scholarly journals Rational Chebyshev series for the Thomas–Fermi function: Endpoint singularities and spectral methods

2013 ◽  
Vol 244 ◽  
pp. 90-101 ◽  
Author(s):  
John P. Boyd
Keyword(s):  
Spectral Methods ◽  
Chebyshev Series ◽  
Thomas Fermi ◽  
Fermi Function ◽  
Rational Chebyshev

scholarly journals Spectral Method for Solving the Nonlinear Thomas-Fermi Equation Based on Exponential Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Raka Jovanovic ◽  
Sabre Kais ◽  
Fahhad H. Alharbi
Keyword(s):  
Spectral Method ◽  
Spectral Methods ◽  
High Rate ◽  
Basis Set ◽  
Basis Sets ◽  
Infinite Domain ◽  
New Approach ◽  
Thomas Fermi ◽  
Wide Range ◽  
Nonlinear Algebraic System

We present an efficient spectral methods solver for the Thomas-Fermi equation for neutral atoms in a semi-infinite domain. The ordinary differential equation has been solved by applying a spectral method using an exponential basis set. One of the main advantages of this approach, when compared to other relevant applications of spectral methods, is that the underlying integrals can be solved analytically and numerical integration can be avoided. The nonlinear algebraic system of equations that is derived using this method is solved using a minimization approach. The presented method has shown robustness in the sense that it can find high precision solution for a wide range of parameters that define the basis set. In our test, we show that the new approach can achieve a very high rate of convergence using a small number of bases elements. We also present a comparison of recently published results for this problem using spectral methods based on several different basis sets. The comparison shows that our method is highly competitive and in many aspects outperforms the previous work.

scholarly journals An efficient numerical method for solving nonlinear Thomas-Fermi equation

2018 ◽  
Vol 10 (1) ◽  
pp. 134-151 ◽  
Author(s):  
Kourosh Parand ◽  
Kobra Rabiei ◽  
Mehdi Delkhosh
Keyword(s):  
Collocation Methods ◽  
Arbitrary Parameter ◽  
Algebraic Equations ◽  
Approximation Solution ◽  
Thomas Fermi ◽  
Linear Algebraic Equations ◽  
Rational Chebyshev Functions ◽  
Approximate Amount ◽  
Rational Chebyshev ◽  
Fourier Spectral Methods

Abstract In this paper, the nonlinear Thomas-Fermi equation for neutral atoms by using the fractional order of rational Chebyshev functions of the second kind (FRC2), ${\rm{FU}}_{\rm{n}}^\alpha \left( {{\rm{t}},{\rm{L}}} \right)$ (t, L), on an unbounded domain is solved, where L is an arbitrary parameter. Boyd (Chebyshev and Fourier Spectral Methods, 2ed, 2000) has presented a method for calculating the optimal approximate amount of L and we have used the same method for calculating the amount of L. With the aid of quasilinearization and FRC2 collocation methods, the equation is converted to a sequence of linear algebraic equations. An excellent approximation solution of y(t), y′ (t), and y ′ (0) is obtained.

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