Chaotic dynamics of a third-order Newton-type method

The dynamics of a classical third-order Newton-type iterative method is studied when it is applied to degrees two and three polynomials. The method is free of second derivatives which is the main limitation of the classical third-order iterative schemes for systems. Moreover, each iteration consists only in two steps of Newton's method having the same derivative. With these two properties the scheme becomes a real alternative to the classical Newton method. Affine conjugacy class of the method when is applied to a differentiable function is given. Chaotic dynamics have been investigated in several examples. Applying the root-finding method to a family of degree three polynomials, we have find a bifurcation diagram as those that appear in the bifurcation of the logistic map in the interval.     

Approximate solution of a mixed problem for a parabolic equation by means of a special basis of functions

Using the eigenfunctions of two Sturm-Liouville problems (with the same operator of the most general form but two different sets of boundary conditions), we propose a method for construction of specific basis functions such that the corresponding expansions of smooth and piecewise smooth functions lead to fast converging series. The last circumstance can be successfully employed for approximate solution of mixed problems for a parabolic equation when the sought function is approximated in the spatial variables by few basis functions. First, the case of one spatial coordinate is elaborated and the two-dimensional case is briefly discussed in Section 5. The method aims primarily at the case when the sought function is piecewise smooth in the spatial variable and the implementation of the method bases on the concept of generalized solution.     

Efficiency and production rate of a transfer line with two machines and a finite storage buffer

A markov model for a transfer line with two unreliable machines separated by a finite storage size buffer is introduced. Service time distribution for the two machines is Erlang whereas failure and repair times are assumed to be exponential random variables. The paper presents an efficient method to solve analytically the steady state probabilities of the system. This method is independent of the buffer size. We also include in the paper a study of the behavior of some systems performance measures such as the efficiency of the two machines and the production rate of the system.     

On a spline collocation method for a singularly perturbed problem

We consider a singularly perturbed boundary value problem with two small parameters. The problem is numerically treated by a quadratic spline collocation method. The suitable choice of collocation points provides the inverse monotonicity enabling utilization of barrier function method in the error analysis. Numerical results give justification of the proposed method. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Identification of a time-dependent source term for a time fractional diffusion problem

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method.     

Guaranteed and locally computable a posteriori error estimate

^** Email: vejchod{at}math.cas.cz A new approach, based on the combination of the equilibratedresidual method and the method of hypercircle, is proposed fora posteriori error estimation. Computer implementation of theequilibrated residual method is fast, but it does not produceguaranteed estimates. On the other hand, the method of hypercircledelivers guaranteed estimates, but it is not fast because itinvolves solving a global linear algebraic system. The combinationof these two methods leads to guaranteed and locally computablea posteriori error estimator. This combined method is appliedto linear elliptic problem in two dimensions with mixed boundaryconditions and non-negative absolute terms.     

一种基于虚拟决策单元的排序方法的完善和扩展

本文对文[1]中提出的基于虚拟决策单元的排序方法进行了完善和扩展。首先,根据CCR模型,给出了两类特殊的DEA模型,分别是仅有投入数据的DEA模型和仅有产出数据的DEA模型;其次,基于这两个模型,应用上述方法实现了对仅有投入(或产出)数据的决策单元的排序;第三,给出了排序方法中参数a的计算方法;最后,通过修正排序模型,有效提高了排序方法的计算精度。改进后的排序方法避免了两个决策单元因为相对效率值过小而不能排序的情形,其应用范围也进一步扩大。     

Iterative methods for a forward-backward heat equation in two-dimension

A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-order central difference scheme at the middle interface are used. Maximum norm error estimate for the procedure is derived. Then an iterative method based on domain decomposition is presented for the numerical scheme and the convergence of the given method is established. Then numerical experiments are presented to support the theoretical analysis.     

Confidence interval estimation by a joint pivot method

The Birnbaum–Saunders distribution is widely used in fields such as survival analysis. In this paper, we use a new method to construct confidence intervals for the shape parameter of the Birnbaum–Saunders distribution and find it superior to some existing methods. Our method can be called the “joint pivot method” as we obtain the confidence interval by uniting two pivots. Furthermore, we find that the joint pivot method can also be used to solve the Behrens–Fisher problem. Simulation studies show that the joint pivot method is comparable with some commonly practiced ones. Finally, we analyse two real data sets to illustrate the proposed procedure.     

Expansion over a system of shifts of pyramidal functions with a square base

A method for approximation of functions of two variables by a linear combination of non-negative piecewise linear functions with a compact support is presented. Two quadratic pyramids are used as generating functions for the system of shifts. The accuracy of this local method is proved to have the same order as the best approximation by piecewise linear functions.     

