Exact Travelling Wave Solutions for the Space-time Fractional Benjamin-Ono Equation with Three Types of Fractional Operators

The space-time fractional Benjamin-Ono equation (STFBOE) is of fundamental importance in ocean science, particularly for modeling wave propagation in deep water. This study investigates the STFBOE employing three distinct fractional operators: the conformable derivative, the beta derivative, and the M-truncated derivative. By applying a fractional traveling wave transformation, the original nonlinear fractional partial differential equation is reduced to an ordinary differential equation. We then utilize three analytical techniques-the fractional functional variable method, the modified Kudryashov method, and the improved F-expansion method to derive novel exact traveling wave solutions. To the best of our knowledge, the exact solutions for the fractional Benjamin-Ono equation in the form considered here have been scarcely studied. A key contribution of this work is the introduction and tailored application of these methods to systematically construct solutions under each fractional derivative definition. Accordingly, we establish specific traveling wave variables corresponding to the conformable, beta, and <i>M</i>-truncated derivatives. A significant advantage of the proposed framework is its flexibility and effectiveness, enabling a straightforward derivation of solutions for both time-fractional and space-fractional versions of the Benjamin-Ono equation. Finally, we present a comparative graphical analysis of the obtained solutions using two- and three-dimensional plots, illustrating the spatio-temporal dynamics under selected parameter values. All the derived solutions are entirely new and extend the current understanding of nonlinear wave phenomena described by the STFBOE.