A novel numerical technique to obtain an accurate solution to the Thomas-Fermi equation

A rational approximation to the boundary layer flow of a non-Newtonian fluid

“…Some of the most important equations in physics, chemistry, mechanics mathematical sciences occur in the semi-infinite or infinite intervals. To solve these equations using spectral methods, various strategies have been proposed [43][44][45][46][47][48][49][50].…”

On the Majorana solution to the Thomas-Fermi equation

“…There has been great interest in the accurate calculation of the solution to the nonlinear differential equation that comes from the Thomas-Fermi model for neutral atoms [1][2][3][4][5][6][7][8][9][10][11][12][13]. Several approaches have been applied for this purpose; for example: Padé Hankel method [1][2][3]5], fractional order of rational Euler functions [6], fractional order of rational Bessel functions collocation method [7], fractional order of rational Jacobi functions [8], rational Chebyshev functions [4], fractional order of rational Chebyshev functions of the second kind [10], a hybrid approach based on the collocation and Newton-Kantorovich methods plus fractional order of rational Legendre functions [11], Newton iteration with spectral algorithms based on fractional order of rational Gegenbauer functions [12] and rational Chebyshev series accelerated through coordinate transformations [13].…”

“…We have just mentioned the most accurate results. Other authors have already obtained less accurate ones and more often than not reported many wrong digits as shown in the tables of some of the papers just mentioned [6][7][8][9][10][11].…”

“…The purpose of this paper is the discussion of a semi-analytical solution to the Thomas-Fermi equation discovered by Majorana in 1928 that remained unpublished for a long time as revealed in an enlightening pedagogical article by Esposito [14]. Although the Majorana solution was mentioned in some of the papers just quoted [4][5][6][7][8][8][9][10] none of those authors used it for testing their calculations.…”

A new numerical approach to solve Thomas–Fermi model of an atom using bio-inspired heuristics integrated with sequential quadratic programming

“…The overview, importance and applications of classical numerical approaches for TFE can be seen (Kirzhnits 1957 ; Bush and Caldwell 1931 ). The research community has shown great interest in the accurate and reliable calculation of the solution for the TFE including sinc-collocation method (Parand et al 2013a ), Laguerre pseudospectral approximation (Liu and Zhu 2015 ), Chebyshev pseudospectral method (Kılıçman et al 2014 ), rational Chebyshev pseudospectral approach (Parand and Shahini 2009 ), hermite collocation method (Bayatbabolghani and Parand 2014 ), rational approximation (Fernández 2011 ), Homotopy Analysis Method (HAM) (Yao 2008 ) improved HAM (Zhao et al 2012 ), rational Bessel functions collocation method (Parand et al 2016a ), rational Euler functions based methods (Parand et al 2016b ), methods based on Jacobi rational functions with Gauss quadrature formula (Bhrawy and El-Soubhy 2015 ) and Padé–Hankel method (Amore et al 2014 ), Beside these there are many other studies for solving Thomas–Fermi models, see (Fernández 2008 ; Liao 2003a ; Filobello-Nino et al 2015 ; Dahmani and Anber 2015 ; Feng et al 2015 ) and the references therein.…”