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An explicit analytic solution to the Thomas-Fermi equation. (English) Zbl 1034.34005

The author considers the boundary value problem for the Thomas-Fermi equation \[ u'' (x) = \sqrt{u^{3} (x) / x}, x > 0, \quad u (0) = L, \quad u (+\infty ) = 0. \tag{1} \] With the homotopy analysis method the explicit analytic solution to problem (1) is given. This solution has the form \[ u (x) = \sum_{k=0}^{\infty } \sum_{n=1}^{4k+1} \alpha _{k, n} (1+x)^{-n},\tag{2} \] where the coefficients can be calculated successive with the help of recurrence formulae. The author gives the results of calculations, when, in place of the precise solution to (2), the corresponding \(m\)th-order approximation \(u(x) = \sum_{k=0}^{m } \sum_{n=1}^{4k+1} \alpha _{k, n} (1+x)^{-n}\) for \(m=40, 60\) is used.

MSC:

34A05 Explicit solutions, first integrals of ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
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