Self-consistent approach to solving the 1D Thomas-Fermi equation using an exponential basis set

In this paper we focus on calculating an approximate solution to the one dimensional Thomas-Fermi equation in the form of an expansion using exponential basis functions. We use a self-consistent approach for finding the expansion coefficients. In practice this results in an iterative algorithm. In this way, the problem of solving a system of nonlinear equations, which is common for other similar methods for finding approximate solutions for the equation of interest, is avoided. The evaluation of this approach has been performed in two directions. First, to see the effect of using the exponential basis set, we compare the quality of found approximate solutions using the proposed algorithm with an analog self-consistent approach based on finite elements. A comparison is also conducted with the use of Padé approximation for solving the one dimensional Thomas-Fermi equation.