The improved homotopy analysis method for the Thomas-Fermi equation

The homotopy analysis method (HAM) is sharpened to solve the Thomas–Fermi equation. Some techniques are employed, including the use of asymptotic analysis to introduce proper transformation, and the use of optimal initial guess and optimal auxiliary linear operator to accelerate the convergence of homotopy approximations. The optimal convergence control parameters are determined by the minimum of the squared residual error. As a result, the initial slop is provided with more-than-10-digit accuracy, which is far more accurate than the results obtained by other authors using the same method. It demonstrates the flexibility and power of the HAM equipped with these techniques. One of the most important nonlinear ordinary differential equations that occurs in mathematical physics is the Thomas– Fermi (TF) equation [1,2] u 00 ðxÞ ¼ ffiffiffiffiffiffiffiffiffiffiffi u 3 ðxÞ x r ; ð1Þ with the boundary conditions uð0Þ ¼ 1; uðþ1Þ ¼ 0 ð2Þ in common case. The above equation is defined in a semi-infinite interval which has a singularity at x ¼ 0 since u 00 ðxÞ ! 1 as x ! 0. Because of the importance of this problem in physics, it has been solved by different methods during the past century, such as the differential analyzer [2], the d-expansion method [3,4], the variational approach [5], Adomian's decomposition method [6], the Chebyshev pseudospectral method [7], and the Hankel–Padé method [8,9]. All of these solutions give the value of the initial slope u 0 ð0Þ, which plays an important role in determining the energy for a neutral atom. The best-known result is given by Kobayashi [10], who employed the inward numerical integration and gave the initial slope u 0 ð0Þ ¼ À1:5880710. So far, the most accurate value of u 0 ð0Þ is À1:588071022611375313, given by Fernández [9] using Hankel–Padé method. As an analytic tool to solve nonlinear differential equations, the homotopy analysis method (HAM) [11] has been successfully used to investigate a variety of nonlinear problems in science and engineering. The HAM enjoys great freedom in choosing auxiliary linear operator and initial guess. In particular, it provides a convenient way to guarantee the convergence of solution series. In most cases, the solution series given by the HAM converge quickly. However, when the HAM is applied to the TF equation [11–15], which has a singularity at x ¼ 0, the approximations of the initial slope u 0 ð0Þ converge rather slowly, as pointed out by Fernández [8,9]. We analyzed …