This paper develops an implicit fractional exponential fitting/adapted back-ward differential formula of second-order (FEABDF2) for solving fractional-order differential equations (FDE) of order \( \alpha \in (0,1) \) by developing new generating functions. Building on the work in Shahbazi and Javidi (J. Comput. Appl. Math. 42(4):179, 2023), where a numerical method was proposed that generates solutions within the space formed by linear combinations of \( \langle 1,\, e^{\lambda x} ,\, xe^{\lambda x}\rangle \) , this paper presents a new numerical method where the solutions are generated within the space formed by \( \langle e^{\lambda x}, 1, x,\ldots , x^{r} \rangle \) . The proposed method is applied to both linear and non-linear FDEs with Riesz space fractional advection–diffusion equations. The FEABDF2 is used to approximate the Riesz space fractional derivative and the finite method is introduced for the Riesz space fractional advection–diffusion equation. The consistency, stability, and convergence of the method are analyzed. Then, the stability regions are delineated. The method’s efficiency is demonstrated through several examples, with the results confirming the accuracy of the presented theory when compared to existing methods.
