtual boundary in the interface and starting with an initial approximate solution on this boundary followed by updating the solution by an iterative procedure. In addition, we show that the time discrete scheme is unconditionally stable and convergent using the energy method. Furthermore, some computational aspects will be suggested to efficiently deal with the formulation of the proposed method. Finally, two forward–backward problems, for which the exact solution is available, will be numerically solved for two different domains to demonstrate the efficiency of the proposed approach.     

Finite element analysis of parabolic integro-differential equations of Kirchhoff type

The aim of this paper is to study parabolic integro-differential equations of Kirchhoff type. We prove the existence and uniqueness of the solution for this problem via Galerkin method. Semidiscrete formulation for this problem is presented using conforming finite element method. As a consequence of the Ritz–Volterra projection, we derive error estimates for both semidiscrete solution and its time derivative. To find the numerical solution of this class of equations, we develop two different types of numerical schemes, which are based on backward Euler–Galerkin method and Crank–Nicolson–Galerkin method. A priori bounds and convergence estimates in spatial as well as temporal direction of the proposed schemes are established. Finally, we conclude this work by implementing some numerical experiments to confirm our theoretical results.     

一类非线性矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在Schur插值问题中遇到的含未知矩阵二次项之逆的非线性矩阵方程转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求非线性矩阵方程的对称解的双迭代算法.双迭代算法仅要求非线性矩阵方程有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定.数值算例表明,双迭代算法是有效的.     

Observations on the tanh-coth expansion method for finding solutions to nonlinear evolution equations

The ‘tanh-coth expansion method’ for finding solitary travelling-wave solutions to nonlinear evolution equations has been used extensively in the literature. It is a natural extension to the basic tanh-function expansion method which was developed in the 1990s. It usually delivers three types of solution, namely a tanh-function expansion, a coth-function expansion, and a tanh-coth expansion. It is known that, for every tanh-function expansion solution, there is a corresponding coth-function expansion solution. It is shown that there is a tanh-coth expansion solution that is merely a disguised version of the coth solution. In many papers, such tanh-coth solutions are erroneously claimed to be ‘new’. However, other tanh-coth solutions may be delivered that are genuinely new in the sense that they would not be delivered via the basic tanh-function method. Similar remarks apply to tan, cot and tan-cot expansion solutions.     

基于近似惯性流形的后验Galerkin的方法

１．引言 为提高用数值方法解非线性发展方程及非线性椭圆边值问题的逼近阶,许多学者例如Ｊ．Ｎｏｖｏ和 Ｅ．Ｔｉｔｉ［４］, Ｍａｒｉｏｎ和 Ｔｅｍａｎ［６］,Ｊ．Ｘｕ［７］以及 Ｗ．Ｌａｙｔｏｎ［９］等人,提出了后验Ｇａｌｅｒｋｉｎ方法、近似惯性流形方法、非线性Ｇａｌｅｒｋｉｎ方法、各种区域分裂法、多重网格法等等．本文根据［１］提出了一种新的高精度的后验 Ｇａｌｅｒｋｉｎ方法．它的逼近阶是经典 Ｇａｌｅｒｋｉｎ方法逼近阶的两倍． 考虑非线性椭圆边值问题这里ｎ是按ｄ＝２,３）上具有分段光滑边界ｒ的有界区域,…     

An Outer Approximation Algorithm Guaranteeing Feasibility of Solutions and Approximate Accuracy of Optimality

We treat a concave programming problem with a compact convex feasible set. Assuming the differentiability of the convex functions which define the feasible set, we propose two solution methods. Those methods utilize the convexity of the feasible set and the property of the normal cone to the feasible set at each point over the boundary. Based on the proposed two methods, we propose a solution algorithm. This algorithm takes advantages over classical methods: (1) the obtained approximate solution is always feasible, (2) the error of such approximate value can be evaluated properly for the optimal value of such problem, (3) the algorithm does not have any redundant iterations.     

Solution methods for the generalized integral volterra equation of the first kind

This paper is devoted to exact and approximate methods (first of all, direct ones) for the solution of integro-operational equations. Themost attention is paid to the theoretical substantiation of the collocation method for the solution of the mentioned equations within the general theory of approximate methods developed by L. V. Kantorovich.     

一类海-气振子ENSO模型的同伦分析解法

研究了一个ENSO海-气时滞振子模型.利用同伦分析方法,得到了该模型解的近似展开式,通过与特殊情况下的精确解比较,得到的二级近似解具有较高的精度.     

对非线性方程近似解的认识和思考

二分法和牛顿法求非线性方程根的近似值已列入中学课程.但它背后的哲学原理(相对真理)/(绝对真理)=0.9,只在林群的新书中说到2^(1/2)时提出来.根据教学需要,通过(不足近似值)/(过剩近似值)=0.9等数值化的公式,来刻画根的近似过程.可以清楚地看到,随着小数点后9的个数的增加,近似解和真实解的误差在不断减小.因此0.9数值化系列公式也可以看做是误差估计的另一种表型形式.     

域外奇源分布法场点解的唯一性

应用奇源解的核函数沿距离递减的条件和积分不等式,可证这类奇源在凸域外分布只要满足以它产生的响应来表达的边界条件时,所引起相应的域内场点解是唯一的。文中给出这类奇源的部分例子,如Kelvin的点力,点圆力偶(PRC)等。并给出PRC分布解回转体扭转问题的场点解的唯一性证明作为应用例。     

On the problem of superconvergence of finite element method algorithms

The coincidence of an approximate solution to the boundary value problem for an ordinary differential equation with the exact solution at mesh nodes is proved for a certain class of the generalized finite element methods.     

差分方程解的数值余量消元法

本文提出一种新的消元方法,该法利用数值的直接迭代产生余量方程,从而构成已消去很多未知量的线性方程组.本文的方法具有求解简便、精确和快速的优点.     

An open problem on the optimality of an asymptotic solution to Duffing’s nonlinear oscillation problem

Based on the renormalization group method, Kirkinis (2012) [8] obtained an asymptotic solution to Duffing’s nonlinear oscillation problem. Kirkinis then asked if the asymptotic solution is optimal. In this paper, an affirmative answer to the open problem is given by means of the homotopy analysis method.     

Convergence study of the Chorin-Marsden formula

Using the fundamental solution of the heat equation, we give an expression of the solutions to two-dimensional initial-boundary value problems of the Navier-Stokes equations, where the vorticity is expressed in terms of a Poisson integral, a Newtonian potential, and a single layer potential. The density of the single layer potential is the solution to an integral equation of Volterra type along the boundary. We prove there is a unique solution to the integral equation. One fractional time step approximation is given, based on this expression. Error estimates are obtained for linear and nonlinear problems. The order of convergence is for the Navier-Stokes equations. The result is in the direction of justifying the Chorin-Marsden formula for vortex methods. It is shown that the density of the vortex sheet is twice the tangential velocity for the half plane, while in general the density differs from it by one additional term.

     

Two regularization methods for inverse source problem on the Poisson equation

This paper is devoted to discuss an inverse problem of determining an unknown source on the Poisson equation. This is a mildly ill-posed problem. Two regularization methods, one based on the mollification of the data and the other based on the modification of the ‘kernel’ of the solution, are proposed to solve this problem. The convergence estimates between the exact solution and the regularization solution are presented using a priori regularization parameter choice rule. Numerical results are presented to illustrate the accuracy and efficiency of the proposed methods.