Solitary and Periodic Wave Solutions of the Space-Time Fractional Extended Kawahara Equation

Abstract

The extended Kawahara (Gardner Kawahara) equation is the improved form of the Korteweg–de Vries (KdV) equation, which is one of the most significant nonlinear evolution equations in mathematical physics. In that research, the analytical solutions of the conformable fractional extended Kawahara equation were acquired by utilizing the Jacobi elliptic function expansion method. The given expansion method was applied to different fractional forms of the extended Kawahara equation, such as the fraction that occurs in time, space, or both time and space by suitably changing the variables. In addition, various types of fractional problems are exhibited to expose the realistic application of the given method, and some of the obtained solutions were illustrated in two- or three-dimensional graphics as proof of the visualization.

Analytical soliton solutions for the beta fractional derivative Gross–Pitaevskii system with linear magnetic and time dependent laser interactions

“…On the other hand as the value of α decreases the number of oscillations or singularities increases for small time values and decreases for large time values. The above behavior has been reported in other nonlinear models like the fractional modified KdV equation and combined fractional KdV-mKdV equation or fractional extended Korteweg-de Vries [78].…”

Extracting the Ultimate New Soliton Solutions of Some Nonlinear Time Fractional PDEs via the Conformable Fractional Derivative

“…Again, using the wave transformation from Equation (37), the other soliton solution of Equation ( 3) is written as…”

“…Consequently, researchers have focused on investigating fractional-order calculus and detecting exact and methodical techniques for finding the perfect solutions for fractional PDEs. Recently, many researchers have investigated these types of equations by applying different methods, such as the Generalized Kudryashov method [25,26], the residual-power-series method [27,28], the exp-function method [29,30], the long-wave method [31], the variational iteration method [32,33], the extended direct algebraic method [34,35], the sine-Gordon expansion approach [36], the Jacobi elliptic function method [37], the Sarder sub-equation method [38], the G ′ G , 1 G -expansion method [39][40][41], and many other techniques. Now, there is a more well-organized method called the [42,43] to solve nonlinear fractional-order PDEs.…”

A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices

“…In [34], the authors followed a conservative difference algorithm to handle the generalized Kawahara equation. In [35], the author developed wave solutions for some Kawahara equations. The authors of [36] applied the Adomian decomposition method for treating the modified Kawahara equation.…”