Abstract:
In this study, we show the superiority of Chebyshev polynomials to Bessel polynomials in solving first order ordinary differential equation with rational coefficients. Shifted Chebyshev polynomials, Bessel polynomials as well as Canonical polynomial were generated in solving the differential equation of our choice in order to show the superiority of Chebyshev polynomials to Bessel polynomial. Numerical examples are given which show the superiority of Chebyshev polynomials to Bessel polynomials.

Abstract:
In this paper, we proved the superiority of Legendre polynomial to Chebyshev polynomial in solving first order ordinary differential equation with rational coefficient. We generated shifted polynomial of Chebyshev, Legendre and Canonical polynomials which deal with solving differential equation by first choosing Chebyshev polynomial T*n (X), defined with the help of hypergeometric series T*n (x) =F ( -n, n, ;X) and later choosing Legendre polynomial P*n (x) define by the series P*n (x) = F ( -n, n+1, 1;X); with the help of an auxiliary set of Canonical polynomials Qk in order to find the superiority between the two polynomials. Numerical examples are given which show the superiority of Legendre polynomials to Chebyshev polynomials. Journal of Applied Sciences and Environmental Management Vol. 9(3) 2005: 121-124

Abstract:
in this paper, an efficient method is presented for solving two dimensional fredholm and volterra integral equations of the second kind. chebyshev polynomials are applied to approximate a solution for these integral equations. this method transforms the integral equation to algebraic equations with unknown chebyshev coefficients. the high accuracy of this method is verified through some numerical examples. mathematical subject classification: 65r20, 41a50, 41a55, 65m70.

Abstract:
The Lane-Emden equation has been used to model several phenomenas in theoretical physics, mathematical physics and astrophysics such as the theory of stellar structure. This study is an attempt to utilize the collocation method with the Rational Chebyshev of Second Kind function (RSC) to solve the Lane-Emden equation over the semi-infinit interval [0; +infinity). According to well-known results and comparing with previous methods, it can be said that this method is efficient and applicable.

Abstract:
We characterize the generalized Chebyshev polynomials of the second kind (Chebyshev-II), and then we provide a closed form of the generalized Chebyshev-II polynomials using the Bernstein basis. These polynomials can be used to describe the approximation of continuous functions by Chebyshev interpolation and Chebyshev series and how to efficiently compute such approximations. We conclude the paper with some results concerning integrals of the generalized Chebyshev-II and Bernstein polynomials.

Abstract:
Chebyshev polynomials are utilized to obtain solutions of a set of p th order linear differential equations with periodic coefficients. For this purpose, the operational matrix of differentiation associated with the shifted Chebyshev polynomials of the first kind is derived. Utilizing the properties of this matrix, the solution of a system of differential equations can be found by solving a set of linear algebraic equations without constructing the equivalent integral equations. The Floquet Transition Matrix (FTM) can then be computed and its eigenvalues (Floquet multipliers) subsequently analyzed for stability. Two straightforward methods, the ‘differential state space formulation’ and the ‘differential direct formulation’, are presented and the results are compared with those obtained from other available techniques. The well-known Mathieu equation and a higher order system are used as illustrative examples.

Abstract:
We extend a collocation method for solving a nonlinear ordinary differential equation (ODE) via Jacobi polynomials. To date, researchers usually use Chebyshev or Legendre collocation method for solving problems in chemistry, physics, and so forth, see the works of (Doha and Bhrawy 2006, Guo 2000, and Guo et al. 2002). Choosing the optimal polynomial for solving every ODEs problem depends on many factors, for example, smoothing continuously and other properties of the solutions. In this paper, we show intuitionally that in some problems choosing other members of Jacobi polynomials gives better result compared to Chebyshev or Legendre polynomials. 1. Introduction The Jacobi polynomials with respect to parameters , (see, e.g., [1, 2]) are sequences of polynomials satisfying the following relation where These polynomials are eigenfunctions of the following singular Sturm-Liouville equation: A consequence of this is that spectral accuracy can be achieved for expansions in Jacobi polynomials so that The Jacobi polynomials can be obtained from Rodrigue's formula as Furthermore, we have that The Jacobi polynomials are normalized such that An important consequence of the symmetry of weight function and the orthogonality of Jacobi polynomial is the symmetric relation that is, the Jacobi polynomials are even or odd depending on the order of the polynomial. In this form the polynomials may be generated using the starting form such that which is obtained from Rodrigue's formula as follows: Following the two seminal papers of Doha [3, 4] let be an infinitely differentiable function defined on [？1, 1]; then we can write and, for the th derivative of , Then, where For the proof of the above, see [3]. The formula for the expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of those of the function is available for expansions in ultraspherical and Jacobi polynomials in Doha [5]. Another interesting formula is with where For the proof see [4]. Chebyshev, Legendre, and ultraspherical polynomials are particular cases of the Jacobi polynomials. These polynomials have been used both in the solution of boundary value problems [6] and in computational fluid dynamics [7, 8]. For the ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function, see [5]. Collocation method is a kind of spectral method that uses the delta function as a test function. Test functions have an important role because these functions are applied to obtain the minimum value for residual by using inner

Abstract:
本文研究了一类高阶多点边值问题的数值解法问题.利用第二类Chebyhsev小波及其积分算子矩阵，将线性与非线性高阶常微分方程多点边值问题转化为代数方程组进行求解.通过与现有文献算法结果的比较，说明了该算法求解高阶多点边值问题的准确性与有效性.扩展了高阶多点边值问题的数值求解方法. In this paper, a numerical algorithm is concerned for solving approximate solutions of high-order multi-point boundary value problems. The second kind Chebyhsev wavelets and operational matrix of integration are used to convert multi-point linear and nonlinear ordinary differential equation to a system of algebraic equations. By comparing with the results of the existing literature, the accuracy and validity of the algorithm for solving the high-order multi-point boundary value problem are explained. The proposed method extends the numerical solution of higher-order multi-point boundary value problems

Abstract:
The onset of convection in a horizontal layer of fluid heated from below in the presence of a gravity field varying across the layer is investigated. The eigenvalue problem governing the linear stability of the mechanical equilibria of the fluid layer in the case of free boundaries is solved using a Galerkin method based on shifted polynomials (Legendre and Chebyshev polynomials).

Abstract:
Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.