Adaptive Techniques for Spline Collocation

We integrate optimal quadratic and cubic spline collocation methods for second-order two-point boundary value problems with adaptive grid techniques, and grid size and error estimators. Some adaptive grid techniques are based on the construction of a mapping function that maps uniform to non-uniform points, placed appropriately to minimize a certain norm of the error. One adaptive grid technique for cubic spline collocation is mapping-free and resembles the technique used in COLSYS (COLNEW) [2], [4]. Numerical results on a variety of problems, including problems with boundary or interior layers, and singular perturbation problems indicate that, for most problems, the cubic spline collocation method requires less computational effort for the same error tolerance, and has equally reliable error estimators, when compared to Hermite piecewise cubic collocation. Comparison results with quadratic spline collocation are also presented.     

On the error analysis of the numerical solution of linear Volterra integral equations of the second kind

In the numerical solution of linear Volterra integral equations, two kinds of errors occur. If we use the collocation method, these errors are the collocation and numerical quadrature errors. Each error has its own effect in the accuracy of the obtained numerical solution. In this study we obtain an error bound that is sum of these two errors and using this error bound the relation between the smoothness of the kernel in the equation and also the length of the integration interval and each of these two errors are considered. Concluded results also are observed during the solution of some numerical examples.     

Application of orthogonal collocation to simulation and control of first order hyperbolic systems

First order hyperbolic equations appear often as control systems' models of such plants as heat exchangers, tubular reactors, metallurgical and ceramic kilns, aerated rivers, inventory and demographical processes.Application of orthogonal collocation to simulation and control of systems governed by these equations is presented in the paper. The collocation method is one of several Methods of Weighted Residuals. It makes possible the easy transition of a partial differential equation to an ordinary one, the solution of which approximates the exact solution in some discrete spatial collocation points.The results of numerical experiments are given. They serve to evaluate usefulness of the method. In the examples the approximation accuracy in dependence of input signal frequency and accuracy of linear-quadratic optimal control solution for different collocation approximations are considered.The best results have been obtained by choosing the collocation points as zeroes of the Legendre orthogonal polynomials. In practice it is sufficient to approximate one scalar partial differential equation by 4 to 5 ordinary ones.So the orthogonal collocation can be recommended to simulation and control of systems governed by first order hyperbolic equations. The formulae and collocation constants given in the appendix make the application easier.     

Family of quadratic spline difference schemes for a convection–diffusion problem

We consider the finite difference approximation of a singularly perturbed one-dimensional convection–diffusion two-point boundary value problem. It is discretized using quadratic splines as approximation functions, equations with various piecewise constant coefficients as collocation equations and a piecewise uniform mesh of Shishkin type. The family of schemes is derived using the collocation method. The numerical methods developed here are non-monotone and therefore apart from the consistency error we use Green's grid function analysis to prove uniform convergence. We prove the almost first order of convergence and furthermore show that some of the schemes have almost second-order convergence. Numerical experiments presented in the paper confirm our theoretical results.     

Performance analysis of error-control B-spline Gaussian collocation software for PDEs

B-spline Gaussian collocation software has been widely used in the numerical solution of boundary value ordinary differential equations (BVODEs) and 1D partial differential equations (PDEs) for several decades. Such packages represent the numerical solution in terms of a piecewise polynomial (B-spline) basis with basis coefficients determined through the use of Gaussian collocation. The software package, BACOL, developed over a decade ago, was the first 1D PDE package of this type to provide both temporal and spatial error control. A recently developed package, BACOLI, improves upon the efficiency of BACOL through the use of new types of spatial error estimation and control. The complexity of the interactions among the component numerical algorithms used by these packages (particularly the spatial and temporal error estimation and control algorithms) implies that extensive testing and analysis of the test results is an essential factor in the ongoing development of these packages In this paper, we investigate the performance of BACOL and BACOLI with respect to several important machine independent algorithmic measures, examine the effectiveness of the new spatial error estimation and control strategies, and investigate the influence of the choice of the degree of the B-spline basis on the performance of the solvers. These results will provide new insights into how to improve BACOLI, potentially lead to improvements in Gaussian collocation BVODE solvers, and guide further development of B-spline Gaussian collocation software with error control for 2D PDEs.     

