A potential rook polynomial integration approach for seventh-order time frame fractional KdV models

Abstract Nonlinear evolution equations are an intriguing and challenging field of mathematics and physics. The Korteweg–de Vries (KdV) model is a prominent nonlinear evolution paradigm extensively explored in computational physics owing to its significance in replicating shallow water waves. Higher-order KdV equations, such as the Sawada-Kotera-Ito, Kaup-Kuperschimdt, and Lax simulations, have advanced wave research applications encompassing hydrodynamics, plasma physics, and nonlinear optical phenomena. Time fractional higher-order KdV models contribute to more effective problem-solving in the actual world. Seventh-order time fractional KdV models are incredibly complex, with higher-order and fractional time derivatives. Traditional numerical approaches may be insufficient to describe these equations’ behavior accurately. Advanced numerical techniques use fewer processing resources and less time. As a result, we developed a novel, efficient numerical strategy from the backdrop of algebraic graph theory to solve these models. This numerical algorithm includes an operational integration matrix based on the Rook polynomials to approximate the KdV models. The Caputo fractional derivatives are incorporated to deal with the time fractional derivatives in the seventh-order KdV models. With the standard collocation points, the emerging equation is turned into an ensemble of nonlinear algebraic equations and solved by Newton’s method to provide an alternative Rook polynomial collocation mechanism (RCM) solution. Numerical examples demonstrate the RCM’s efficacy. Absolute and residual errors are computed and compared with the recent, efficient methods to confirm the viability of the RCM algorithm. The comparative tabular results and graphical analysis witnessed the dynamic behaviors in the fractional domain and the RCM’s compatibility with the accurate results. The study demonstrates that the proposed methodology is an effective and convenient solution for fractional KdV models and generates scope for solving other variable order KdV models.