An explicit analytic solution to the Thomas-Fermi equation

A new kind of analytic technique, namely the homotopy analysis method, is employed to give an explicit analytic solution of the Thomas-Fermi equation and the related recurrence formulae of constant coefficients. This solution can be regarded as the definition of the exact solution of the Thomas-Fermi equation.

[1]  Shigehiro Kobayashi.  Some Coefficients of the Series Expansion of the TFD Function , 1955 .

[2]  S. Liao,et al.  A Simple Approach of Enlarging Convergence Regions of Perturbation Approximations , 1999 .

[3]  Bernard J. Laurenzi.  An analytic solution to the Thomas–Fermi equation , 1990 .

[4]  S. Liao.  A kind of approximate solution technique which does not depend upon small parameters — II. An application in fluid mechanics , 1997 .

[5]  A. Cedillo.  A perturbative approach to the Thomas–Fermi equation in terms of the density , 1993 .

[6]  S. N. Venkatarangan,et al.  Modification of Adomian's decomposition method to solve equations containing radicals , 1995 .

[7]  Abdul-Majid Wazwaz,et al.  The modified decomposition method and Padé approximants for solving the Thomas-Fermi equation , 1999, Appl. Math. Comput..

[8]  A. MacLeod.  Chebyshev series solution of the Thomas-Fermi equation , 1992 .

[9]  C. Bender,et al.  A new perturbative approach to nonlinear problems , 1989 .

[10]  L. H. Thomas.  2 – The Calculation of Atomic Fields , 1927 .

[11]  S. Liao,et al.  Application of Homotopy Analysis Method in Nonlinear Oscillations , 1998 .

[12]  Y. Hon,et al.  A constructive solution for a generalized Thomas-Fermi theory of ionized atoms , 1987 .

[13]  Vimal Singh,et al.  Perturbation methods , 1991 .

[14]  B. Burrows,et al.  A variational-iterative approximate solution of the Thomas-Fermi equation , 1984 .

[15]  H. Weitzner,et al.  Perturbation Methods in Applied Mathematics , 1969 .

[16]  Francis P. Keenan,et al.  Electron impact excitation of Ni-like gadolinium , 1999 .

[17]  G. A. Baker.  Essentials of Padé approximants , 1975 .

[18]  S. Liao.  An analytic approximation of the drag coefficient for the viscous flow past a sphere , 2002 .

[19]  George Adomian,et al.  Solving Frontier Problems of Physics: The Decomposition Method , 1993 .

[20]  S. Liao.  AN EXPLICIT TOTALLY ANALYTIC APPROXIMATION OF BLASIUS VISCOUS FLOW PROBLEMS , 1999 .

[21]  Faruk Civan,et al.  On the solution of the Thomas-Fermi equation by differential quadrature , 1984 .

[22]  J. Boyd.  Pade´ approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain , 1997 .

[23]  Mu-Shiang Wu,et al.  Modified variational solution of the Thomas-Fermi equation for atoms , 1982 .

[24]  Shijun Liao,et al.  Analytic solutions of the temperature distribution in Blasius viscous flow problems , 2002, Journal of Fluid Mechanics.

[25]  Shigehiro Kobayashi,et al.  Accurate Value of the Initial Slope of the Ordinary TF Function , 1955 .

[26]  S. Liao.  An explicit, totally analytic approximate solution for Blasius’ viscous flow problems , 1999 .

[27]  C. Y. Chan,et al.  A constructive method for the Thomas-Fermi equation , 1986 .

[28]  G. J. Pert,et al.  Approximations for the rapid evaluation of the Thomas-Fermi equation , 1999 .

[29]  S. Liao.  An approximate solution technique not depending on small parameters: A special example , 1995 .

[30]  S. H. Caldwell,et al.  Thomas-Fermi Equation Solution by the Differential Analyzer , 1931 .

[31]  L. H. Thomas.  The calculation of atomic fields , 1927, Mathematical Proceedings of the Cambridge Philosophical Society.

[32]  Shijun Liao,et al.  AN APPROXIMATE SOLUTION TECHNIQUE WHICH DOES NOT DEPEND UPON SMALL PARAMETERS: A SPECIAL EXAMPLE , 1995 .

[33]  C. D. Luning,et al.  An iterative technique for solution of the Thomas-Fermi equation utilizing a nonlinear eigenvalue problem , 1977 .

[34]  S. Liao.  A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate , 1999, Journal of Fluid Mechanics.