A collocation method for solving singular integro-differential equations

This paper describes a collocation method for numerically solving Cauchy-type linear singular integro-differential equations. The numerical method is based on the transformation of the integro-differential equation into an integral equation, and then applying a collocation method to solve the latter. The collocation points are chosen as the Chebyshev nodes. Uniform convergence of the resulting method is then discussed. Numerical examples are presented and solved by the numerical techniques.     

An explicit spectral collocation method using nonpolynomial basis functions for the time‐dependent Schrödinger equation

We propose a spectral collocation method for the numerical solution of the time‐dependent Schrödinger equation, where the newly developed nonpolynomial functions in a previous study are used as basis functions. Equipped with the new basis functions, various boundary conditions can be imposed exactly. The preferable semi‐implicit time marching schemes are employed for temporal discretization. Moreover, the new basis functions build in a free parameter λ intrinsically, which can be chosen properly so that the semi‐implicit scheme collapses to an explicit scheme. The method is further applied to linear Schrödinger equation set in unbounded domain. The transparent boundary conditions are constructed for time semidiscrete scheme of the linear Schrödinger equation. We employ spectral collocation method using the new basis functions for the spatial discretization, which allows for the exact imposition of the transparent boundary conditions. Comprehensive numerical tests both in bounded and unbounded domain are performed to demonstrate the attractive features of the proposed method.     

用摄动配置方法求解含时薛定谔方程

将摄动配置方法应用到含时薛定谔方程,在计算实现的基础上结合摄动配置的特征提出了一类新的数值积分方法,并给出了一个2级2阶和一个3级4阶的辛摄动配置方法对含时薛定谔方程的数值算例.为了检验新的数值积分方法,我们还给出了与两个辛摄动配置格式在理论上等价的辛龙格-库塔方法以及同阶的非辛方法的数值模拟.展示了一些数值结果,并给出了一些分析.     

Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations

Summary. We prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation by a (periodic) boundary value approach. This equation arises, e.g., in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations. We obtain convergence results for the standard collocation algorithm and for two variants. In particular, estimates of the difference between the collocation solution and the true solution are derived. For the standard collocation scheme the convergence results are “unconditional”, that is, they do not require mesh-ratio restrictions. Numerical results that support the theoretical findings are also given. Received June 9, 2000 / Revised version received December 14, 2000 / Published online October 17, 2001     

Linear barycentric rational collocation method for solving heat conduction equation

The linear barycentric rational collocation method for solving heat conduction equation is presented. The matrix form of discrete heat conduction equation by collocation method is also obtained. With the help of convergence rate of the barycentric interpolation, the convergence rate of linear barycentric rational collocation method for solving heat conduction equation is proved. At last, several numerical examples are provided to validate the theoretical analysis.     

A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel

This work is concerned with the numerical solution of a nonlinear weakly singular Volterra integral equation. Owing to the singular behavior of the solution near the origin, the global convergence order of product integration and collocation methods is not optimal. In order to recover the optimal orders a hybrid collocation method is used which combines a non-polynomial approximation on the first subinterval followed by piecewise polynomial collocation on a graded mesh. Some numerical examples are presented which illustrate the theoretical results and the performance of the method. A comparison is made with the standard graded collocation method.     

A numerical approach for solving initial-boundary value problem describing the process of cooling of a semi-infinite body by radiation

In this article, we have introduced a Taylor collocation method, which is based on collocation method for solving initial-boundary value problem describing the process of cooling of a semi-infinite body by radiation. This method is based on first taking the truncated Taylor expansions of the solution function in the fractional differential equation and then substituting their matrix forms into the equation. Using collocation points, we have the system of nonlinear algebraic equation. Then, we solve the system of nonlinear algebraic equation using Maple 13 and we have the coefficients of Taylor expansion. In addition, numerical results are presented to demonstrate the effectiveness of the proposed method.     

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.     

混合广义Jacobi和Chebyshev谱配置法求解时间分数阶对流扩散方程

研究时间Caputo分数阶对流扩散方程的高效高阶数值方法.对于给定的时间分数阶偏微分方程,在时间和空间方向分别采用基于移位广义Jacobi函数为基底和移位Chebyshev多项式运算矩阵的谱配置法进行数值求解.这样得到的数值解可以很好地逼近一类在时间方向非光滑的方程解.最后利用一些数值例子来说明该数值方法的有效性和准确性.     

