Quintic splines method for second-order boundary value problems

Abstract

We develop a numerical method for computing smooth approximations to the solution of a system of second-order boundary value problems associated with obstacle, unilateral and contact problems based on uniform mesh quintic splines. It is shown that this method gives better approximations than those produced by other collocation, finite-difference and spline methods. A numerical example is given to illustrate the applicability of the new method.     

多层积分值三次样条拟插值

目的 在实际问题中,某些插值问题结点处的函数值往往是未知的,而仅仅知道一些连续等距区间上的积分值。为此提出了一种基于未知函数在连续等距区间上的积分值和多层样条拟插值技术来解决函数重构。该方法称之为多层积分值三次样条拟插值方法。方法 首先,利用积分值的线性组合来逼近结点处的函数值;然后,利用传统的三次B-样条拟插值和相应的误差函数来实现多层三次样条拟插值;最后,给出两层积分值三次样条拟插值算子的多项式再生性和误差估计。结果 选取无穷次可微函数对多层积分值三次样条拟插值方法和已有的积分值三次样条拟插值方法进行对比分析。数值实验印证了本文方法在逼近误差和数值收敛阶均稍占优。结论本文多层三次样条拟插值函数能够在整体上很好的逼近原始函数,一阶和二阶导函数。本文方法较之于已有的积分值三次样条拟插值方法具有更好的逼近误差和数值收敛阶。该方法对连续等距区间上积分值的函数重构具有普适性。     

Necessary and sufficient conditions for the existence of a positive definite solution of a nonlinear matrix equation

Some necessary and sufficient conditions for the existence of a positive definite solution of the nonlinear matrix equation X+A*X ^?α A=Q (0<α≤1) are given. By the way an iterative method is presented. Furthermore, the convergence and error estimation of the iterative algorithm are derived. The illustrative numerical examples due to Peng are worked out.     

Alternating group explicit method with exponential-type for the diffusion–convection equation

Based on the semi-explicit asymmetric exponential schemes constructed by the author, a new alternating group explicit (AGE) method with exponential-type for the numerical solution of the convection–diffusion equation is derived in the paper. The method has the obvious property of parallelism and is unconditionally stable. The results of numerical examples are given to show the effectiveness of the present methods that are in preference to the Evans and Abdullah' method in [Evans, D.J. and Abdullah, A.R., 1985, A new explicit method for the diffusion–convection equation. Computers & Mathematics with Application, 11, 145–154].     

On Accelerated Iterative Methods for the Solution of Systems of Linear Equations

In literature, it has been reported that the convergence of some preconditioned stationary iterative methods using certain type upper triangular matrices as preconditioners are faster than the basic iterative methods. In this paper, a new preconditioned iterative method for the numerical solution of linear systems has been introduced, and the convergence analysis of the proposed method and an existing one have been done. Some numerical examples have also been given, which show the effectiveness of both of the methods.     

Existence,Uniqueness and Applications of the Spline Method for Functional-Differential Equation of Neutral Type

This paper presents a class of numerical method for the approximate solution for the functional-differential equation of neutral type. These methods are essentially based on the spline functions. The study of existence and uniqueness are considered of 3h-step spline function of degree m =4. Numerical examples and comparisons with other methods are given.     

A Study On Computation Of Component Centers In The Degree- n Bifurcation Set

The governing equation locating component centers in the degree- n bifurcation set is a polynomial with a very high degree and its root-finding lacks numerical accuracy. The equation is transformed to have its degree reduced by a factor (n - 1) . Newton's method applied to the transformed equation improves the accuracy with properly chosen initial values. The numerical implementation is done with Maple V using a large number of computational precision digits. Many cases are studied for 2\leq n \leq 25 and show a remarkably improved computation. Our study extends the results given by Peitgen and Richter [11].     

连续等距区间上积分值的二次样条插值

目的 在现实中,某些插值问题结点处的函数值往往是未知的,而仅仅已知一些区间上的积分值。为此提出一种给定已知函数在连续等距区间上的积分值构造二次样条插值函数的方法。方法 首先,利用二次B样条基函数的线性组合去满足给定的积分值和两个端点插值条件,该插值问题等价于求解n+2个方程带宽为3的线性方程组。然后,运用算子理论给出二次样条插值函数的误差估计,继而得到二次样条函数逼近结点处的函数值时具有超收敛性。最后,通过等距区间上积分值的线性组合逼近两个端点的函数值方法实现了不带任何边界条件的积分型二次样条插值问题。结果 选取低频率函数,对积分型二次样条插值方法和改进方法分别进行数值测试,发现这两种方法逼近效果都是良好的。同样,选取高频率函数对积分型二次样条插值方法进行数值实验,得到数值收敛阶与理论值相一致。结论 实验结果表明,本文算法相比已有的方法更简单有效,对改进前后的二次样条插值函数在逼近结点处的函数值时的超收敛性得到了验证。该方法对连续等距区间上积分值的函数重构具有普适性。     

Correction of Numerical Integration as an Optimal Control Problem

The drift of numerical solution of a dynamical system off the integral and constraint surface is eliminated with the help of a gradient supplement to the equations of motion. This supplement makes the image point in the phase space to move along the normal to a given surface. The solution is compared with those obtained by integrating the generalized Hamilton–Dirac dynamics equations. The calculations were performed in Maple V R5 with the use of the standard subroutine dsolve/numeric for solving ordinary differential equations by the fourth-order Runge–Kutta method.     

