Numerical Solutions via Shifted Pell Polynomials for Third-Order Rosenau–Hyman and Gilson–Pickering Equations

This paper introduces a collocation algorithm for numerically solving the third-order Gilson–Pickering equation (GPE) and the classical Rosenau–Hyman equation (RHE). We employ newly developed shifted Pell polynomials as basis functions. Novel formulas for these polynomials are devised and utilized in constructing the proposed algorithm. Specifically, we establish a new power form and its inversion formula, along with an explicit formula for derivatives of the shifted Pell polynomials, from which the operational matrices of derivatives (OMDs) are derived. These matrices facilitate the conversion of nonlinear dispersive models into systems of algebraic equations, efficiently solved using Newton’s iterative technique. The error analysis of the shifted Pell expansion is discussed in depth. Several numerical examples, including the RHE, its fourth-order variant, and the Fornberg–Whitham equation, are provided to demonstrate the method’s performance and accuracy. Comparative results are also reported.