This study proposes novel spectral algorithms employing the Lucas coefficient polynomials to solve two significant nonlinear models: The static beam problem and the fractional Riccati equation. Our suggested approaches are based on developing novel theoretical findings that we derive for the introduced polynomials. These results include formulae for inversion, moment, derivatives, and linearization. Two methodologies are followed to treat the two nonlinear problems. The nonlinear fourth-order integro-differential static beam problem is treated using the collocation method, while the nonlinear fractional Riccati equation is treated using the tau method. Rigorous convergence and error analysis for the Lucas coefficient expansions are given. Compared to previous methods, the suggested algorithms exhibit exponential convergence and high accuracy, as verified by numerical testing. The findings demonstrate that spectral algorithms may effectively handle nonlinear differential equations using Lucas coefficient polynomials.

This work examined the two-dimensional, steady-state flow of a non-Newtonian nanofluid past an impermeable stretching sheet, incorporating temperature-dependent density, nonlinear rheology, nanoparticle transport mechanisms (thermophoresis/Brownian motion), and thermal radiation. The model was formed by nonlinear equations for mass, momentum, heat, and particle transport, and the surface heating condition was applied before using similarity transformations to reduce the system to ordinary differential equations. The numerical solution employs an innovative approach using merged Fibonacci-Lucas polynomials combined with least squares approximation, transforming the equations into algebraic form solved via the Newton iteration method. Rigorous convergence testing and error analysis verified the method's precision and reliability. The findings demonstrate that higher density and convection parameters substantially improve all transport processes, with heat transfer rates increasing by more than double. However, the Williamson parameter and Brownian motion show opposing influences, in which they decrease both surface friction and thermal transfer while simultaneously enhancing mass transport efficiency. Further, elevating the density parameter from 0.0 to 1.0 increases the skin-friction coefficient from 0.96084 to 1.18692 while simultaneously boosting both reduced Nusselt and Sherwood numbers. Conversely, augmenting the Williamson parameter from 0.0 to 0.6 reduces the skin-friction coefficient from 1.12885 to 0.96097, accompanied by moderate variations in heat and mass transfer rates. Extensive benchmarking against published numerical results demonstrated the scheme's accuracy, with close matching to existing solutions substantiating the reliability of our proposed approach.