Solution of the Thomas–Fermi equation with a convergent approach

The explicit analytic solution of the Thomas–Fermi equation thorough a new kind of analytic technique, namely the homotopy analysis method, was employed by Liao [1]. However, the base functions and the auxiliary linear differential operator chosen were such that the convergence to the exact solution was fairly slow. New base functions and auxiliary linear operator to form a better homotopy are the main concern of the present paper. Optimum convergence control parameter concept is used together with a mathematical proof of the convergence.

► A new approach based on the homotopy analysis method is adopted for the Thomas–Fermi equation.
► Fast convergence and better accuracy is obtained by selecting more appropriate auxiliary parameters.
► The convergence of the resulting homotopy series is proven mathematically.
► Comparisons with the published ones of Liao [1] shows the accuracy of the method.