A note on the Thomas-Fermi equation

This work is conceived as a theoretical study.In this paper we consider the nonlinear differential equation of the second order, which is known in applications as the Thomas-Fermi equation.We deal with setting the conditions under which this equation exhibits singularity in 0, ∞ .In the available literature, the authors investigate the singularity of the above equation at point 0, using numerical solution methods.The investigated equation has a wide application in quantum mechanics.The Thomas-Fermi equation represents models of atomic ions with a finite charge cloud and an overall positive charge.The boundary of the charge cloud is defined by the condition y = 0, where y represents the charge density at a given point .We illustrate the existence of singular solutions numerically with two examples.In addition, to its role in quantum mechanics, the equation also has potential applications in biochemistry.For example, if the atomic ions, e.g.sodium and chlorine ions penetrate the bacterium, they can cause the death of the bacterium.