A meshless method for solving mKdV equation

A meshless method based on a global collocation with radial basis functions for the numerical solution of the modified Korteweg–de Vries (mKdV) equation is presented. Standard types of radial basis functions are applied in the method of collocation. The stability analysis of the method is dealt with using a linearized stability analysis. The method?s accuracy and efficiency are examined by the simulation of a single soliton and interaction of two solitary waves. The four invariants of the motion are evaluated to determine the conservation properties of the method. A comparison with some earlier reported results is also carried out.     

网格缩减的自适应hp伪谱法

针对控制变量不连续的最优控制问题,本文提出一种自适应更新的忉伪谱法,这种方法在(Legendre Gauss Radau,LGR)点处取配点,能够以较小的网格规模获得较高的精度.通过计算相对误差估计,判断网格规模是增加还是缩减,若相对容许误差大于给定值,则增加网格区间数或网格配点数提高解的精度,反之则合并网格或减小网格配点数缩减网格规模提高计算效率.将hp伪谱法应用于最优控制问题,仿真验证了hp伪谱法的优越性.     

一类生物系统的参数辨识方法及其应用

针对S-型生物系统的参数辨识问题,给出了以极小化误差函数为优化目标的参数辨识模型。基于修正配置和B样条插值,提出了一种有效的参数辨识模型求解方法。计算研究表明,该方法可以获得更为精确的参数辨识结果。将该方法应用到甘油生物发酵系统中,进一步验证了该方法的有效性。     

基于改进的PHT-样条自适应等几何配点法

将传统等几何配点法扩展至任意高阶单元并且满足自适应局部细分功能,提出一种基于改进的PHT样条单元的自适应等几何配点法.改进的PHT样条单元依然具有传统PHT样条单元局部细分功能,但因为传统PHT样条函数在层级网格划分后需要对部分基函数的定义域进行截断处理,所以在层级细分过于频繁区域,部分函数可能因为严重变形而影响计算稳...     

A Galerkin collocation method for some integral equations of the first kind

Here we present a certain modified collocation method which is a fully discretized numerical method for the solution of Fredholm integral equations of the first kind with logarithmic kernel as principal part. The scheme combines high accuracy from Galerkin's method with the high speed of collocation methods. The corresponding asymptotic error analysis shows optimal order of convergence in the sense of finite element approximation. The whole method is an improved boundary integral method for a wide class of plane boundary value problems involving finite element approximations on the boundary curve. The numerical experiments reveal both, high speed and high accuracy.     

Application of the hybrid functions to solve neutral delay functional differential equations

This paper presents a computational technique for the solution of the neutral delay differential equations with state-dependent and time-dependant delays. The properties of the hybrid functions which consist of block-pulse functions plus Legendre polynomials are presented. The approach uses these properties together with the collocation points to reduce the main problems to systems of nonlinear algebraic equations. An estimation of the error is given in the sense of Sobolev norms. The efficiency and accuracy of the proposed method are illustrated by several numerical examples.     

Improved accuracy of He’s energy balance method for analysis of conservative nonlinear oscillator

In this paper, the accuracy of He’s energy balance method for the analysis of conservative nonlinear oscillator is improved based on combining features of collocation method and Galerkin–Petrov method. In order to demonstrate the effectiveness of proposed method, Duffing oscillator with cubic nonlinearity, double-well Duffing oscillator, and nonlinear oscillation of pendulum attached to a rotating support are considered. Comparison of results with ones achieved utilizing other techniques shows improved energy balance method can very effectively reduce the error of simple energy balance method. Also, results show in large amplitude of oscillation, and improved energy balance method yields better accuracy rather than second-order energy balance method based on collocation and second-order energy balance method based on Galerkin method. Improved energy balance method can be successfully used for accurate analytical solution of other conservative nonlinear oscillator.     

Numerical treatment for the modified burgers equation

In this paper, we consider the solution of the modified Burger's equation by using the collocation method with quintic splines. Applying the Von-Neumann stability analysis method we show that the proposed method is unconditionally stable. By conducting a comparison between the absolute error for our numerical results and the analytic solution of the modified Burger's equation we will test the accuracy of the proposed method.     

