Abstract
AbstractThe purpose of this article is to construct some novel exact travelling and solitary wave solutions of the time fractional (2 + 1) dimensional Konopelchenko–Dubrovsky equation, and two different forms of integration schemes have been utilized in this context. As a result, a variety of bright and dark solitons, kink- and antikink-type solitons, hyperbolic functions, trigonometric functions, elliptic functions, periodic solitary wave solutions and travelling wave solutions are obtained, and the sufficient conditions for the existence of solution are also discussed. Moreover, some of the obtained solutions are illustrated as two- and three-dimensional graphical images by using computational software Mathematica. These types of solutions have a wide range of applications in applied sciences and mathematical physics. The proposed methods are very useful for solving nonlinear partial differential equations arising in physical science and engineering.
Highlights
We have introduced two interesting algorithms for the extraction of travelling and solitary wave solutions of nonlinear evaluation equations (NLEEs), which demonstrate a wide range of applications in mathematical physics, plasma wave chemical physics, in fluid mechanics and many other nonlinear sciences
We have successfully applied two interesting algorithms that are unified Riccati equation expansion and modified extended auxiliary equation mapping method to compute the exact travelling and solitary wave solutions of the time fractional (2 + 1) dimensional coupled Konopelchenko–Dubrovsky equation (KDE). This system of coupled KDE describes the evolution of nonlinear wave, which is the extension of Kadomtsev–Petviashvili and modified Kadomtsev–Petviashvili
New families of traveling and solitary wave solutions are recovered in the form of bright and dark solitons, kink- and antikink-type solitons, hyperbolic functions, trigonometric functions and elliptic functions, and for details see Figures 1–13
Summary
Over the last few decades, nonlinear phenomena have been observed to have fascinating characteristics in. The unified Riccati equation expansion method and the modified extended auxiliary equation mapping method are successfully employed to construct a variety of new travelling wave solutions to a coupled time fractional (2 + 1) dimensional Konopelchenko–Dubrovsky system [20,21]: Dtα Υ. Due to abundant features of fractional differential equations (FDEs), it has become one of the most interesting fields of research. For this purpose, various techniques have been developed to formulate exact and travelling wave solutions of FDEs. Recently, a new definition of fractional calculus has been introduced by Jumarie’s modified Riemann–Liouville (mRL) of order α as [38,39]:.
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