An efficient numerical scheme for the simulation of time-fractional nonhomogeneous Benjamin-Bona-Mahony-Burger model

Abstract

The Benjamin-Bona-Mahony-Burger (BBM-Burger) equation is important for explaining the unidirectional propagation of long waves in nonlinear dispersion systems. This manuscript proposes an algorithm based on cubic B-spline basis functions to study the nonhomogeneous time fractional model of BBM-Burger via Caputo derivative. The discretization of fractional derivative is achieved by L1 formula, while the temporal and spatial derivatives are interpolated by means of Crank-Nicolson and forward finite difference scheme together with B-spline basis functions. The performance of the Cubic B-spline scheme (CBS) is examined by three test problems with homogeneous initial and boundary conditions. The obtained results are found to be in good agreement with the exact solutions. The behaviour of travelling wave is studied and presented in the form of tables and graphics for various values of α and t. A linear stability analysis, based on the von Neumann scheme, shows that the CBS is unconditionally stable. Moreover, the accuracy of the scheme is quantified by computing error norms.