Fourth order Runge-Kutta method is commonly used to solve the differential equations in engineering due to its high precision and easy convergence. In this paper, based on the Feynman et al.’s calculated method, the classical fourth order Runge-Kutta method was applied to solve the Thomas-Fermi-Dirac (TFD) equation and more accurate results have been obtained. Using this method, the numerical solution of the TFD equation for the element Cu was solved, and the electron density values of some common elements at the Wigner-Seitz radius were calculated.