A numerical scheme for solutions of the Chen system

In this paper, we will develop the Bessel collocation method to find approximate solutions of the Chen system, which is a three‐dimensional system of ODEs with quadratic nonlinearities. This scheme consists of reducing the problem to a nonlinear algebraic equation system by expanding the approximate solutions by means of the Bessel polynomials with unknown coefficients. By help of the collocation points and the matrix operations of derivatives, the unknown coefficients of the Bessel polynomials are calculated. The accuracy and efficiency of the proposed approach are demonstrated by two numerical examples and performed with the aid of a computer code written in MAPLE. In addition, comparisons between our method and the homotopy perturbation method numerical solutions are made with the accuracy of solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

An Adaptive Greedy Algorithm for Solving Large RBF Collocation Problems

The solution of operator equations with radial basis functions by collocation in scattered points leads to large linear systems which often are nonsparse and ill-conditioned. But one can try to use only a subset of the data for the actual collocation, leaving the rest of the data points for error checking. This amounts to finding sparse approximate solutions of general linear systems arising from collocation. This contribution proposes an adaptive greedy method with proven (but slow) linear convergence to the full solution of the collocation equations. The collocation matrix need not be stored, and the progress of the method can be controlled by a variety of parameters. Some numerical examples are given.     

Chebyshev Wavelet collocation method for solving generalized Burgers–Huxley equation

In this paper, new and efficient numerical method, called as Chebyshev wavelet collocation method, is proposed for the solutions of generalized Burgers–Huxley equation. This method is based on the approximation by the truncated Chebyshev wavelet series. By using the Chebyshev collocation points, algebraic equation system has been obtained and solved. Approximate solutions of the generalized Burgers–Huxley equation are compared with exact solutions. These calculations demonstrate that the accuracy of the Chebyshev wavelet collocation solutions is quite high even in the case of a small number of grid points. Copyright © 2015 John Wiley & Sons, Ltd.     

A Newton-Picard Collocation Method for Periodic Solutions of Delay Differential Equations

This paper presents a collocation method with an iterative linear system solver to compute periodic solutions of a system of autonomous delay differential equations (DDEs). We exploit the equivalence of the linearized collocation system and the discretization of the linearized periodic boundary value problem (BVP). This linear BVP is solved using a variant of the Newton-Picard method [Int. J. Bifurcation Chaos, 7 (1997), pp. 2547–2560]. This method combines a direct method in the low-dimensional subspace of the weakly stable and unstable modes with an iterative solver in the high-dimensional orthogonal complement. As a side effect, we also obtain good estimates for the dominant Floquet multipliers. We have implemented the method in the DDE-BIFTOOL environment to test our algorithm. AMS subject classification (2000) 65J15, 65P30, 65Q05     

On the stability of the spline collocation method for a class of integral equations of the first kind

This paper is concerned with the stability of the spline collocation method for a class of integral equations of the first kind with logarithmic kernels. It is shown that a proper choice of the mesh size can be made in the numerical computation so that one will obtain an optimal rate of convergence for the approximate solutions.     

Geometric meshes in collocation methods for Volterra integral equations with proportional delays

In this paper we analyse the local superconvergence propertiesof iterated piecewise polynomial collocation solutions for linearsecond-kind Volterra integral equations with (vanishing) proportionaldelays qt (0 < q < 1). It is shown that on suitable geometricmeshes depending on q, collocation at the Gauss points leadsto almost optimal superconvergence at the mesh points. Thiscontrasts with collocation on uniform meshes where the problemregarding the attainable order of local superconvergence remainsopen.     

Boundary collocation method for a cracked rectangular plate with double external tension

In this article, the boundary collocation method is employed to investigate the problems of a central crack in a rectangular plate which applied double external tension on the outer boundary under the assumption that the dimensions of the plate are much larger than that of the crack. A set of stress functions has also been proposed based on the theoretical analysis which satisfies the condition that there is no external force on the crack surfaces. It is only necessary to consider the condition on the external boundary. Using boundary collocation method, the linear algebra equations at collocation points are obtained. The least squares method is used to obtain the solution of the equations, so that the unknown coefficients can be obtained. According to the expression of the stress intensity factor at crack tip, we can obtain the numerical results of stress intensity factor. Numerical experiments show that the results coincide with the exact solution of the infinite plate. In particular, this case of the double external tension applied on the outer boundary is seldom studied by boundary collocation method.     

