On Clenshaws's method and a generalisation to faber series

Summary The method of expanding the solution of linear ordinary differential equations in power or laurent series is classical, and is usually associated with the name of Frobenius. In the early days of electronic computation, it was appreciated that expansions in Chebyshev series are often of more practical use, and the necessary techniques developed by Clenshaw. (This is usually carried out in order to approximate special functions defined by ordinary differential equations, rather than as a technique for actually solving such equations in general, for which finite difference methods are to be preferred.) In this paper we show that by means of a conformal map the problem of expansion in Chebyshev series can in fact be reduced to that of expansion in a Laurent series, yielding a method which is usually computationally simpler. Moreoveer, the method can be generalised to the case of Faber expansions on regions of the complex plane. A non-trivial example is explored in order to illustrate the method, and we also use the technique to generalise an identity relating Chebyshev polynomials to the Faber case.This paper is dedicated to the memory of Prof. Peter Henrici, a mentor, collegue and friend who will be greatly missed by all those who had the privilege of knowing and working with him     

The use of Chebyshev series for approximate analytic solution of ordinary differential equations

Application of Chebyshev series to solve ordinary differential equations is described. This approach is based on the approximation of the solution to a given Cauchy problem and its derivatives by partial sums of shifted Chebyshev series. The coefficients of the series are determined by an iterative process using Markov quadrature formulas. It is shown that the proposed approach can be applied to formulate an approximate analytical method for solving Cauchy problems. A number of examples are considered to illustrate the obtaining of approximate analytical solutions in the form of partial sums of shifted Chebyshev series.     

A series method for solving nonlinear two-point boundary value problems

Summary A new method for solving nonlinear boundary value problems based on Taylor-type expansions generated by the use of Lie series is derived and applied to a set of test examples. A detailed discussion is given of the comparative performance of this method under various conditions. The method is of theoretical interest but is not applicable, in its present form, to real life problems; in particular, because of the algebraic complexity of the expressions involved, only scalar second order equations have been discussed, though in principle systems of equations could be similarly treated. A continuation procedure based on this method is suggested for future investigation.     

Application of orthogonal expansions for approximate integration of ordinary differential equations

An approximate method to solve the Cauchy problem for normal and canonical systems of second-order ordinary differential equations is proposed. The method is based on the representation of a solution and its derivative at each integration step in the form of partial sums of series in shifted Chebyshev polynomials of the first kind. A Markov quadrature formula is used to derive the equations for the approximate values of Chebyshev coefficients in the right-hand sides of systems. Some sufficient convergence conditions are obtained for the iterative method solving these equations. Several error estimates for the approximate Chebyshev coefficients and for the solution are given with respect to the integration step size.     

An approximate method for integration of ordinary differential equations

An approximate analytic method of solving a Cauchy problem for normal systems of ordinary differential equations is considered. The method is based on the approximation of the solution by partial sums of shifted Chebyshev series. The coefficients of the series are determined by an iterative process using Markov quadrature formulas.     

An adaptive Chebyshev iterative method\newline for nonsymmetric linear systems based on modified moments

Summary. Large, sparse nonsymmetric systems of linear equations with a matrix whose eigenvalues lie in the right half plane may be solved by an iterative method based on Chebyshev polynomials for an interval in the complex plane. Knowledge of the convex hull of the spectrum of the matrix is required in order to choose parameters upon which the iteration depends. Adaptive Chebyshev algorithms, in which these parameters are determined by using eigenvalue estimates computed by the power method or modifications thereof, have been described by Manteuffel [18]. This paper presents an adaptive Chebyshev iterative method, in which eigenvalue estimates are computed from modified moments determined during the iterations. The computation of eigenvalue estimates from modified moments requires less computer storage than when eigenvalue estimates are computed by a power method and yields faster convergence for many problems. Received May 13, 1992/Revised version received May 13, 1993     

Strict Chebyshev approximation for general systems of linear equations

Summary An ascent exchange algorithm for computing the strict Chebyshev solution to general systems of linear equations is presented. It uses generalized exchange rules to ensure convergence and splits up the entire system into subsystems by means of a canonical decomposition of a matrix obtained by Gaussian elimination methods. All updating procedures are developed and several numerical examples illustrate the efficiency of the algorithm.     

The solution of high-order nonlinear ordinary differential equations by Chebyshev Series

By the use of the Chebyshev series, a direct computational method for solving the higher order nonlinear differential equations has been developed in this paper. This method transforms the nonlinear differential equation into the matrix equation, which corresponds to a system of nonlinear algebraic equations with unknown Chebyshev coefficients, via Chebyshev collocation points. The solution of this system yields the Chebyshev coefficients of the solution function. An algorithm for this nonlinear system is also proposed in this paper. The method is valid for both initial-value and boundary-value problems. Several examples are presented to illustrate the accuracy and effectiveness of the method.     

Boundary value problems for first order differential equations of mixed type

We apply the monotone iterative method to formulate existence results for integro-differential equations with nonlinear boundary conditions. Some results for integro-differential inequalities are also given. Examples illustrate obtained results.     