Dynamic strength around a coated nanowire with surfaces/interfaces in a half solid

In this paper, the multiple scattering of anti-plane shear waves around a coated nanowire with surfaces/interfaces embedded in a half solid is studied, and the dynamic stress at the two surfaces/interfaces is presented. The boundary condition at the edge of the half solid is satisfied by the image method. The analytical solutions of displacements in the two half solids, in the coating layers, and inside the nanowire are expressed by wave function expansion method. The expanded mode coefficients are determined by satisfying the boundary conditions at the two surfaces/interfaces of the coated nanowire and the straight edge of the structure. The addition theorem for cylindrical wave function is employed to accomplish the superposition of displacement fields in the two half solids. Analyses show that the properties of the outer and inner interfaces show different effect on the dynamic stress around the nanowire. The dynamic stress distribution around the nanowire is also significantly related to the interfacial properties at the edge and the position of the nanowire.     

Computation of a rotating flow past a permeable bed

Summary The rotating flow of a viscous incompressible fluid between two disks is studied when there is a porous layer on the lower disk. The motion relative to a rotating frame is caused by a differential rotation of the disks. Generalised Darcy's law represents the flow. The numerical solution is obtained using a shooting method.     

A parameterized Newton method and a quasi-Newton method for nonsmooth equations

This paper presents a parameterized Newton method using generalized Jacobians and a Broyden-like method for solving nonsmooth equations. The former ensures that the method is well-defined even when the generalized Jacobian is singular. The latter is constructed by using an approximation function which can be formed for nonsmooth equations arising from partial differential equations and nonlinear complementarity problems. The approximation function method generalizes the splitting function method for nonsmooth equations. Locally superlinear convergence results are proved for the two methods. Numerical examples are given to compare the two methods with some other methods.This work is supported by the Australian Research Council.     

The reconstruction of a solely time-dependent load in a simply supported non-homogeneous Euler–Bernoulli beam

In this paper, the theoretical and numerical determination of a solely time-dependent load distribution is investigated for a simply supported non-homogeneous Euler–Bernoulli beam. The missing source is recovered from an additional “local” integral measurement. The existence and uniqueness of a solution to the corresponding variational problem is proved by employing Rothe’s method. This method also reveals a time-discrete numerical scheme based on the backward Euler method to approximate the solution. Corresponding error estimates are proved and assessed by two numerical experiments.     

A Geometrical Solution to a Tree Network Problem

A problem in tree networks, that of minimizing the cost of a pipeline tree connecting a source to a number of sinks, is discussed. A method of solution in which the optimal solution is synthesised from optimized subsystems is described. The method is built on the geometrical properties of the simplest subsystem; that with one source and two sinks. Experience with the method suggests that it is a practical alternative to numerical-iterative methods used hitherto.     

一类非线性波动方程的势对称分类

先给出了含有一个任意函数的线性波动方程的古典和势对称的完全分类.然后,在此基础上给出了含有两个任意函数的一类非线性波动方程的两种情形势对称分类,得到了该方程的新势对称.在处理对称群分类问题的难点-求解确定方程组时我们提出了微分形式吴方法算法,克服了以往难于处理的困难.在整个计算过程中反复使用了吴方法,吴方法起到了关键的作用.     

The transmission of a plane acoustic wave through a non-uniform thermoelastic layer

The reflection and refraction of a plane acoustic wave by a thermoelastic plane layer, non-uniform in thickness, bounded by non-viscous heat-conducting liquids, generally different, is considered. The system of equations for small perturbations of the thermoelastic medium is reduced to a system of ordinary differential equations, the boundary-value problem for which is solved by two methods: the spline-collocation method and the power-series method. Analytic expressions are obtained which describe the wave fields outside the layer. The results of calculations of the intensity transmission coefficient of the acoustic wave are presented.     

A posteriori error estimation for a defect correction method applied to conduction convection problems

We present a posteriori error estimate for a defect correction method for approximating solutions of the stationary conduction convection problems in two dimension. The defect correction method is aiming at small viscosity ν. A reliable a posteriori error estimation is derived for the defect correction method. Finally, two numerical examples validate our theoretical results. The first example is a problem with known solution and the second example is a physical model of square cavity stationary flow. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Finite-Difference Method for a System of Third-Order Boundary-Value Problems

In this paper, we develop a two-stage numerical method for computing the approximate solutions of third-order boundary-value problems associated with odd-order obstacle problems. We show that the present method is of order two. A numerical example is presented to illustrate the applicability of the new method. A comparison is also given with previously known results.     

变刚度矩形薄板的弯曲与稳定计算

本文研究两对边简支,其它两边任意支承,板的刚度沿简支边方向按任意规律变化的矩形板.采用文[1]提出的有限板条元素法求解,其特点与习用的有限元法或有限单条法不同.不是先建立单元或单条的刚度矩阵,然后拼装总刚阵再求解,而是确立各个板条元素的变位和内力的传播关系.实例计算表明,该法是一个简便有效的方法.