On the convergence of spline collocation methods for solving fractional differential equations

In the first part of this paper we study the regularity properties of solutions of initial value problems of linear multi-term fractional differential equations. We then use these results in the convergence analysis of a polynomial spline collocation method for solving such problems numerically. Using an integral equation reformulation and special non-uniform grids, global convergence estimates are derived. From these estimates it follows that the method has a rapid convergence if we use suitable nonuniform grids and the nodes of the composite Gaussian quadrature formulas as collocation points. Theoretical results are verified by some numerical examples.     

A collocation approach for solving linear complex differential equations in rectangular domains

In this paper, a collocation method is presented to find the approximate solution of high‐order linear complex differential equations in rectangular domain. By using collocation points defined in a rectangular domain and the Bessel polynomials, this method transforms the linear complex differential equations into a matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. The proposed method gives the analytic solution when the exact solutions are polynomials. Numerical examples are included to demonstrate the validity and applicability of the technique and the comparisons are made with existing results. The results show the efficiency and accuracy of the present work. All of the numerical computations have been performed on a computer using a program written in MATLAB v7.6.0 (R2008a). Copyright © 2012 John Wiley & Sons, Ltd.     

A Space-Time Polynomial Collocation Method for Solving 3D Burgers Equations北大核心CSCD

Burgers方程是一类应用广泛的非线性偏微分方程,方程中的非线性项难以处理。该文提出一种新的时空多项式配点法——多项式特解法求解三维Burgers方程。求解过程分为两步:第一步,对三维Burgers方程中的线性导数项(包括时间导数项),求出相应的多项式特解。第二步,将求出的多项式特解作为基函数,对三维Burgers方程中剩余的非线性项进行迭代求解。与时空多项式函数作为基函数对三维Burgers方程进行直接求解相比,该算法简单易行,得到的近似解精度非常高,算法极其稳定,对于教学过程中提高学生的编程能力,加深对高维Burgers方程的理解能力以及Burgers方程的实际应用具有重要意义。     

A collocation approach for the numerical solution of certain linear retarded and advanced integrodifferential equations with linear functional arguments

This article presents a Taylor collocation method for the approximate solution of high‐order linear Volterra‐Fredholm integrodifferential equations with linear functional arguments. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points. Some numerical examples, which consist of initial and boundary conditions, are given to show the properties of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A collocation method for differential equations with one‐point Taylor initial conditions

This paper presents a new multistep collocation method for nth order differential equations. The interval of interest is first divided into N subintervals. By determining interval conditions in Taylor interpolation in each subinterval, Taylor polynomials are calculated with different step lengths. Then the obtained solutions in each subinterval are used as initial conditions in the next step. Optimal error is assessed using Peano lemma, which shows that the errors decay by decreasing the step length. In particular, for fixed step length h, the error is of O(m^?m), where m is the number of collocation points in each subinterval. Meanwhile, for fixed m, the error is of O(h^m). Numerical examples demonstrate the validity and high accuracy of the proposed method even for stiff problems.     

On the Divergence of Collocation Solutions in Smooth Piecewise Polynomial Spaces for Volterra Integral Equations

In this paper we present a characterization of those smooth piecewise polynomial collocation spaces that lead to divergent collocation solutions for Volterra integral equations of the second kind. The key to these results is an equivalence result between such collocation solutions and collocation solutions in slightly smoother spaces for initial-value problems for ordinary differential equations. For the latter problems Mülthei (1979/1980) established a complete divergence (and convergence) theory. Our analysis can be extended to furnish divergence results for smooth collocation solutions to Volterra integral equations of the first kind. AMS subject classification (2000) 65R20, 65L20, 65L60.Received May 2004. Accepted September 2004. Communicated by Tom Lyche.Hermann Brunner: This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).     

