  Far as we know there are not exact solutions to the equation of motion for a
relativistic harmonic oscillator. In this paper, the relativistic harmonic
oscillator equation which is a nonlinear ordinary differential equation is
studied by means of a combined use of the Adomian Decomposition Method and the
Laplace Transform (LADM). The results that we have obtained, a series of powers
of functions, have never been reported and show a very good match when compared
with other approximate solutions, obtained by different methods. The method
here proposed works with high degree of accuracy and because it requires less
computational effort, it is very convenient to solve this kind of nonlinear
differential equations.
