This study is devoted to studying the existence and uniqueness of solutions for Hadamard implicit fractional differential equations with generalized Hadamard fractional integro-differential boundary conditions by utilizing the contraction principle of the Banach and Leray–Schauder fixed point theorems. Moreover, with two different approaches, the Hyers–Ulam stabilities are also discussed. Different ordinary differential equations of the third order with different boundary conditions (e.g., initial, anti periodic and integro-differential) can be obtained as a special case for our proposed model. Finally, for verification, an example is presented, and some graphs for the particular variables and particular functions are drawn using MATLAB.

In this paper, a numerical approach employing radial basis functions has been applied to solve time-fractional FitzHugh–Nagumo equation. Spatial approximation is achieved by combining radial basis functions with the collocation method, while temporal discretization is accomplished using a finite difference scheme. To evaluate the effectiveness of this method, we first conduct an eigenvalue stability analysis and then validate the results with numerical examples, varying the shape parameter c of the radial basis functions. Notably, this method offers the advantage of being mesh-free, which reduces computational overhead and eliminates the need for complex mesh generation processes. To assess the method’s performance, we subject it to examples. The simulated results demonstrate a high level of agreement with exact solutions and previous research. The accuracy and efficiency of this method are evaluated using discrete error norms, including ${\mathrm{L}}_2$, ${\mathrm{L}}_{\infty }$, and ${\mathrm{L}}_{rms}$.