Eighth-Kind Chebyshev Polynomials Collocation Algorithm for the Nonlinear Time-Fractional Generalized Kawahara Equation

In this study, we present an innovative approach involving a spectral collocation algorithm to effectively obtain numerical solutions of the nonlinear time-fractional generalized Kawahara equation (NTFGKE). We introduce a new set of orthogonal polynomials (OPs) referred to as "Eighth-kind Chebyshev polynomials (CPs)". These polynomials are special kinds of generalized Gegenbauer polynomials. To achieve the proposed numerical approximations, we first derive some new theoretical results for eighth-kind CPs, and after that, we employ the spectral collocation technique and incorporate the shifted eighth-kind CPs as fundamental functions. This method facilitates the transformation of the equation and its inherent conditions into a set of nonlinear algebraic equations. By harnessing Newton's method, we obtain the necessary semi-analytical solutions. Rigorous analysis is dedicated to evaluating convergence and errors. The effectiveness and reliability of our approach are validated through a series of numerical experiments accompanied by comparative assessments. By undertaking these steps, we seek to communicate our findings comprehensively while ensuring the method's applicability and precision are demonstrated.