In this work, a numerical scheme based on combined Lucas and Fibonacci polynomials is proposed for one- and two-dimensional nonlinear advection–diffusion–reaction equations. Initially, the given partial differential equation (PDE) reduces to discrete form using finite difference method and $$\theta -$$θ- weighted scheme. Thereafter, the unknown functions have been approximated by Lucas polynomial while their derivatives by Fibonacci polynomials. With the help of these approximations, the nonlinear PDE transforms into a system of algebraic equations which can be solved easily. Convergence of the method has been investigated theoretically as well as numerically. Performance of the proposed method has been verified with the help of some test problems. Efficiency of the technique is examined in terms of root mean square (RMS), $$L_2$$L2 and $$L_\infty $$L∞ error norms. The obtained results are then compared with those available in the literature.

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 ${\mathrm{L}}_{rms}$.