Two-Parameter General-Shift Lucas Polynomial Spectral Collocation for Time-Fractional Nonlinear Partial Integro-Differential Equations

The present study introduces a two-parameter general-shift Lucas basis from the classical Lucas family via an affine argument transformation, which then forms the basis of a spectral collocation approach for the numerical treatment of nonlinear time-fractional partial integro-differential equations. This parametrized development of the ordinary Lucas basis provides a unified framework for both the classical Lucas polynomials and the standard shifted forms of the Lucas basis. We obtain closed-form power and inverse-power representations in terminating hypergeometric form, which are used to derive the operational matrices of derivatives for ordinary and Caputo fractional differentiation. A tensor-product expansion discretizes the space–time variables. Quadrature approximation is applied to the Volterra integral using Gauss–Legendre quadrature, while collocation of the governing equation, along with its underlying conditions, leads to a system of nonlinear algebraic equations that can be handled by Newton’s method. To evaluate the performance of our proposed basis in terms of accuracy, sensitivity to parameters, and computational efficiency, we provide error bounds and conduct numerical experiments. Our results demonstrate that our proposed basis achieves high levels of accuracy while offering the flexibility provided by its two affine parameters.