Perturbed Chebyshev rational approximation

In this work, we give a perturbed Chebyshev rational approximation for a function f (x) which has a Chebyshev expansion. This approximation contains a perturbation parameter ~ which is calculated so that the perturbed Chebyshev rational approximation agrees with the Chebyshev expansion to a certain number of terms. Also, we introduce a perturbed Chebyshev rational approximation for the definite integral of a function f (x) having Chebyshev expansion and show that this method can be used iteratively to approximate the multiple integral of the considered function. The method has been applied to approximate some functions and their definite integrals.     

Uncertain Method for Optimal Control Problems With Uncertainties Using Chebyshev Inclusion Functions

In this paper, a new uncertain analysis method is developed for optimal control problems, including interval variables (uncertainties) based on truncated Chebyshev polynomials. The interval arithmetic in this research is employed for analyzing the uncertainties in optimal control problems comprising uncertain‐but‐bounded parameters with only lower and upper bounds of uncertain parameters. In this research, the Chebyshev method is utilized because it generates sharper bounds for meaningful solutions of interval functions, rather than the Taylor inclusion function, which is efficient in handling the overestimation derived from the wrapping effect due to interval computations. For utilizing the proposed interval method on the optimal control problems with uncertainties, the Lagrange multiplier method is first applied to achieve the necessary conditions and then, by using some algebraic manipulations, they are converted into the ordinary differential equation. Afterwards, the Chebyshev inclusion method is employed to achieve the solution of the system. The final results of the Chebyshev inclusion method are compared with the interval Taylor method. The results show that the proposed Chebyshev inclusion function based method better handle the wrapping effect than the interval Taylor method.     

Polynomial solutions of certain differential equations

In this paper, the Chebyshev matrix method is applied generalisations of the Hermite, Laguerre, Legendre and Chebyshev differential equations which have polynomial solution. The method is based on taking the truncated Chebyshev series expansions of the functions in equation, and then substituting their matrix forms into the result equation. Thereby the given equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Chebyshev coefficients.     

A boundary value problem for the KdV equation: Comparison of finite-difference and Chebyshev methods

Solutions of a boundary value problem for the Korteweg–de Vries equation are approximated numerically using a finite-difference method, and a collocation method based on Chebyshev polynomials. The performance of the two methods is compared using exact solutions that are exponentially small at the boundaries. The Chebyshev method is found to be more efficient.     

A modified Chebyshev pseudospectral DD algorithm for the GBH equation

In this paper, a Chebyshev spectral collocation domain decomposition (DD) semi-discretization by using a grid mapping, derived by Kosloff and Tal-Ezer in space is applied to the numerical solution of the generalized Burger’s-Huxley (GBH) equation. To reduce roundoff error in computing derivatives we use the above mentioned grid mapping. In this work, we compose the Chebyshev spectral collocation domain decomposition and Kosloff and Tal-Ezer grid mapping, elaborately. Firstly, the theory of application of the Chebyshev spectral collocation method with grid mapping and DD on the GBH equation is presented. This method yields a system of ordinary differential algebraic equations (DAEs). Secondly, we use a fourth order Runge-Kutta formula for the numerical integration of the system of DAEs. Application of this modified method to the GBH equation show that this method (M-DD) is faster and more accurate than the standard Chebyshev spectral collocation DD (S-DD) method.     

板类结构动力检测与控制中的一种新方法

曲率模态在结构动力检测中具有对动力结构损伤部位非常敏感的特性，传统方法主要是运用中心差分法求解曲率模态，由于中心差分法的计算精度依赖于测点分布的紧密程度，这样就使动力检测结果具有很大的误差，本文利用函数的契贝雪夫多项式的展开式具有很高的逼近特性，提出了板类结构动力检测的曲率模态算法——契贝雪夫多项式算法，构造出了板类结构振型的契贝雪夫多项式函数，对该函数进行求二阶偏导得到x和y方向的曲率模态，进而求出结构损伤前后的曲率模态差，该方法可为结构损伤检测提供可靠的数据，从而达到良好的检测控制效果。     

Chebyshev Approximation by Exponential-Power Expression

The properties of the Chebyshev approximation by exponential-power expressions with four unknown parameters are investigated. The condition for the existence and uniqueness of such approximation with the smallest relative error is established. A method to determine the parameters of the Chebyshev approximation is proposed and justified. The error of the Chebyshev approximation by the exponential–power expression is estimated.     

切比契夫谱元素局部混合基函数构造

针对切比契夫谱方法,该文首次构造了两类局部混合基函数,据此发展了一种新的谱元素方法:在元素端点采用局部拉格朗日插值基,元素内部采用经调整后的切比契夫多项式。这里的两类混合基函数在计算精度上可与传统的拉格朗日基相媲美,而且元素矩阵具有稀疏特征和数据重用性。该文给出的局部混合基函数对传统的谱元素方法进行了扩充。     

Frobenius-Chebyshev polynomial approximations with a priori error bounds for nonlinear initial value differential problems

In this paper, we present a method for approximating the solution of initial value ordinary differential equations with a priori error bounds. The method is based on a Chebyshev perturbation of the original differential equation together with the Frobenius method for solving the equation. Chebyshev polynomials in two variables are developed. Numerical results are presented.     

