<p>This paper develops a numerical method for a one-dimensional nonlinear time–space fractional partial differential equation involving the Caputo time derivative, the Riesz space-fractional derivative, a nonlinear memory term, and a nonlinear convection term. The Caputo derivative is approximated by a direct quadratic polynomial interpolation, while the Riesz derivative is discretized by a fractional centered difference scheme on a uniform spatial mesh. The main contribution is the unified construction and analysis of the resulting semi-discrete and fully discrete schemes for a nonlinear model that simultaneously contains temporal memory and spatial nonlocality. Stability and convergence analyses are established under clearly stated regularity and boundedness assumptions. Numerical examples, including an additional test problem, confirm the theoretical accuracy, clarify the observed convergence orders, and demonstrate the effectiveness of the method in the presence of nonlinear and memory effects.</p>