Numerical solution of some initial optimal control problems using the reproducing kernel Hilbert space technique

ABSTRACT

In this paper, we present a general technique for solving a class of linear/nonlinear optimal control problems. In fact, an analytical solution of the state variable is represented in the form of a series in a reproducing kernel Hilbert space. Sometimes with the aid of this series form, we can also present the optimal control variable in a series form. An iterative method is given to obtain the approximate optimal control and state variables and the cost functional is numerically obtained. Convergence analysis of the method is also provided. Several numerical examples are tested to demonstrate the applicability and efficiency of the method.     

Partially symmetrical derivative-free Liu–Storey projection method for convex constrained equations

ABSTRACT

Applying Powell symmetrical technique to the Liu–Storey conjugate gradient method, a partially symmetrical Liu–Storey conjugate gradient method is proposed and extended to solve nonlinear monotone equations with convex constraints, which satisfies the sufficient descent condition without any line search. By using some line searches, the global convergence is proved merely by assuming that the equations are Lipschitz continuous. Moreover, we prove the R-linear convergence rate of the proposed method with an additional assumption. Finally, compared with one existing method, the performance of the proposed method is showed by some numerical experiments on the given test problems.     

Preconditioning technique for symmetric M-matrices

Abstract Linear systems with M-matrices occur in a wide variety of areas including numerical partial differential equations, input-output production and growth models in economics, linear complementarity problems in operations research and Markov chains in stochastic analysis.In this paper, we propose a new preconditioner for solving a system with symmetric positive definite M-matrix by the preconditioned conjugate gradient (PCG) method. We show that our preconditioner increases the convergence rate of the PCG method and reduces the operation cost. Numerical results are given.     

A split least-squares characteristic mixed element method for nonlinear nonstationary convection–diffusion problem

In this paper, a split least-squares characteristic mixed finite element method is proposed for solving nonlinear nonstationary convection–diffusion problem. By selecting the least-squares functional property, the resulting least-squares procedure can be split into two independent symmetric positive definite sub-schemes. The first sub-scheme is for the unknown variable u, which is the same as the standard characteristic Galerkin finite element approximation. The second sub-scheme is for the unknown flux σ. Theoretical analysis shows that the method yields the approximate solutions with optimal accuracy in L ²(Ω) norm for the primal unknown and in H(div; Ω) norm for the unknown flux, respectively. Some numerical examples are given to confirm our theory results.     

Computational Aspects Of Numerical Integration Based On Optimal Nodal Splines

The numerical construction of nodal spline integration rules is considered for the evaluation both of weakly singular integrals and of certain integrals defined in the Hadamard finite part sense. A bound for the quadrature sum condition number is also given.     

Numerical methods for minimizing real functions unimodal on a set

Abstract An extension of the classical concept of unimodality was recently proposed in [4] and slightly modified in [5]. Here we present a numerical method based on the idea of bisection for determining the minimum points of a real unimodal function on a set. A serial and a parallel algorithm are given.     

Maximization of eigenvalues using topology optimization

Topology optimization is used to optimize the eigenvalues of plates. The results are intended especially for MicroElectroMechanical Systems (MEMS) but can be seen as more general. The problem is not formulated as a case of reinforcement of an existing structure, so there is a problem related to localized modes in low density areas. The topology optimization problem is formulated using the SIMP method. Special attention is paid to a numerical method for removing localized eigenmodes in low density areas. The method is applied to numerical examples of maximizing the first eigenfrequency. One example is a practical MEMS application; a probe used in an Atomic Force Microscope (AFM). For the AFM probe the optimization is complicated by a constraint on the stiffness and constraints on higher order eigenvalues. Received June 10, 1999     

PHoM – a Polyhedral Homotopy Continuation Method for Polynomial Systems

PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedral-linear homotopy functions, based on the polyhedral homotopy theory, from input data for a given system of polynomial equations f(x)=0. The second module CMPSc traces the solution curves of the homotopy equations to compute all isolated solutions of f(x)=0. The third module Verify checks whether all isolated solutions of f(x)=0 have been approximated correctly. We describe numerical methods used in each module and the usage of the package. Numerical results to demonstrate the performance of PHoM include some large polynomial systems that have not been solved previously.AMS Subject Classification: Primary: 65H10 system of equations, secondary: 65H20 global methods, including homotopy approaches.     

Clinical Pure Types as a Fuzzy Partition

Abstract

This paper discusses the applicability of the ideas of fuzzy sets and grades of membership to problems encountered in the quantification of clinical (i.e., diagnostic and prognostic) judgment. The methods of constrained maximum likelihood are used to derive consensus estimates of grades of membership given a set of categorical data and an a priori set of specified pure types. A numerical example is given.     

The numerical solution of fourth order mildly quasi-linear parabolic initial boundary value problem of second kind

In this article, we report on three-level implicit stable finite difference formulas of O(k ² + h ²) and O(k ² + h ⁴) for the numerical integration of certain mildly quasi-liner fourth order parabolic partial differential equations in one-space dimension. The numerical solution of u _xx is obtained as a by-product of the method. In all cases, we use only (3 + 3 + 3)-grid points and a single computational cell. Difference schemes for the fourth order linear parabolic equation in polar coordinates are also discussed. The stability analysis for the model linear problem is given as a representative example. Numerical results are presented to demonstrate the order and accuracy of the proposed methods.