Algorithms for the integration and derivation of Chebyshev series

General formulas for the mth integral and derivative of a Chebyshev polynomial of the first or second kind are presented. The result is expressed as a finite series of the same kind of Chebyshev polynomials. These formulas permit to accelerate the determination of such integrals or derivatives. Besides, it is presented formulas for the mth integral and derivative of finite Chebyshev series and a numerical algorithm for the direct evaluation of the mth derivative of such a series.     

Some sixth order zero-finding variants of Chebyshev-Halley methods

A zero-finding technique in which the order of convergence is improved and nonlinear equations are solved more efficiently than they are solved by traditional iterative methods is derived. Composing a modified Chebyshev-Halley method with a variant of this method that just introduces one evaluation of the function the iterative methods presented are obtained. By carrying out this procedure the output numerical results show that the new methods compete in both order and efficiency with the modified Chebyshev-Halley methods.     

一种基于Chebyshev迭代解非线性方程组的方法

本文根据牛顿迭代和Chebyshev迭代法给出了一种新的迭代方法,该方法有较高的收敛阶,并在理论上给予了证明.最后给出了四个实例,将本文的实验结果与现有的几种方法的实验结果进行比较,表明我们的方法迭代次数少,有明显的优势.     

Numerical solution of integral equations in Chebyshev series

In this paper we discuss the Chebyshev series method with Newton iterations for the numerical solution of nonlinear integral equations. An existence theorem for nonlinear integral equations is given using a functional analytic approach. A method to compute and error bound to an approximate solution is discussed on the basis of the theorem.     

基于正交多项式的解不适定算子方程的隐式迭代法

该文研究了基于Chebyshev和Jacobi多项式的解不适定算子方程的隐式迭代法.建立了隐式迭代法和由Hanke提出的显式迭代法之间的关系. 给出了与Chebyshev第一和第二多项式相关的迭代格式的残差有理式的一个重要引理. 对精确和扰动的数据, 研究了方程的收敛性和收敛速率. 利用Morozov残差原则, 给出了一个可执行的强健的正则化算法.最后还给出了一些数值例子, 数值结果与理论分析基本一致.     

Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions

This paper describes the method of quasilinearization for first-order nonlinear impulsive functional differential equations with anti-periodic boundary conditions. A monotone iterative technique coupled with lower and upper solutions is employed to obtain sequences of approximate solutions converging monotonically and quadratically to the unique solution of the problem at hand.     

Calculation of expansion coefficients of series in Chebyshev polynomials for a solution to a Cauchy problem

Two approaches are proposed to determine an initial approximation for the coefficients of an expansion of the solution to a Cauchy problem for ordinary differential equations in the form of series in shifted Chebyshev polynomials of the first kind. This approximation is used in an analytical method to solve ordinary differential equations using orthogonal expansions.     

Solving Theodorsen's integral equation for conformal maps with the fast fourier transform and various nonlinear iterative methods

Summary We investigate several iterative methods for the numerical solution of Theodorsen's integral equation, the discretization of which is either based on trigonometric polynomials or function families with known attenuation factors. All our methods require simultaneous evaluations of a conjugate periodic function at each step and allow us to apply the fast Fourier transform for this. In particular, we discuss the nonlinear JOR iteration, the nonlinear SOR iteration, a nonlinear second order Euler iteration, the nonlinear Chebyshev semi-iterative method, and its cyclic variant. Under special symmetry conditions for the region to be mapped onto we establish local convergence in the case of discretization by trigonometric interpolation and give simple formulas for the optimal parameters (e.g., the underrelaxation factor) and the asymptotic convergence factor. Weaker related results for the general non-symmetric case are presented too. Practically, our methods extend the range of application of Theodorsen's method and improve its effectiveness strikingly.This work was partially supported by NRC (Canada) grant No. A8240 while, in 1976, the author was at the Dept. of Computer Science, University of British Columbia, Vancouver, B.C., Canada. It is part of a thesis submitted for Habilitation at ETH Zurich     

Adomian polynomials: A powerful tool for iterative methods of series solution of nonlinear equations

In this article, we illustrate how the Adomian polynomials can be utilized with different types of iterative series solution methods for nonlinear equations. Two methods are considered here: the differential transform method that transforms a problem into a recurrence algebraic equation and the homotopy analysis method as a generalization of the methods that use inverse integral operator. The advantage of the proposed techniques is that equations with any analytic nonlinearity can be solved with less computational work due to the properties and available algorithms of the Adomian polynomials. Numerical examples of initial and boundary value problems for differential and integro-differential equations with different types of nonlinearities show good results.     

Chebyshev-Secant-type Methods for Non-differentiable Operators

In this paper we give a semilocal convergence theorem for a family of iterative methods for solving nonlinear equations defined between two Banach spaces. This family is obtained as a combination of the well known Secant method and Chebyshev method. We give a very general convergence result that allow the application of these methods to non-differentiable problems.     

Periodic boundary value problems for second-order impulsive integro-differential equations

By developing a new comparison result and using the monotone iterative technique, we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for second-order nonlinear impulsive integro-differential equations of mixed type.