An explicit analytic solution to the Thomas-Fermi equation.
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.
nonlinear ordinary differential equation
scientific article; zbMATH DE number 1970655
scientific article