The Thomas Fermi model of atoms and nuclei is used in many applications of atomic and nuclear physics. The ODE related to this model is: $$\frac{d^2}{dx^2}\phi(x)=x^{-\frac{1}{2}}\phi(x)^{3/2}$$ with boundary conditions: $$\phi(0)=1$$ and: $$\phi(\infty)=0$$ The best attempt to solve it is given by Majorana: http://arxiv.org/pdf/physics/0111167v1.pdf but unfortunatly also Majorana wasn't able to solve it analytically. Is there today some advanced method to find an analytical solution of this equation or the only possible way to solve it is a numerical method? Thanks.
2026-03-29 02:30:27.1774751427
Analytical solutions of Thomas Fermi equation
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