New Similarity Solutions of a Generalized Variable-Coefficient Gardner Equation with Forcing Term

Abstract

The compatibility method is used for a generalized variable-coefficient Gardner equation (GVGE) with a forcing term. By the compatibility of the considered equation and a non-classical symmetry of a given form, four types of symmetry are obtained. Then, by solving the characteristic equations of symmetry, the GVGE is reduced to variable coefficients ordinary differential equations, and rich varieties of new similarity solutions are presented. Our results show that the compatibility method can be employed for variable coefficients nonlinear evolution equations with forcing terms.

Asymptotic analysis of the angular motion of a projectile based on Lyapunov artificial-small-parameter perturbation

“…Next, to solve Equation ( 10), we substitute Equation (11) into Equation (10). Note that the order of magnitude of   0  A v v is 10 0 and that of  …”

“…The equation of angular motion for a projectile is a second-order non-homogeneous ordinary differential equation with variable coefficients. For this kind of equation, research has shown that it is possible to obtain an accurate analytical solution only when the equation is in a specific form [10][11][12]. However, the angular motion equation is not in a specific form.…”

Symmetry reductions of the ( 3 + 1 ) $(3+1)$ -dimensional modified Zakharov–Kuznetsov equation

“…The Lie-group method, originally proposed by Sophus Lie, is a classical method to determine the symmetry reduction of partial differential equations (PDEs) [13][14][15][16]. During the past several decades, there have been many extensions of the Lie-group method such as the nonclassical Lie group method [17], the CK direct method [18], the direct symmetry method [19], and so on [20][21][22][23][24]. Among them, the direct symmetry method is an effective approach for seeking symmetry reductions.…”