Monotone methods for the Thomas-Fermi equation

Abstract

Abstract.The boundary-value problem for the ionized atom case of the ThomasFermi equation is transformed to a certain convex nonlinear boundary-value problem. Two iterative procedures, previously developed for such problems, are constructed for the ionized atom problem. A comparative analysis of the efficiency of the iteration schemes is presented. The existence and uniqueness of a solution is established and the solution is shown to have monotonic dependence on the boundary conditions. Numerical bounds are obtained for a specific problem.

Constructive existence theorems for problems of Thomas‐Fermi Type

“…Iterative procedures guaranteed to approximate a solution of the Thomas [lo]. [13]). The approach used here has the advantage of yielding existence, ordering and boundedness results for the minimal solution and of giving iteration schemes which provide upper and lower solution bounds for any b > 0 in (2.10).…”

A constructive solution for a generalized Thomas-Fermi theory of ionized atoms

“…Approximate solutions of the problem (1.1) and (1.3) were given, for example, by Bush and Caldwell [3], Sommerfeld [12], Ramnath [10], and more recently by Anderson and Arthurs [1], and Burrows and Core [2] using the variational approach. The case of the ionized atom given by (1.1) subject to (1.4) was studied by Mooney [8] using monotone methods with both (modified) Picard and Newton algorithms, and recently by Chan and Du [4], In this paper, we generalize (1.1) to the form y" +(b/x)y' = cx'y", (1.5) where b, c, p, and q are constants such that 0 ^ b < 1, c > 0, p > -2, and q > 1. We study it under the boundary condition (1.4).…”

A constructive method for the Thomas-Fermi equation

“…The problem (1.1) and (1.2) was studied by Mooney [2] using monotone methods with both the Picard and Newton algorithms. Mooney showed that a solution y of the problem (1.1) and (1.2) satisfies the inequalities 0 < y < 1 -x/a for 0 < x < a, and by letting u = 1 -x/a -y, it follows that 0 < u < 1 -x/a for 0 < x < a, and the problem (1.1) and (1.2) becomes…”