Waves Propagation for the Conformable Time-Fractional Benjamin-Bona-Mahony-Burgers Equation Arising on the Surface of Water

This study investigates solitary wave solutions of the conformable time-fractional Benjamin-Bona-Mahony-Burgers (TFBBMB) equation. This is a special type of Sobolev-type equation that can describe long waves on the surface of water in a channel with a minimal amplitude. Three techniques are employed: The generalized projective Riccati equation (GPRE) approach, the modified auxiliary equation (MAE) technique, and the generalized exponential rational function (GERF) approach. We obtain formally generated expressions that are symbolically verified by direct substitution into the original equation. The solutions are obtained in the form of exponential functions, rational functions, hyperbolic functions, and trigonometric forms. Real nonsingular solutions are identified and presented with appropriate parameter restrictions, while physically admissible solutions are highlighted for their relevance to wave propagation phenomena. Moreover, linear stability analysis is discussed for the classical BBMB equation. Unique physical problems (initial and boundary conditions) are constructed for real-valued, nonsingular solutions that can serve as benchmark problems for numerical methods.