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.     

Chebyshev polynomial and heat conduction equation in cylindrical geometry

The solutions of the unsteady heat conduction equations in cylindrical geometry in one and two dimensions are obtained using the Chebyshev polynomial expansions in the spatial domain. Equations are discretized in the time domain using the trapezoidal rule. The resulting differential equations are reduced to backward recurrence relations for the coefficients occurring in the Chebyshev polynomial expansions, which are then solved using the Tau method. It is shown that the Chebyshev polynomial solutions produce results to the machine-precision accuracy in the spatial domain using only a modest number of terms, and are, therefore, excellent alternatives to the other techniques used.     

A note on the extension of t chebyshev method to quasi linear parabolic p.d.e.s with mixed boundary conditions

Dew [1] proposed a method for computing the numerical solution to quasi-linear parabolic p.d.e.s using a Chebyshev method. The purpose of this note is to extend the method to problems with mixed boundary conditions. An error analysis for the linear problem is given and a global element Chebyshev method is described.     

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.     

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。     

Solution of initial value problems by the differential quadrature method with Hermite bases

The differential quadrature method (DQM) is used to solve the first-order initial value problem. The initial condition is given at the beginning of the interval. The derivative of a space-independent variable at a sampling grid point within the interval can be defined as a weighted linear sum of the given initial conditions and the function values at the sampling grid points within the defined interval. Hermite polynomials have advantages compared with Lagrange and Chebyshev polynomials, and so, unlike other work, they are chosen as weight functions in the DQM. The proposed method is applied to a numerical example and it is shown that the accuracy of the quadrature solution obtained using the proposed sampling grid points is better than solutions obtained with the commonly used Chebyshev–Gauss–Lobatto sampling grid points.     

PARALLEL PSEUDOSPECTRAL SOLUTION OF FINANCIAL PARTIAL DIFFERENTIAL EQUATIONS

Abstract

We apply the Pseudospectral method to two fundamental financial equations: the Black-Scholes equation and the Cox Ingersoil Ross model of the term structure of interest rates. The former is used to price a European Call Option and the latter to price a zero coupon bond. Chebyshev polynomials are used as the basis functions and Chebyshev collocation points for the space discretisation. The Crank-Nicolson scheme is used for the time differencing. We have developed a C++ program to solve general second order linear parabolic equations, A parallel quasi-minimal residual version of the Bi-Conjugate Gradient stabilised algorithm is applied to solve the linear system on the AP3000, a parallel computer. The regular space domain and the smooth solutions often encountered in finance suggest the suitability of using this higher order technique.     

Chebyshev polynomial acceleration for block SOR methods for solving the rank-deficient least-squares problem

In this article, we give the acceleration of the block successive overrelaxation (SOR) method for solving the rank-deficient least-squares problem. Santos and Silva proposed the two-block SOR method and the three-block SOR method. Here, we consider the acceleration of the two-block SOR method and the three-block SOR method using the Chebyshev polynomial and derive what we term the C-2-block SOR method and the C-3-block SOR method. The advantage of our methods is that we can get good results with very small iteration number. The comparison between the C-2-block method and the C-3-block method is presented. Finally, numerical examples are given.     

A new fractional Chebyshev FDM: an application for solving the fractional differential equations generated by optimisation problem

In this paper, we introduce a new numerical technique which we call fractional Chebyshev finite difference method. The algorithm is based on a combination of the useful properties of Chebyshev polynomial approximation and finite difference method. We implement this technique to solve numerically the non-linear programming problem which are governed by fractional differential equations (FDEs). The proposed technique is based on using matrix operator expressions which applies to the differential terms. The operational matrix method is derived in our approach in order to approximate the Caputo fractional derivatives. This operational matrix method can be regarded as a non-uniform finite difference scheme. The error bound for the fractional derivatives is introduced. The application of the method to the generated FDEs leads to algebraic systems which can be solved by an appropriate method. Two numerical examples are provided to confirm the accuracy and the effectiveness of the proposed method. A comparison with the fourth-order Runge–Kutta method is given.     

A new second-order accurate time discretization method for the nonlinear cubic Schrödinger equation

A new second-order accurate semi-analytical time discretization method is introduced for the numerical solution of the one-dimensional nonlinear cubic Schrödinger equation. This method is based on the combination of the method of lines, Crank–Nicolson method, Newton method and Lanczos’ Tau method. It is a self-starting averaged two-time-level scheme that has proved to be stable, accurate and energy conservative for long time integration periods. At each time level, approximate solutions are sought on a segmented spatial interval as finite expansions in terms of a given orthogonal polynomial basis mapped appropriately onto each spatial subsegment. We have carried out numerical simulation concerning several cases for the propagation, collision and the bound states of solitons. Accurate results have been obtained using Chebyshev and Legendre polynomials. These results are well comparable with other published results obtained by the use of various standard numerical methods.     

