Abstract
We utilize the modified Riemann–Liouville derivative sense to develop careful arrangements of time-fractional simplified modified Camassa–Holm (MCH) equations and generalized (3 + 1)-dimensional time-fractional Camassa–Holm–Kadomtsev–Petviashvili (gCH-KP) through the potential double G ′ / G , 1 / G -expansion method (DEM). The mentioned equations describe the role of dispersion in the formation of patterns in liquid drops ensued in plasma physics, optical fibers, fluid flow, fission and fusion phenomena, acoustics, control theory, viscoelasticity, and so on. A generalized fractional complex transformation is appropriately used to change this equation to an ordinary differential equation; thus, many precise logical arrangements are acquired with all the freer parameters. At the point when these free parameters are taken as specific values, the traveling wave solutions are transformed into solitary wave solutions expressed by the hyperbolic, the trigonometric, and the rational functions. The physical significance of the obtained solutions for the definite values of the associated parameters is analyzed graphically with 2D, 3D, and contour format. Scores of solitary wave solutions are obtained such as kink type, periodic wave, singular kink, dark solitons, bright-dark solitons, and some other solitary wave solutions. It is clear to scrutinize that the suggested scheme is a reliable, competent, and straightforward mathematical tool to discover closed form traveling wave solutions.
Highlights
In recent years, fractional calculus (FC) assumed a basic part of a capable, catalyst, and rudimentary hypothetical structure for more sufficient displaying of multifaceted powerful cycles
Researchers are increasingly interested in seeking exact solutions to nonlinear fractional differential equations (NLFDEs), which play a significant role in nonlinear science
Where C1 and C2 stand for constants and those are arbitrary, we talk about the principle part of proposed methods to take exact traveling wave solutions to the NLFDE is as the form pu, Dαt u, Dβxu, Dαt Dαt u, Dαt Dβxu, DβxDβx, . . . , 0, 0 < α ≤ 1, 0 < β ≤ 1, (16)
Summary
Fractional calculus (FC) assumed a basic part of a capable, catalyst, and rudimentary hypothetical structure for more sufficient displaying of multifaceted powerful cycles. FC and nonlinear fractional differential equations (NLFDEs) have recently been used to solve problems in plasma physics, protein chemistry, cell biology, mechanical engineering, signal processing and systems recognition, electrical transmission, control theory, economics, and fractional dynamics. Mathematical Problems in Engineering to know the inner mechanism of the mentioned complex tangible phenomena, investigation of exact solutions of NLFDEs are very much important. In this way, numerous authors have been interested in studying the FC and finding precise and productive techniques for comprehending nonlinear fractional partial differential equations (NFPDEs). The point of this investigation is to build up some fresh and further broad precise solutions for the previously mentioned condition utilizing the DEM.
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