Exact Traveling Wave Solutions for the (1 + 1)-Dimensional Compound KdVB Equation via the Novel (G'/G)-Expansion Method

Abstract

In this work, while applying a new and novel (G'/G)-expansion version technique, we identify four families of the traveling wave solutions to the (1 + 1)-dimensional compound KdVB equation. The exact solutions are derived, in terms of hyperbolic, trigonometric and rational functions, involving various parameters. When the parameters are tuned to special values, both solitary, and periodic wave models are distinguished. State of the art symbolic algebra graphical representations and dynamical interpretations of the obtained solutions physics are provided and discussed. This in turn ends up revealing salient solutions features and demonstrating the used method efficiency.

New Exact Solutions of the Conformable Space-Time Sharma–Tasso–Olver Equation Using Two Reliable Methods

“…In Figure 1, we show different plots of the exact solution u 1 9 (x, t) in Equation (28) using the following parameter values, µ = 0.5, λ = 1, v = 0.5, d = 1, A = 0.5, and B = 1. In particular, Figure 1a-c shows the 3-D plot, the 2-D plot with t fixed at t = 1, and the contour plot of solution (28), respectively, when the set of the fractional orders {β = 1, α = 1} is used. Using the same parameter values as shown above except using {β = 0.8, α = 0.2}, the 3-D graph, the 2-D graph while t is held fixed at t = 1 and the contour graph of solution (28) are plotted in Figure 1d-f, respectively.…”

“…In particular, Figure 1a-c shows the 3-D plot, the 2-D plot with t fixed at t = 1, and the contour plot of solution (28), respectively, when the set of the fractional orders {β = 1, α = 1} is used. Using the same parameter values as shown above except using {β = 0.8, α = 0.2}, the 3-D graph, the 2-D graph while t is held fixed at t = 1 and the contour graph of solution (28) are plotted in Figure 1d-f, respectively. Proceeding in a manner analogous to the above plots except using the fractional order set {β = 0.5, α = 0.5}, we obtain the 3-D plot, the 2-D plot with t = 1, and the contour plot of solution (28) in Figure 1g-i, respectively.…”

Two Reliable Methods for Solving the (3 + 1)‐Dimensional Space‐Time Fractional Jimbo‐Miwa Equation

“…Searching for exact explicit solutions to nonlinear fractional partial differential equations (NFPDEs) is a research field of active interest. Nowadays, many approaches with the help of symbolic software packages have been developed to efficiently provide exact solutions of NFPDEs, for example, the improved extended tanhcoth method [34], the improved generalized exp-function method [35], the fractional Riccati expansion method [36], the ( / , 1/ )-expansion method [37][38][39][40][41], and the novel ( / )-expansion method [42][43][44][45]. The common idea of these mentioned methods is based on the homogeneous balance principle.…”

Microtubules Nonlinear Models Dynamics Investigations through the exp(−Φ(ξ))-Expansion Method Implementation

“…To date solving NPDEs exactly or approximately, a plethora of methods have been in use. These include, but are not limited to, (G 1 /G)-expansion [1][2][3][4][5][6], Frobenius decomposition [7], local fractional variation iteration [8], local fractional series expansion [9], multiple exp-function algorithm [10,11], transformed rational function [12], exp-function method [13,14], trigonometric series function [15], inverse scattering [16], homogeneous balance [17,18], first integral [19][20][21][22], F-expansion [23][24][25], Jacobi function [26][27][28][29], Sumudu transform [30][31][32], solitary wave ansatz [33][34][35][36], novel (G 1 /G) -expansion [37][38][39][40][41][42], modified direct algebraic method [43,44], and last but not least, the expp´Φpξqq-Expansion Method [45]…”