An incomplete factorization iterative technique for elliptic equations

This paper describes efficient iterative techniques for solving the large sparse symmetric linear systems that arise from application of finite difference approximations to self-adjoint elliptic equations. We use an incomplete factorization technique with the method of D'Yakonov type, generalized conjugate gradient and Chebyshev semi-iterative methods. We compare these methods with numerical examples. Bounds for the 4-norm of the error vector of the Chebyshev semi-iterative method in terms of the spectral radius of the iteration matrix are derived.     

A Chebyshev pseudospectral multidomain method for the soliton solution of coupled nonlinear Schrödinger equations

The collision of solitary waves is an important problem in both physics and applied mathematics. In this paper, we study the solution of coupled nonlinear Schrödinger equations based on pseudospectral collocation method with domain decomposition algorithm for approximating the spatial variable. The problem is converted to a system of nonlinear ordinary differential equations which will be integrated in time by explicit Runge–Kutta method of order four. The multidomain scheme has much better stability properties than the single domain. Thus this permits using much larger step size for the time integration which fulfills stability restrictions. The proposed scheme reduces the effects of round-of-error for the Chebyshev collocation and also uses less memory without sacrificing the accuracy. The numerical experiments are presented which show the multidomain pseudospectral method has excellent long-time numerical behavior and preserves energy conservation property.     

Superconvergence of a Chebyshev Spectral Collocation Method

We reveal the relationship between a Petrov–Galerkin method and a spectral collocation method at the Chebyshev points of the second kind (±1 and zeros of U _k ) for the two-point boundary value problem. Derivative superconvergence points are identified as the Chebyshev points of the first kind (Zeros of T _k ). Super-geometric convergent rate is established for a special class of solutions. This work was supported in part by the US National Science Foundation grants DMS-0311807 and DMS-0612908.     

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.     

切比雪夫正交基神经网络的权值直接确定法

经典的BP神经网络学习算法是基于误差回传的思想.而对于特定的网络模型,采用伪逆思想可以直接确定权值进而避免以往的反复迭代修正的过程.根据多项式插值和逼近理论构造一个切比雪夫正交基神经网络,其模型采用三层结构并以一组切比雪夫正交多项式函数作为隐层神经元的激励函数.依据误差回传(BP)思想可以推导出该网络模型的权值修正迭代公式,利用该公式迭代训练可得到网络的最优权值.区别于这种经典的做法,针对切比雪夫正交基神经网络模型,提出了一种基于伪逆的权值直接确定法,从而避免了传统方法通过反复迭代才能得到网络权值的冗长训练过程.仿真结果表明该方法具有更快的计算速度和至少相同的工作精度,从而验证了其优越性.     

Adomian decomposition method by Gegenbauer and Jacobi polynomials

In this paper, orthogonal polynomials on [–1,1] interval are used to modify the Adomian decomposition method (ADM). Gegenbauer and Jacobi polynomials are employed to improve the ADM and compared with the method of using Chebyshev and Legendre polynomials. To show the efficiency of the developed method, some linear and nonlinear examples are solved by the proposed method, results are compared with other modifications of the ADM and the exact solutions of the problems.     

切比契夫序列多点估值的两个新方法

自1.引言众所周知,切比契夫多项式在工程实践中使用非常普遍,广泛应用于计算流体力学、计算空气动力学和计算电磁学等科学和工程计算中.这些问题通常可以归结为偏微分方程数值解,而切比契夫序列估     

Construction of optimal feedback control for nonlinear systems via Chebyshev polynomials

A method is proposed to determine the optimal feedback control law of a class of nonlinear optimal control problems. The method is based on two steps. The first step is to determine the open-hop optimal control and trajectories, by using the quasilinearization and the state variables parametrization via Chebyshev polynomials of the first type. Therefore the nonlinear optimal control problem is replaced by a sequence of small quadratic programming problems which can easily be solved. The second step is to use the results of the last quasilinearization iteration, when an acceptable convergence error is achieved, to obtain the optimal feedback control law. To this end, the matrix Riccati equation and another n linear differential equations are solved using the Chebyshev polynomials of the first type. Moreover, the differentiation operational matrix of Chebyshev polynomials is introduced. To show the effectiveness of the proposed method, the simulation results of a nonlinear optimal control problem are shown.     

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.     

A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order

We are concerned with linear and nonlinear multi-term fractional differential equations (FDEs). The shifted Chebyshev operational matrix (COM) of fractional derivatives is derived and used together with spectral methods for solving FDEs. Our approach was based on the shifted Chebyshev tau and collocation methods. The proposed algorithms are applied to solve two types of FDEs, linear and nonlinear, subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Numerical results with comparisons are given to confirm the reliability of the proposed method for some FDEs.