Spectral solutions for nonlinear static beam and fractional Riccati problems using new Lucas coefficient polynomials

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.