Shifted Pell–Lucas Polynomial-Based Algorithms for Fractional and Classical Rosenau–Hyman Models

This paper introduces two numerical algorithms for solving the fractional and classical Rosenau–Hyman equations. Certain shifted Pell–Lucas polynomials (SPLPs) are employed. New formulas for these polynomials are developed and used to analyze our proposed algorithms, together with the application of the typical collocation method. The established operational matrices for both integer and fractional derivatives are used to convert the problems, along with their governing conditions, into matrix systems that can be treated effectively. The convergence of the shifted Pell–Lucas expansion is thoroughly investigated by developing new estimates. The high accuracy and efficiency of the proposed numerical algorithms are tested through numerical experiments supported by comparisons with some other techniques in the literature.