Advanced Graph Polynomial Technique for Time‐Fractional Degasperis‐Procesi and Camassa‐Holm Models Describe Wave Propagation in Shallow Water

The article introduces an innovative approach to the shallow water wave models, the modified Degasperis–Procesi Equation (MD–PE) and modified Camassa–Holm Equation (MC–HE) with a time‐fractional derivative. A functional matrix of Integration of a unique class of graph polynomials, such as the Complete graph's characteristic polynomial, is constructed to approximate the solution of the models. The Caputo sense addresses the fractional time derivatives in the MD–PE and MC–HE models. The conventional collocation points switch the models to a set of nonlinear algebraic equations. Newton's approach discovers the unknown coefficients of the characteristic polynomials, resulting in the novel Characteristic polynomial collocation method () solution. Despite the conventional perturbation approach, the procedure isn't contingent on certain presumptions or variable constraints. Some numerical instances serve, and their outcomes are juxtaposed visually in 2D and 3D setups. Compared to typical hybrid interpreters that produce series solutions, this methodology generates exact outcomes utilizing less computational power and in a more straightforward manner. We compute absolute and other error norms to confirm the efficacy of our suggested method.