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A constructive solution for a generalized Thomas-Fermi theory of ionized atoms. (English) Zbl 0639.34021

This paper studies the problem: \(y''+(b/x)y'=cx^ py^ q,\) \(y(0)=1\), \(y(a)=0\), where b,c, and q are constants such that \(0\leq b<1\), \(c>0\), \(p>-2\) and \(q>1\). Modified Bessel functions of the first and the second kinds are used to construct Green’s function of a related linear problem, which is used in each successive approximation to the nonlinear problem. A sequence of lower bounds and a sequence of upper bounds are constructed explicitly such that each sequence converges to obtain existence of a nonnegative solution. Uniqueness follows from the maximum principle. The dependence of the solution on the size of the interval [0,a] is also established. To illustrate the computational method, two numerical examples, including the Thomas-Fermi model for the ionized atom, are given.
Reviewer: C.Y.Chan

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65J99 Numerical analysis in abstract spaces
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