A practical chebyshev collocation method for differential equations

This paper describes a method for solving ordinary and partial differential equations in Chebyshev series. The main feature of the method, which is based on the collocation principle, (Lanczos [8]) is that it solves the problem of differentiating a Chebyshev series directly by the use of a stable recurrence relation. As a practical consequence the method is very simple and can easily be coded into a general-purpose program for solving some differential equations.     

Integration of Dynamic Equation of Spring-Mass-Damper Systems via Chebyshev Polynomials

A numerical scheme based on Chebyshev polynomials for the determination of the response of spring-mass-damper systems is presented. The state vector of the differential equation of the spring-mass-damper system is expanded in terms of Chebyshev polynomials. This expansion reduces the original differential equations to a set of linear algebraic equations where the unknowns are the coefficient of Chebyshev polynomials. A formal procedure to generate the coefficient matrix and the right-hand side vector of this system of algebraic equations is discussed. The numerical efficiency of the proposed method is compared with that of Runge-Kutta method. It is shown that this scheme is accurate and is computationally efficient.     

Practical Fabrication of Discrete Chebyshev Nets

We propose a computational and practical technique to allow home users to fabricate discrete Chebyshev nets for various 3D models. The success of our method relies on two key components. The first one is a novel and simple method to approximate discrete integrable, unit-length, and angle-bounded frame fields, used to model discrete Chebyshev nets. Central to our field generation process is an alternating algorithm that takes turns executing one pass to enforce integrability and another pass to approach unit length while bounding angles. The second is a practical fabrication specification. The discrete Chebyshev net is first partitioned into a set of patches to facilitate manufacturing. Then, each patch is assigned a specification on pulling, bend, and fold to fit the nets. We demonstrate the capability and feasibility of our method in various complex models.     

Numerical solution of fourth-order integro-differential equations using Chebyshev cardinal functions

A numerical technique is presented for the solution of fourth-order integro-differential equations. This method uses the Chebyshev cardinal functions. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. Using the operational matrix of derivative, we reduce the problem to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results.     

A Chebyshev Finite Difference Method For Solving A Class Of Optimal Control Problems

A new Chebyshev finite difference method for solving class of optimal control problem is proposed. The algorithm is based on Chebyshev approximations of the derivatives arising in system dynamics. In the performance index, we use Chebyshev approximations for integration. The numerical examples illustrate the robustness, accuracy and efficiency of the proposed technique.     

On a functional approximation for inversion of Laplace transforms via shifted Chebyshev series

A functional representation for inversion of the Laplace transform of a function is considered. The function is given as a shifted Chebyshev series expansion. Using special operational properties, each Laplace transform is converted into a set of simultaneous linear algebraic equations that are then easily solved to give the coefficients of the Chebyshev series. The method is simple and very suitable for computer programming. Applications to rational and irrational Laplace transforms are presented to demonstrate the satisfactory results that the method provides.     

Using Chebyshev Polynomials to Approximate Partial Differential Equations: A Reply

Caporale and Cerrato (Comput Econ 35(3):235–244, 2010) propose a simple method based on Chebyshev approximation and Chebyshev nodes to approximate partial differential equations (PDEs). However, they suggest not to use Chebyshev nodes when dealing with optimal stopping problems. Here, we use the same optimal stopping example to show that Chebyshev polynomials and Chebyshev nodes can still be successfully used together if we solve the model in a matrix environment.     

A Chebyshev technique for solving nonlinear optimal controlproblems

A numerical technique for solving nonlinear optimal control problems is introduced. The state and control variables are expanded in the Chebyshev series, and an algorithm is provided for approximating the system dynamics, boundary conditions, and performance index. Application of this method results in the transformation of differential and integral expressions into systems of algebraic or transcendental expressions in the Chebyshev coefficients. The optimum condition is obtained by applying the method of constrained extremum. For linear-quadratic optimal control problems, the state and control variables are determined by solving a set of linear equations in the Chebyshev coefficients. Applicability is illustrated with the minimum-time and maximum-radius orbit transfer problems     

Spectral viscosity for convection dominated flow

A stabilized treatment of convection dominated flow problems with a high order spectral viscosity method is presented. This method stabilizes the spectral scheme and remains the high spectral accuracy by introducing some viscosity only to the highest Fourier or Chebyshev modes. In the practical computation the method is employed to the Chebyshev pseudospectral (or collocation) discretization of some singular perturbation problems.     

基于Chebyshev小波的分布参数系统辨识

工程实际和社会系统中广泛存在着分布参数系统,因而研究分布参数系统的辨识与控制具有重要意义.但由于其复杂性,对分布参数系统的辨识研究十分困难.借助于Chebyshev多项式的逼近性质,以及小波的时频特性,构造了Chebyshev小波,并利用其积分运算矩阵,运用于分布参数系统的辨识,从而将一类分布参数系统的辨识问题转化为一般代数问题.并且考虑了初始条件和边界条件对辨识结果的影响,因此具有较好的适用性,仿真结果证实了该方法的有效性.     

Using Chebyshev Polynomials to Approximate Partial Differential Equations

This paper suggests a simple method based on a Chebyshev approximation at Chebyshev nodes to approximate partial differential equations (PDEs). It consists in determining the value function by using a set of nodes and basis functions. We provide two examples: pricing a European option and determining the best policy for shutting down a machine. The suggested method is flexible, easy to programme and efficient. It is also applicable in other fields, providing efficient solutions to complex systems of PDEs.