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.