Monotonous enclosures for the Thomas-Fermi equation in the isolated neutral atom case.

*(English)*Zbl 0854.65069This paper is concerned with the calculation of enclosures (i.e. two-sided bounds) to the solution of a boundary value problem (BVP) for the Thomas-Fermi equation on a semi-infinite interval.

The original BVP is defined by: (1) \(x^{(1/2)} u''(x) - (u(x))^{(3/2)} = 0\), \(x \in (0, +\infty)\), \(u(0) = 1\), \(\lim_{x \to +\infty} u(x) = 0\), but it can be written in other equivalent forms that allow to apply comparison principles for two point BVPs [see e.g. M. H. Protter and H. F. Weinberger: Maximum-principles in differential equations (1967; Zbl 0153.13602)]. In this setting, and taking into account the asymptotic behaviour of the solutions of (1), analytical upper and lower bounds are derived in Section 2. Next the authors consider the derivation of monotone enclosures for discretizations of (1). The main tool to derive these bounds is the so-called monotone discretization technique widely used by one of the authors for BVPs in a finite interval, and extended to the problem under consideration paying special attention to the asymptotic behaviour of the solution. A careful derivation of the enclosures for the discrete solution is presented including interesting remarks on the refinement of the (non uniform) grids.

Finally, the results of some numerical experiments are presented for the cases of a finite interval and a smooth extension to confirm the accuracies of the computed enclosures.

The original BVP is defined by: (1) \(x^{(1/2)} u''(x) - (u(x))^{(3/2)} = 0\), \(x \in (0, +\infty)\), \(u(0) = 1\), \(\lim_{x \to +\infty} u(x) = 0\), but it can be written in other equivalent forms that allow to apply comparison principles for two point BVPs [see e.g. M. H. Protter and H. F. Weinberger: Maximum-principles in differential equations (1967; Zbl 0153.13602)]. In this setting, and taking into account the asymptotic behaviour of the solutions of (1), analytical upper and lower bounds are derived in Section 2. Next the authors consider the derivation of monotone enclosures for discretizations of (1). The main tool to derive these bounds is the so-called monotone discretization technique widely used by one of the authors for BVPs in a finite interval, and extended to the problem under consideration paying special attention to the asymptotic behaviour of the solution. A careful derivation of the enclosures for the discrete solution is presented including interesting remarks on the refinement of the (non uniform) grids.

Finally, the results of some numerical experiments are presented for the cases of a finite interval and a smooth extension to confirm the accuracies of the computed enclosures.

Reviewer: M.Calvo (Zaragoza)

##### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

81Q05 | Closed and approximate solutions to the SchrĂ¶dinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

65L70 | Error bounds for numerical methods for ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |