Approximate periodic solutions for the non-linear relativistic harmonic oscillator via differential transformation method

The relativistic harmonic oscillator equation is a nonlinear ordinary differential equation given by: x ¨ + ( 1 - x ˙ 2 ) 3 / 2 x = 0 . In this paper, the differential transformation method (DTM) and a relatively new technique, known as aftertreatment technique, are proposed to obtain new approximate periodic solutions for the relativistic harmonic oscillator equation under the initial conditions x ( 0 ) = 0 , x ˙ ( 0 ) = β .