Abstract
Nonlinear partial differential equations (NLPDEs) are an inevitable mathematical tool to explore a large variety of engineering and physical phenomena. Due to this importance, many mathematical approaches have been established to seek their traveling wave solutions. In this study, the researchers examine the Gardner equation via two well-known analytical approaches, namely, the improved tanΘϑ-expansion method and the wave ansatz method. We derive the exact bright, dark, singular, and W-shaped soliton solutions of the Gardner equation. One can see that the methods are relatively easy and efficient to use. To better understand the characteristics of the theoretical results, several numerical simulations are carried out.
Highlights
Applications of the Gardner Equation via improved tan(Θ(θ))-expansion method (ITEM)We will examine ITEM for equation (1). To find the traveling solutions for equation (1), we define the wave transformation as u U(θ), where θ μx − θt, μ ≠ 0, and θ ≠ 0 to be determined later
Introduction eGardner equation is given as [1,2,3]ut + 2αuux + 3βu2ux + cuxxx 0, (1)where α, β, and (c > 0) are constant values
The improved tan(Θ(θ))-expansion method (ITEM) [13,14,15,16] and the wave ansatz method [17,18,19,20,21] have been exploited to integrate a variety of nonlinear partial differential evolution equations (NLPDEs)
Summary
We will examine ITEM for equation (1). To find the traveling solutions for equation (1), we define the wave transformation as u U(θ), where θ μx − θt, μ ≠ 0, and θ ≠ 0 to be determined later. We apply the ITEM to obtain traveling wave solutions of the Gardner equation (1). According to this method, the solution of equation (26) can be written in the form of equation (4). Setting these values in categories 2, 6, 10, and 14 of Section 2, respectively, we acquire the following solutions: u1(θ). Where a2 + b2 − c2 > 0 and θ is given by (32) Setting these values in categories 3, 5, and 6 of Section 2, respectively, we obtain α 1 + tanh a2 + b2 /2θ a2 + b2 + a. Where b is an optional and β < 0 must be held Setting these values in categories 1, 6, and 13 of Section 2, respectively, we obtain α (tanh(b/2θ) + 1). It is worth to note that one can find some more new exact solitary solutions from solutions (31), (38), and (45)
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