NEW APPLICATIONS OF THE FRACTIONAL DERIVATIVE TO EXTRACT ABUNDANT SOLITON SOLUTIONS OF THE TIME FRACTIONAL ZAKHAROV–KUZNETSOV–BENJAMIN–BONA–MAHONY EQUATION IN MATHEMATICAL PHYSICS

This paper explores the time fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation as a framework for various phenomena, including water wave mechanics, shallow water waves, quantum mechanics, ion-acoustic waves in plasma, and electro-hydro-dynamical models for local electric fields and signal processing waves are transmitted over optical cables. Extended Direct Algebraic Technique (EDAT) and Improved Generalized Tanh-Coth Technique are used to find new accurate traveling-wave solutions with appropriate physical free parameter values. The fractional traveling-wave transformation is used to convert the equation into a nonlinear ordinary differential equation, where the fractional derivative is assessed in a conformable manner. Trigonometric and hyperbolic functions are the forms in which the solutions can be obtained. The suggested techniques can achieve periodic, mixed dark-bright soliton, bright soliton, dark soliton, M-shaped, Compacton soliton, bell-type soliton, smooth mixed dark-bright soliton and W-shaped soliton. Some of the obtained solutions are graphically represented as 3D and contour plots. Meanwhile, the impacts of the fractional parameter are shown in 2D plots. The above techniques are effective and reliable, and it may be utilized as a substitute to develop new solutions for many fractional differential equation types employed in mathematical physics.