Soliton dynamics in fractional Benjamin-Bona-Mahony-Burgers-Type Equations: A two-step Laplace Adams-Bashforth algorithm with machine learning verification

This paper presents a comprehensive numerical investigation of the time-dependent fractional Benjamin-Bona-Mahony-Burgers (BBMB) equation, which governs nonlinear dispersive wave phenomena and admits stable soliton solutions that maintain their shape during propagation. We developed and compared three distinct computational frameworks for simulating soliton dynamics under fractional order operators: (i) a finite difference method employing discretization for the Caputo fractional derivative, (ii) a novel two-step Laplace Adams-Bashforth (TSLAB) hybrid scheme that combines Laplace transformation with fractional predictor-corrector methodology, and (iii) a neural network (NN) incorporating fractional operator differentiation. The TS-LAB method demonstrates robust stability properties and improved computational efficiency for multi-term fractional partial differential equations (PDEs), utilizing Laplace transforms for accurate starting values and aiding soliton evolution over extended domains. The neural network implementation, optimized via the Levenberg-Marquardt algorithm, achieves exceptional convergence with a mean squared error of [Formula: see text] in only 12 epochs, while accurately capturing the characteristic shape-preserving property of solitary waves. Moreover, numerical experiments are conducted for both the fractional BBM and BBMB equations across varying fractional orders [Formula: see text]. Results reveal that as a approaches to unity, systematic amplitude reduction occurs alongside improved numerical accuracy, with the soliton profile remaining well-resolved despite amplitude attenuation. The BBMB equation exhibits a sharp error reduction for [Formula: see text], indicating optimal performance of the TS-LAB scheme near integer-order derivatives.