We deem the time-fractional Benjamin-Ono (BO) equation out of the Riemann–Liouville (RL) derivative by applying the Lie symmetry analysis (LSA). By first using prolongation theorem to investigate its similarity vectors and then using these generators to transform the time-fractional BO equation to a nonlinear ordinary differential equation (NLODE) of fractional order. We complete the solutions by utilizing the power series method (PSM). Lie symmetry method provides an effective tool for deriving the analytic solutions of the nonlinear partial differential equations (NLPDEs) [1-4]. In recent years, many authors have studied the nonlinear fractional differential equations (NLFDEs) because these equations express many nonlinear physical phenomena and dynamic forms in physics, electrochemistry, and viscoelasticity [5-9].
Time-fractional NLDEs arise from classical NLPDEs by replacing its time derivative with the fractional derivative. The methods applied to derive the analytic solutions of NLFPDEs are the exp-function, the G'⁄G expansion, fractional sub-equation, Lie symmetry method, and many more [10-19].
The one-dimensional Benjamin-Ono equation is considered here as follow (see [20]):
u_t+hu_xx+uu_x=0 (1)
In fact, the BO equation describes one-dimensional internal waves in deep water. We consider LSA for the analytic solutions by using PS expansion for the time-fractional BO equation:
〖u^α〗_t+hu_xx+uu_x=0 , 0