Numerical investigation of variable-order fractional Benjamin–Bona–Mahony–Burgers equation using a pseudo-spectral method

This article introduces a new local variable-order (VO) fractional differentiation called the VO conformable fractional derivative. This derivative is employed to define the VO time fractional Benjamin–Bona–Mahony–Burgers (BBMB) equation. A pseudo-spectral algorithm using the Chebyshev cardinal functions (CCFs) is adopted to find a numerical solution for this equation. The presented method with the help of the CCFs derivative matrices (which are obtained in this research) turns problem solving into solving an algebraic system of equations. The proposed approach is applied on some test problems, and its accuracy is examined in terms of L_∞ and L₂ error norms. A numerical comparison is performed between the achieved results of the method with the results obtained from the cubic B-spline method and the hybrid method generated using the quintic Hermite approach and the finite difference technique.