Oscillators with a fractional-order restoring force: Higher-order approximations for motion via a modified Ritz method

In this paper conservative single-degree-of-freedom oscillators with a fractional-order restoring force are considered. In order to obtain the solution for motion in higher approximations, the Ritz method is adjusted by introducing an approximate Lagrangian and by using the exact value of the frequency of vibrations. Explicit expressions for the amplitudes of the second and third approximations are derived. The results obtained are compared with numerical results and with the existing approximate result for the first approximation.