Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique

Abstract

Nonlinear fractional partial differential equations are highly applicable for representing a wide variety of features in engineering and research, such as shallow-water, oceanography, fluid dynamics, acoustics, plasma physics, optical fiber system, turbulence, nonlinear biological systems, and control theory. In this research, we chose to construct some new closed form solutions of traveling wave of fractional order nonlinear coupled type Boussinesq–Burger (BB) and coupled type Boussinesq equations. In beachside ocean and coastal engineering, the suggested equations are frequently used to explain the spread of shallow-water waves, depict the propagation of waves through dissipative and nonlinear media, and appears during the investigation of the flow of fluid within a dynamic system. The subsidiary extended tanh-function technique for the suggested equations is solved for achieve new results by conformable derivatives. The fractional order differential transform was used to simplify the solution process by converting fractional differential equations to ordinary type differential equations by using the mentioned method. Using this technique, some applicable wave forms of solitons like bell type, kink type, singular kink, multiple kink, periodic wave, and many other types solution were accomplished, and we express our achieve solutions by 3D, contour, list point, and vector plots by using mathematical software such as MATHEMATICA to express the physical sketch much more clearly. Moreover, we assured that the suggested technique is more reliable, pragmatic, and dependable, that also explore more general exact solutions of close form traveling waves.

Exploring fractional-order new coupled Korteweg-de Vries system via improved Adomian decomposition method

“…Analytical solutions for non-linear evolution equations (NLEEs) play a crucial role in understanding nonlinear problems across various applied sciences [1][2][3][4]. The pursuit of analytical solutions for diverse NLEEs is essential, and recent literature reflects a notable effort to obtain traveling wave solutions [5][6][7][8], typically relying on variables associated with traveling waves [9,10]. However, obtaining analytical solutions for these differential equations can be challenging, leading to the introduction of semi-analytical techniques for representation in series form.…”