New exact solutions for the variable coefficient modified KdV equation using direct reduction method

Abstract

In this paper, the direct reduction method combined with the homogeneous balance method is used to construct new exact solutions for the variable coefficient modified KdV model in fluid-filled elastic tubes.The variable coefficient modified KdV is reduced to a Ricatti ordinary equation that has many exact solutions. The solutions obtained in this manuscript are new and recover other solutions obtained before in many literature.

New rogon waves for the nonautonomous variable coefficients Schrö dinger equation

“…Recently, many new methods have been constructed to obtain new solutions for nonlinear partial differential equations like symmetry groups, tanh method, trial equation method, sin-Gordon equation method, etc. (Chen et al 2019(Chen et al , 2020Hua et al 2019;Liu et al 2019;Mao et al 2019;Xia et al 2020;Xu et al 2020) (El-Sayed et al 2015, 2014, 2021, 2018aEl-Shiekh and Rehab 2018b;El-Shiekh 2017, 2015, 2013El-Shiekh and Gaballah 2020b;Moatimid et al 2013;Moussa et al 2012;El-Shiekh 2010, 2011;Moatimid and El Shikh 2008;Moussa and El Shikh 2006;Chen et al 2021;He et al 2021;Lü and Ma 2016;Xia et al 2020;Yin et al 2020).…”

“…In the following we are going to apply one of the similarity techniques, the direct similarity reduction method (El-Shiekh 2019, 2017, 2015, 2013, 2018aEl-Shiekh and Gaballah 2020b;El-Shiekh 2008, 2011) where ′ denotes the derivative with respect to . Collect the U coefficient and its derivatives, also, the real and imaginary parts together, assuming that e i( (z,t)) ≠ 0, we have…”

Invariant and simulation analysis to the time fractional Abrahams–Tsuneto reaction diffusion system

“…In the current work, we consider Lie symmetry analysis which is a very important, numerical approximations by means of residual power series (RPS) method [63][64][65] of the time fractional Abrahams-Tsuneto reaction diffusion system (ATRDS) [66] for the first time. It is widely known that, symmetry analysis is one of the potent way that provide more information for the systems of differential equations [67][68][69][70][71][72][73][74][75][76][77][78][79].…”

New similarity solutions for the generalized variable-coefficients KdV equation by using symmetry group method

“…In fluid mechanics, by considering the blood as incompressible fluid, the evolution equation is obtained as a variable-coefficient mKdV (vcmKdV) equation [14] U…”

“…In many real physical backgrounds, the variable-coefficient nonlinear evolution equations often can provide more powerful and realistic models than their constant-coefficient counterparts when the inhomogeneities of media and nonuniformities of boundaries are considered [10][11][12][13][14][15][16][17][18][19][20][21][22].Thereby, some inhomogeneous KdV models with the time-dependent coefficients have been derived to describe a variety of interesting and significant phenomena in fluid mechanics, ocean dynamics and plasma physics, as follows:…”