scholarly journals The Rational Third-Kind Chebyshev Pseudospectral Method for the Solution of the Thomas-Fermi Equation over Infinite Interval

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Majid Tavassoli Kajani ◽  
Adem Kılıçman ◽  
Mohammad Maleki
Keyword(s):  
Numerical Solutions ◽  
Pseudospectral Method ◽  
Nonlinear Ordinary Differential Equation ◽  
Collocation Methods ◽  
Infinite Interval ◽  
Algebraic Equations ◽  
Chebyshev Pseudospectral Method ◽  
Thomas Fermi ◽  
Present Solution ◽  
Chebyshev Pseudospectral

We propose a pseudospectral method for solving the Thomas-Fermi equation which is a nonlinear ordinary differential equation on semi-infinite interval. This approach is based on the rational third-kind Chebyshev pseudospectral method that is indeed a combination of Tau and collocation methods. This method reduces the solution of this problem to the solution of a system of algebraic equations. Comparison with some numerical solutions shows that the present solution is highly accurate.

scholarly journals A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
E. H. Doha ◽  
D. Baleanu ◽  
A. H. Bhrawy ◽  
R. M. Hafez
Keyword(s):  
Numerical Solutions ◽  
Rational Functions ◽  
Absolute Error ◽  
Delay Systems ◽  
Infinite Interval ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
Linear And Nonlinear ◽  
Pseudospectral Scheme

A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small.

scholarly journals Prediction of Dynamic Stability Using Mapped Chebyshev Pseudospectral Method

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Jae-Young Choi ◽  
Dong Kyun Im ◽  
Jangho Park ◽  
Seongim Choi
Keyword(s):  
Unsteady Flow ◽  
Three Dimensional ◽  
Pseudospectral Method ◽  
Accurate Method ◽  
Spectral Approach ◽  
Fourier Pseudospectral Method ◽  
Chebyshev Pseudospectral Method ◽  
Fourier Pseudospectral ◽  
Chebyshev Pseudospectral ◽  
Good Agreement

A mapped Chebyshev pseudospectral method is extended to solve three-dimensional unsteady flow problems. As the classical Chebyshev spectral approach can lead to numerical instabilities due to ill conditioning of the spectral matrix, the Chebyshev points are evenly redistributed over the domain by an inverse sine mapping function. The mapped Chebyshev pseudospectral method can be used as an alternative time-spectral approach that uses a Chebyshev collocation operator to approximate the time derivative terms in the unsteady flow governing equations, and the method can make general applications to both nonperiodic and periodic problems. In this study, the mapped Chebyshev pseudospectral method is employed to solve three-dimensional periodic problem to verify the spectral accuracy and computational efficiency with those of the Fourier pseudospectral method and the time-accurate method. The results show a good agreement with both of the Fourier pseudospectral method and the time-accurate method. The flow solutions also demonstrate a good agreement with the experimental data. Similar to the Fourier pseudospectral method, the mapped Chebyshev pseudospectral method approximates the unsteady flow solutions with a precise accuracy at a considerably effective computational cost compared to the conventional time-accurate method.

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