Polynomial spectral collocation method for space fractional advection–diffusion equation

This article discusses the spectral collocation method for numerically solving nonlocal problems: one‐dimensional space fractional advection–diffusion equation; and two‐dimensional linear/nonlinear space fractional advection–diffusion equation. The differentiation matrixes of the left and right Riemann–Liouville and Caputo fractional derivatives are derived for any collocation points within any given bounded interval. Several numerical examples with different boundary conditions are computed to verify the efficiency of the numerical schemes and confirm the exponential convergence; the physical simulations for Lévy–Feller advection–diffusion equation and space fractional Fokker–Planck equation with initial δ‐peak and reflecting boundary conditions are performed; and the eigenvalue distributions of the iterative matrix for a variety of systems are displayed to illustrate the stabilities of the numerical schemes in more general cases. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 514–535, 2014     

热传导型半导体问题的配置方法及误差分析

热传导型半导体瞬态问题的数学模型是一类非线性偏微分方程的初边值问题．电子位势方程是椭圆型的，电子、空穴浓度方程及热传导方程是抛物型的．该文给出求解的配置方法，得到次优犔２模误差估计，并将配置法和Ｇａｌｅｒｋｉｎ有限元方法进行数值结果比较．     

A spline collocation approach for the numerical solution of a generalized nonlinear Klein-Gordon equation

In this paper, a finite element collocation approach using cubic B-splines is employed for the numerical solution of a generalized form of the nonlinear Klein-Gordon equation. The efficiency of the method is tested on a number of examples that represent special cases of the extended equation including the sine-Gordon equation. The numerical results are compared with existing numerical and analytic solutions and the outcomes confirm that the scheme yields accurate and reliable results even when few nodes are used at the time levels.     

Bessel collocation approach for solving continuous population models for single and interacting species

In this paper, we present a collocation method to obtain the approximate solutions of continuous population models for single and interacting species. By using the Bessel polynomials and collocation points, this method transforms population model into a matrix equation. The matrix equation corresponds to a system of nonlinear equations with the unknown Bessel coefficients. The reliability and efficiency of the proposed scheme are demonstrated by two numerical examples and performed on the computer algebraic system Maple.     

Numerical solutions of KdV equation using radial basis functions

Numerical solution of the Korteweg-de Vries equation is obtained by using the meshless method based on the collocation with radial basis functions. Five standard radial basis functions are used in the method of the collocation. The results are compared for the numerical experiments of the propagation of solitons, interaction of two solitary waves and breakdown of initial conditions into a train of solitons.     

Algorithms for numerical solution of the modified equal width wave equation using collocation method

Quintic B-spline collocation algorithms for numerical solution of the modified equal width wave (MEW) equation have been proposed. The algorithms are based on Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. Quintic B-spline collocation method over the finite intervals is also applied to the time split MEW equation and space split MEW equation. Results for the three algorithms are compared by studying the propagation of the solitary wave, interaction of the solitary waves, wave generation and birth of solitons.     

Collocation and iterated collocation methods for a class of weakly singular Volterra integral equations

We discuss the convergence properties of spline collocation and iterated collocation methods for a weakly singular Volterra integral equation associated with certain heat conduction problems. This work completes the previous studies of numerical methods for this type of equations with noncompact kernel. In particular, a global convergence result is obtained and it is shown that discrete superconvergence can be achieved with the iterated collocation if the exact solution belongs to some appropriate spaces. Some numerical examples illustrate the theoretical results.     

A COLLOCATION METHOD FOR THE CONDUCTIVITY PROBLEM WITH DISCONTINUOUS COEFFICIENT

In this paper, a new collocation BEM for the Robin boundary value problem of the conductivity equation ▽(γ▽u) = 0 is discussed, where the 7 is a piecewise constant function. By the integral representation formula of the solution of the conductivity equation on the boundary and interface, the boundary integral equations are obtained. We discuss the properties of these integral equations and propose a collocation method for solving these boundary integral equations. Both the theoretical analysis and the error analysis are presented and a numerical example is given.     

Stability estimates based on numerical ranges with an application to a spectral method

This paper is concerned with the stability of numerical processes for solving initial value problems. We present a stability result which is related to a well-known theorem by von Neumann, but the requirements to be satisfied are less severe and easier to verify.As an illustration we consider a simple convection-diffusion equation. For the spatial discretization we use a spectral collocation method (based on so-called Legendre-Gauss-Lobatto points). We show that the fully discretized numerical process is stable, provided that the temporal step size is bounded by a constant depending only on the convection-diffusion equation, the number of collocation points and the time-stepping method under consideration.This research has been supported by the Netherlands Organization for Scientific Research (N.W.O.).     

Piecewise polynomial collocation for linear boundary value problems of fractional differential equations

We consider a class of boundary value problems for linear multi-term fractional differential equations which involve Caputo-type fractional derivatives. Using an integral equation reformulation of the boundary value problem, some regularity properties of the exact solution are derived. Based on these properties, the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed. In particular, we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points. Theoretical results are verified by two numerical examples.     

变系数Burgers方程的一种预处理谱方法

本文用预处理Legendre-Galerkin-Chebyshev配置方法来求解二阶变系数Burgers方程,该方法首先对方程进行预处理,然后在空间方向应用Legendre-Galerkin-Chebyshev配置方法,对时间方向采用leap-frog/Crank-Nicolson格式进行离散,这样可使得变系数项和非线性项能够显式计算.我们对该方法进行了误差估计,并通过数值算例验证了算法的有效性和理论分析的正确性.