Abstract
This study investigates the solitary wave solutions of the nonlinear fractional Jimbo–Miwa (JM) equation by using the conformable fractional derivative and some other distinct analytical techniques. The JM equation describes the certain interesting (3+1)-dimensional waves in physics. Moreover, it is considered as a second equation of the famous Painlev’e hierarchy of integrable systems. The fractional conformable derivatives properties were employed to convert it into an ordinary differential equation with an integer order to obtain many novel exact solutions of this model. The conformable fractional derivative is equivalent to the ordinary derivative for the functions that has continuous derivatives up to some desired order over some domain (smooth functions). The obtained solutions for each technique were characterized and compared to illustrate the similarities and differences between them. Profound solutions were concluded to be powerful, easy and effective on the nonlinear partial differential equation.
Highlights
During the last five decades, the nonlinear fractional partial differential equations (NLFPD)have been used for modeling many of the nonlinear phenomena in various fields
Fractional calculus is a generalization of ordinary calculus, where derivatives and integrals of arbitrary real or complex order are defined
According to the fundamental role of fractal models, an extensive study is applied on the fractional calculus to discover new fractional derivatives that have been defined such as the Riemann–Liouville, Caputo, Hadamard, Riesz, Grünwald–Letnikov, Marchaud, etc. [1–5]
Summary
During the last five decades, the nonlinear fractional partial differential equations (NLFPD). Based on the conformable derivative definition, the fractional models convert to nonlinear ordinary differential equations with integer order. In this step, the contribution of both analytical and numerical schemes have started to study explicit solutions for these models and for this purpose, many analytical and approximate schemes have been derived such as exp (−φ(Θ))-expansion, improved F-expansion, extended ( GG )-expansion, extended tanh- function, simplest and extended simplest equation, generalized Riccati expansion and sinh-Gordon expansion, Riccati–Bernoulli Sub-ODE, and modified Khater methods to discover more physical and dynamical properties of these models. The rest of the paper is organized as follows: Section 2 applies the above-mentioned techniques to the nonlinear time fractional JM equation to obtain exact and solitary wave solutions.
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