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 twopoint boundary value problem on the half-line r is an element of [0, +infinity) and on an interval r is an element of [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 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 is an element of [0, +infinity), with the parameter 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 is an element of [0, 1] at a rate higher than that for an earlier constructed similar representation. An efficient analyticalnumerical method is presented that computes the solution of the Thomas-Fermi problem on the halfline with any prescribed accuracy not only in an neighborhood of r = +infinity, but also at any point of the half-line r is an element of [0, +infinity. 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 is an element of [0, R] are satisfied with a suitable R > 0. An efficient analytical-numerical method for solving this Cauchy problem is constructed and numerically implemented.