Symmetry and Convergence Analysis of the Time-Fractional Benjamin–Bona–Mahony Equation in the Riemann–Liouville Derivative Sense

In this paper, a complete Lie symmetry analysis of the time-fractional Benjamin–Bona–Mahony (BBM) equation in the Riemann–Liouville sense is presented. We give a full symmetry classification and two infinitesimal generators Q1 and Q2, unlike previous studies concerned with the construction of particular solutions. These symmetries are then used to reduce the governing fractional partial differential equation to a system of ordinary fractional differential equations. The reduced equation is solved analytically by the power series method, and the convergence of the series solution is rigorously proved by the implicit function theorem. Moreover, the systematic derivation of conservation laws is carried out by the nonlinear self-adjointness method. The results offer an analytical method to investigate fractional BBM-type equations, including convergent series representations and related conservation laws. The proposed framework can be applied to the mathematical modelling of systems with nonlinear dispersive and memory effects. A numerical example accompanied by a table and figure is provided to complement the theoretical results.