Numerical investigation of the fractional perturbed Korteweg–de Vries equation with the ABC fractional derivative using the Bernstein-shifted Legendre spectral method

In this work, the propagation behavior of the fractional perturbed Korteweg–de Vries (pKdV) equation is investigated numerically using the shifted Legendre–Bernstein spectral method. The fractional derivative is considered in the Atangana–Baleanu–Caputo sense and is approximated in terms of Bernstein-shifted Legendre polynomials. By employing the corresponding operational matrices, the proposed problem is transformed into a solvable set of algebraic equations. In comparison with recursive approaches such as ADM and HPM, the proposed shifted Legendre–Bernstein spectral method converts the fractional nonlinear pKdV equation directly into a finite nonlinear algebraic system. This makes the method suitable for treating the ABC fractional derivative and nonlinear dispersive terms with relatively simple implementation. The reliability of the current approach is demonstrated through the presentation of the necessary definitions and properties, together with a convergence analysis. Furthermore, the efficiency of the numerical scheme is examined through several examples associated with different initial conditions. In addition, the influence of the perturbation parameter on the dynamics of the pKdV equation is investigated for various choices of the fractional order. The obtained numerical approximations and the associated findings support the capability of the adopted scheme to produce accurate and stable solutions with relatively simple implementation. The findings of the paper confirm the capability of the adapted numerical scheme to investigate other fractional nonlinear equations.