The extended Laplace transform method for mathematical analysis of the Thomas-Fermi equation

The recently developed method by the authors, known as the extended Laplace transform method (ELTM), is used to obtain an explicit semi-analytical, semi-numerical solution for the Thomas-Fermi (TF) equation for neutral atoms. The method makes it possible to calculate the Laplace transform of the intrinsic nonlinearity of the TF equation namely N(ϕ)=ϕ3 , as a novel and unique advantage. We demonstrate that the obtained solution is remarkably accurate and it is superior to the previous solutions in the literature, particularly those by the homotopy analysis method, in terms of the computational efficiency.