Numerical simulations of Rosenau–Burgers equations via Crank–Nicolson spectral Pell matrix algorithm

The current work addresses the use of the numerical Crank–Nicolson technique along with spectral collocation to seek the approximate solution of a class of high-order nonlinear partial differential equations known as the Rosenau–Burgers equation. First, the Crank–Nicolson approach is applied to reduce the Rosenau–Burgers equation to a set of linear equations. The solvability and uniqueness of the semi-discretized form is established. Then, the spectral based collocation method by using the Pell functions is employed to approximate the derived system of linearized equations in terms of unknown coefficients. The theoretical part and error analysis are comprehensively studied in the weighted \(L^2\) $L^2$ -norm. Finally, four illustrative test cases are conducted to showcase the efficiency of the suggested combined technique. The results are well compared with existing results and exact solutions in the literature to demonstrate the advantages of the applied novel approach.