Nonlinear dynamic wave pattern analysis of the time-fractional Benjamin–Ono equation in ion-acoustic wave

The study of nonlinear fractional-order partial differential equations (PDEs) and its numerous analyses of soliton solutions have been helpful in the development of engineering and physical science. The fractional nonlinear PDEs play a significant character in the sectors of electromagnetic fields, quantum fluctuations, black hole transmission, image processing, nonlinear optics, electromagnetic interactions, and so on. For the time-fractional Benjamin–Ono equation, several pioneering and more general closed-form solitary as well as traveling wave solutions have obtained the new auxiliary equation technique by using the truncated M-fractional derivative, which describes the propagation of internal waves in deep water, like internal waves, shock waves and solitons, conservation laws, marine engineering, mediational instability, rogue waves, and so many. Several widely recognized soliton waveforms, including multiple solitons, multiple periodic, kink, flat kink, anti-kink, and flat soliton, and other sorts of solutions, are illustrated with the use of computational software and draw the 3D and contour plots with definite free parametric values. These solutions were checked and found to be correct with the use of computational software. The suggested scheme establishes more broadly applicable solitary wave solutions, which are reliable, efficient, trustworthy, productive, effective, and appealing from a computational program.