This manuscript introduces a semi-analytical and numerical solutions for the Benjamin Bona Mahony equation (BBME) in the form of convergent series. The BBME holds significance in diverse scientific and engineering applications.  Especially to study issues on shallow water waves, solitons and their importance in modern physics. The Fuzzy Homotopy Perturbation Transform Method (FHPTM) and Differential Quadrature Method (DQM) are utilized to obtain the solutions for the BBME. In DQM, grid point based on Shifted Legendre Polynomials (SLP) have been used to solve the BBME. Additionally, we address the uncertainty in the initial condition by representing it in terms of an interval. The interval BBME (iBBME) is subsequently tackled using the FHPTM providing both lower and upper interval solutions. The convergence of the interval solution is validated considering crisp case. The outcomes obtained through FHPTM for BBME are compared with the exact solution and results obtained in this study are exhibiting good agreement. The numerical outcomes by FHPTM are compared with results obtained by DQM.  Additionally, we presented the time fractional BBME and developed a fuzzy model for it, accounting for uncertainties in the coefficients associated with wave velocity. To analyze the behavior of the fuzzy time fractional BBME, and examined various numerical results using a double parametric approach.