Rational tau free download

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Lisboa Tamil Extra Bold Vanarchiv. Lisboa Tamil Book Vanarchiv. Lisboa Tamil Bold Vanarchiv. Lisboa Sans Tamil Light Vanarchiv. Lisboa Sans Tamil Book Vanarchiv. Lisboa Sans Tamil Normal Vanarchiv. Lisboa Sans Tamil Regular Vanarchiv. Lisboa Sans Tamil Medium Vanarchiv. Different spectral methods have been proposed for solving problems in unbounded domains. The most common method is through the use of polynomials that are orthogonal over unbounded domains, such as the Hermite spectral method and the Laguerre spectral method [1—9].

Guo [10—12] proposed a method that proceeds by mapping the original problem in an unbounded domain to a problem in a bounded domain, and then using suitable Jacobi polynomials to approximate the resulting problems. This method is named domain truncation [13]. Parand , alireza. Rezaei , amirtaghavims2 yahoo. Taghavi 1 Member of research group of Scientific Computing.

Christov [14] and Boyd [15, 16] developed some spectral methods on unbounded intervals by using mutually orthogonal systems of rational functions.

Boyd [16] defined a new spectral basis, named rational Chebyshev functions on the semi-infinite interval, by mapping to the Chebyshev polynomials. Guo et al. They applied a spectral scheme using the rational Legendre functions for solving the Korteweg-de Vries equation on the half line.

Boyd et al. Parand et al. Their approach was based on rational Tau and colloca- tion method. Among these, an approach consists of using the collocation method or the pseudospectral method based on the nodes of Gauss formulas related to unbounded intervals [9].

Collocation method has become increasingly popular for solving differential equations also they are very useful in providing highly accurate solutions to differential equations. We aim to compare rational Chebyshev collocation RCC approach and Hermite func- tions collocation HF collocation approach to solve a population growth of a species within a closed system. This paper is arranged as follows: in subsection 1. In sections 2 and 3, we describe the properties of rational Chebyshev and Hermite functions.

This equation is first converted to an equivalent nonlinear ordinary differential equation and then our methods are applied to solve this new equation, and then a comparison is made with existing methods that were reported in the literature. The numerical results and advantages of the methods are discussed in the final section.

The coefficient c indicates the essential behavior of the population evolution before its level falls to zero in the long term. The only equilibrium solution of Eq. The solution of Eq. Although a closed form solution has been achieved in [26, 27], it was formally shown that the closed form solution cannot lead to any insight into the behavior of the population evolution [26].

In [26], the successive approximations method was suggested for the solution of Eq. The author scaled out the parameters of Eq. In [28], several numerical algorithms namely Euler method, modified Euler method, clas- sical fourth-order Runge-Kutta method and Runge-Kutta-Fehlberg method for the solution of Eq. Moreover, a phase-plane analysis is implemented. In [28], the nu- merical results are correlated to give insight on the problem and its solution without using perturbation techniques.

However, the performance of the traditional numerical techniques is well known in that it provides grid points only, and in addition, it requires large amounts of calculations. In [29] Adomian decomposition method and Sinc-Galerkin method were compared for the solution of some mathematical population growth models. This approach is based on a rational Tau method. They obtained the operational matrices of derivative and the product of rational Chebyshev and Legendre functions and then applied these matrices together with the Tau method to reduce the solution of this problem to the solution of a system of algebraic equations.

They first converted the model to a nonlinear ordinary differential equation and then the new SDMM applied to solve this equation. In [32] the approach is based upon composite spectral functions approximations. These hybrid functions consist of block-pulse and Lagrange-interpolating polynomials. Momani et al. In total, in recent years, numerous works have been focusing on the development of more advanced and efficient methods for initial value problems especially for stiff systems.

Rational Chebyshev Functions This section is devoted to introducing rational Chebyshev functions which we denote RC and expressing some basic properties of them that will be used to construct the RC collocation RCC method. The constant parameter L sets the length scale of the mapping.

We have the following theorem for the convergence: r Theorem 1. A complete proof is given by Guo et al. This theorem shows that the rational Chebyshev approximation has exponential conver- gence. First we note that the Hermite polynomials are generally not suitable in practice due to their wild asymptotic behavior at infinities [38].

We now introduce the Gauss quadrature associated with the Hermite functions approach. Thus, r! Also same theorem has been proved by Shen et al. Also we know approximations can be constructed for infinite, semi-infinite and finite inter- vals. Therefore, the approximate solution of y t , in Eq. It has already been mentioned in [42] that when using a spectral approach on the whole real line R one can possibly increase the accuracy of the computation by a suitable scaling of the underlying time variable t.