Spectral Methods for Substantial Fractional Differential Equations

In this paper, a non-polynomial spectral Petrov–Galerkin method and its associated collocation method for substantial fractional differential equations are proposed, analyzed, and tested. We modify a class of generalized Laguerre polynomials to form our trial basis and test basis. After a proper scaling of these bases, our Petrov–Galerkin method results in diagonal and well-conditioned linear systems for certain types of fractional differential equations. In the meantime, we provide superconvergence points of the Petrov–Galerkin approximation for associated fractional derivative and function value of true solution. Additionally, we present explicit fractional differential collocation matrices based upon Laguerre–Gauss–Radau points. It is noteworthy that the proposed methods allow us to adjust a parameter in the basis according to different given data to maximize the convergence rate. All these findings have been proved rigorously in our convergence analysis and confirmed in our numerical experiments.     

Error analysis of the Chebyshev collocation method for linear second-order partial differential equations

The purpose of this study is to apply the Chebyshev collocation method to linear second-order partial differential equations (PDEs) under the most general conditions. The method is given with a priori error estimate which is obtained by polynomial interpolation. The residual correction procedure is modified to the problem so that the absolute error may be estimated. Finally, the effectiveness of the method is illustrated in several numerical experiments such as Laplace and Poisson equations. Numerical results are overlapped with the theoretical results.     

Incremental attribute evaluation through recursive procedures

Incremental semantic analysis in a programming environment based on Attribute Grammars is performed by an Incremental Attribute Evaluator (IAE). Current IAEs are either table-driven or make extensive use of graph structures to schedule reevaluation of attributes. A method of compiling an Ordered Attribute Grammar into mutually recursive procedures is proposed. These procedures form an optimal time Incremental Attribute Evaluator for the attribute grammar, which does not require any graphs or tables.     

汉字语法语义智能输入法搭配库的设计与实现

为提高输入法的智能性,对供输入法使用的搭配知识库进行了研究.介绍了汉字语法语义智能输入法及其改进的功能,在对语料库中搭配知识分析的基础上,结合输入法中搭配知识的使用,对现有的统计语言模型进行了改进,并给出了词语搭配知识库,语法搭配知识库和语义搭配知识库的关键结构,利用改进后的统计语言模型和语法语义搭配知识,对各搭配知识库进行了算法实现,最后给出了各个搭配知识库的部分实验结果并对其进行了分析.     

A meshless method for numerical solution of the coupled Schrödinger-KdV equations

This paper presents meshless method using RBF collocation scheme for the coupled Schrödinger-KdV equations. Instead of traditional mesh oriented methods such as finite element method (FEM) or finite difference method (FDM), this method requires only a scattered set of nodes in the domain. For this scheme, error estimates and stability analysis are studied. L ₂ and L _∞ error norms between the results and exact solution is used as a performance measure. Moreover the results of numerical experiments are presented, and are compared with the findings of Finite Element method, finite difference Crank–Nicolson (CN) scheme and analytical solution to confirm the good accuracy of the presented scheme.     

Numerical results of Emden–Fowler boundary value problems with derivative dependence using the Bernstein collocation method

In this paper, we propose an efficient numerical technique based on the Bernstein polynomials for the numerical solution of the equivalent integral form of the derivative dependent Emden–Fowler boundary value problems which arises in various fields of applied mathematics, physical and chemical sciences. The Bernstein collocation method is used to convert the integral equation into a system of nonlinear equations. This system is then solved efficiently by suitable iterative method. The error analysis of the present method is discussed. The accuracy of the proposed method is examined by calculating the maximum absolute error and the \(L_{2}\) error of four examples. The obtained numerical results are compared with the results obtained by the other known techniques.

     

Stability and convergence of the two parameter cubic spline collocation method for delay differential equations

In this paper, we propose the cubic spline collocation method with two parameters for solving delay differential equations (DDEs). Some results of the local truncation error and the convergence of the spline collocation method are given. We also obtain some results of the linear stability and the nonlinear stability of the method for DDEs. In particular, we design an algorithm to obtain the ranges of the two parameters α,β which are necessary for the P-stability of the collocation method. Some illustrative examples successfully verify our theoretical results.