Abstract
In this paper, the ( G ′ / G , 1 / G ) -expansion method is applied to acquire some new, exact solutions of certain interesting, nonlinear, fractional-order partial differential equations arising in mathematical physics. The considered equations comprise the time-fractional, (2+1)-dimensional extended quantum Zakharov-Kuznetsov equation, and the space-time-fractional generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) system in the sense of the conformable fractional derivative. Applying traveling wave transformations to the equations, we obtain the corresponding ordinary differential equations in which each of them provides a system of nonlinear algebraic equations when the method is used. As a result, many analytical exact solutions obtained of these equations are expressed in terms of hyperbolic function solutions, trigonometric function solutions, and rational function solutions. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions, solitary wave solutions of kink type, and so on. The method is very efficient, powerful, and reliable for solving the proposed equations and other nonlinear fractional partial differential equations with the aid of a symbolic software package.
Highlights
Nonlinear evolution equations (NLEEs), which can be described using partial differential equations (PDEs), play a significant role for understanding qualitative behaviors of many real-world phenomena
The selected exact solutions of Equation (1), which are expressed in Equations (25), (28) and (35), are plotted for the three-dimensional representations
In 2019, Ali et al, [37] analytically solved Equation (1) using the modified Kudryashov method and the ( G 0 /G2 )-expansion method. They found that the former method provided the two exact solutions written in terms of the reciprocal of exponential function solutions
Summary
Nonlinear evolution equations (NLEEs), which can be described using partial differential equations (PDEs), play a significant role for understanding qualitative behaviors of many real-world phenomena. The associated equations of the generalized Hirota-Satsuma coupled KdV system have been solved using different methods as follows. In 2007, Zhang [44] used the direct algebraic method to construct the exact solutions for the first-order generalized Hirota-Satsuma coupled KdV systems. In 2010, Zigao et al, [45] applied the improved F-expansion method to the variable-coefficient first-order generalized Hirota-Satsuma coupled KdV system for obtaining the new exact solutions. In 2017, Khater et al, [46] found the exact traveling wave solutions of the system using the modified simple equation method, while the time-fractional generalized Hirota-Satsuma coupled KdV system was solved using the direct algebraic method by Neirameh [41] in 2015.
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