Parametrization of solutions to the Emden–Fowler equation and the Thomas–Fermi model of compressed atoms

For the nonlinear Emden–Fowler equation, a singular Cauchy problem and singular two-point boundary value problem on the half-line $r\in [0,+\infty)$ and on an interval $r\in[0,R]$ with a Dirichlet boundary condition at the origin and with a Robin boundary condition at the right endpoint of the interval are considered. For special parameter values, the given boundary value problem corresponds to the Thomas–Fermi model of the charge density distribution inside a spherically symmetric cooled heavy atom occupying a confined or infinite space, where $R$ denotes the boundary of a compressed atom and grows to infinity for a free atom. For the boundary value problem on the half-line, a new parametric representation of the solution is obtained that covers the entire range of argument values, i.e., the half-line $r\in [0,+\infty)$, with the parameter $t$ running over the unit interval. For analytic functions involved in this representation, an algorithm for explicit computation of their Taylor coefficients at $t=0$ is described. As applied to the Thomas–Fermi problem for a free atom, corresponding Taylor series expansions are given and they are shown to converge exponentially on the unit interval $t\in [0,1]$ at a rate higher than that for an earlier constructed similar representation. An efficient analytical-numerical method is presented that computes the solution of the Thomas–Fermi problem on the half-line with any prescribed accuracy not only in an neighborhood of $r=+\infty$, but also at any point of the half-line $r\in[0,+\infty)$. For the Cauchy problem set up at the origin, a new formula for the critical value of the derivative that corresponds to the solution of the problem on the half-line is derived. It is shown in a numerical experiment that this formula is more efficient than the Majorana formula. For the solution of the Cauchy problem with a positive derivative at the origin, a parametrization is obtained that ensures that the boundary conditions of the singular boundary value problem on the interval $r\in[0,R]$ are satisfied with a suitable $R>0$. An efficient analytical-numerical method for solving this Cauchy problem is constructed and numerically implemented.