The Linear Rational Pseudospectral Method for Boundary Value Problems

We consider the version of the pseudospectral method for solving boundary value problems which replaces the differential operator with a matrix constructed from the elementary differentiation matrices whose elements are the derivatives of the Lagrange fundamental polynomials at the collocation points. The iterative solution of the resulting system of equations then requires the recurrent application of that differentiation matrix. Since global polynomial interpolation on the interval only gives useful approximants for points which accumulate in the vicinity of the extremities, the matrix is ill-conditioned. To reduce this drawback, we use Kosloff and Tal-Ezer's suggestion to shift the collocation points closer to equidistant by a conformal map. However, instead of applying their change of variable setting, we extend to stationary equations the linear rational collocation method introduced in former work on partial differential equations. Numerically about as efficient, this does not require any new coding if one starts from an efficient program for the polynomial differentiation matrices.This revised version was published online in October 2005 with corrections to the Cover Date.     

一类带弱奇异核偏积分微分方程空间谱配置方法的全局性

借助拉普拉斯变换,运用谱配置方法研究一类线性偏积分微分方程的半离散问题,这类问题出现在粘弹性模型中.它是一种基于Gauss-Lobatto求积节点的配置方法.我们得到了空间半离散解的稳定性和收敛性结果.     

A collocation method based on the Bessel functions of the first kind for singular perturbated differential equations and residual correction

In this paper, a collocation method is given to solve singularly perturbated two‐point boundary value problems. By using the collocation points, matrix operations and the matrix relations of the Bessel functions of the first kind and their derivatives, the boundary value problem is converted to a system of the matrix equations. By solving this system, the approximate solution is obtained. Also, an error problem is constructed by the residual function, and it is solved by the presented method. Thus, the error function is estimated, and the approximate solutions are improved. Finally, numerical examples are given to show the applicability of the method, and also, our results are compared by existing results. Copyright © 2014 JohnWiley & Sons, Ltd.     

An exponential matrix method for solving systems of linear differential equations

This paper presents an exponential matrix method for the solutions of systems of high‐order linear differential equations with variable coefficients. The problem is considered with the mixed conditions. On the basis of the method, the matrix forms of exponential functions and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown coefficients are determined and thus the approximate solutions are obtained. Also, an error estimation based on the residual functions is presented for the method. The approximate solutions are improved by using this error estimation. To demonstrate the efficiency of the method, some numerical examples are given and the comparisons are made with the results of other methods. Copyright © 2012 John Wiley & Sons, Ltd.     

THE COLLOCATION METHODS FOR SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNELS

1 Introduction Singular integral equations (SIEs) with Cauchy kernels Of the formoften arise in mathematical models of physical phenomena. Since closed-form solutions to SIEsare generally not available, much att.ntion has been focused on numerical methods of solution.In the past twenty years, various collocation methods for SIEs have been the topic of a greatmany of papers, most of which can be found in two surveys[213]. The early works in the fieldis to study tile numerical solutions for…     

A Hermite collocation method for the approximate solutions of high‐order linear Fredholm integro‐differential equations

In this study, a Hermite matrix method is presented to solve high‐order linear Fredholm integro‐differential equations with variable coefficients under the mixed conditions in terms of the Hermite polynomials. The proposed method converts the equation and its conditions to matrix equations, which correspond to a system of linear algebraic equations with unknown Hermite coefficients, by means of collocation points on a finite interval. Then, by solving the matrix equation, the Hermite coefficients and the polynomial approach are obtained. Also, examples that illustrate the pertinent features of the method are presented; the accuracy of the solutions and the error analysis are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1707–1721, 2011     

Superconvergence of the iterated collocation methods for Hammerstein equations

In this paper, we analyse the iterated collocation method for Hammerstein equations with smooth and weakly singular kernels. The paper expands the study which began in [16] concerning the superconvergence of the iterated Galerkin method for Hammerstein equations. We obtain in this paper a similar superconvergence result for the iterated collocation method for Hammerstein equations. We also discuss the discrete collocation method for weakly singular Hammerstein equations. Some discrete collocation methods for Hammerstein equations with smooth kernels were given previously in [3, 18].