New exact solutions for Kudryashov–Sinelshchikov equation

Abstract

In this paper, we firstly change the auxiliary second order ordinary differential equation in the G G-polynomial expansion method to the Riccati equation. By solving the Riccati equation, we obtain more exact solutions to the auxiliary equation and thus obtain more new exact solutions to the Kudryashov-Sinelshchikov equation, which mainly include three types of solutions with parameters: hyperbolic function traveling wave solutions, trigonometric function traveling wave solutions, and rational function traveling wave solutions. At last, some examples and figures are given to demonstrate the solutions.

On some new soliton solutions of (3 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation using two different methods

“…Herein, we present some figures in the two-dimensional, three-dimensional and countours to illustrate the solutions that we have got. Some of the analytical solutions are presented in Figures 1-8, while the accuracy of the methods was compaBlack to the solution given by Duan & Lu, (2021) and Lu (2018) as shown in Figures 1-8, respectively. In Figure 1, we introduce the graph of case 1 for Eq.…”

“…Solitons exhibit the particle-like properties because the energy is at any instant confined to a limited region of space (Ghanbari & Nisar, 2020;Nisar et al, 2021;Zafar, Ali, Raheel, Jafar, & Nisar, 2020). With the development of soliton theory, many powerful methods for obtaining exact solutions of NLPDEs have been presented, such as homotopy perturbation method, nonperturbative method, homogeneous balance method, Backlund transformation, Darboux transformation, extended tanh-function method, extended F-expansion method, G 0 G method, exp-function method, sine-cosine method, Jacobi elliptic function method, extended Riccati equation rational expansion method, extended auxiliary function method and other methods (Achab et al, 2020;Akbar, Alam, & Hafez, 2016;Alam, Hafez, Akbar, & Roshid, 2015;Ali Akbar, Ali, & Tarikul Islam, 2019;Duan & Lu, 2021;Hong & Lu, 2013;Islam, Akbar, & Khan, 2018;Kazi Sazzad Hossain & Ali Akbar, 2017;Lu, 2018;Mahmud, Samsuzzoha, & Akbar, 2017;Mohyud-Din, Nawaz, Azhar, & Akbar, 2017;Nur Alam1, Ali Akbar, & Fazlul Hoque, 2014;Shafiqul Islam, Khan, Ali Akbar, & Mastroberardino, 2014). In this article, we used the sine Gordon expansion method and extended tanh function method to the (3 þ 1) dimensional Boiti-Leon-Manna-Pempinelli equation (BLMP) which is used to describe incompressible liquid in fluid mechanics (Wazwaz, 2019).…”

Studying on Kudryashov-Sinelshchikov dynamical equation arising in mixtures liquid and gas bubbles

“…Several methods are used and applied to obtain traveling and oscillating wave solutions. G G -polynomial expansion method [3], modification of truncated expansion method applied [4], Lie symmetry analysis [5,21], F -expansion method and its improved method [6,22], Riccati-Bernoulli sub-ODE method [7], the method of simplest equation [8,9], generalized Kudryashov method [10,11], the approach of dynamical systems [12], Bernoulli sub-equation function method [13,14], modified method of simplest equation [15,28], Expansion function method [16], sine-Gordon expansion method [17], Bifurcation method, improved F-expansion method and modified exp-function method [18,20,24], solutions obtained from a generalized Korteweg-de Vries equation [19], radial basis function method [23], and Long wave limit method [25].…”

Investigation of Exact Solutions of some Nonlinear Evolution Equations via an Analytical Approach

“…Kudryashov and Sinelshchikov [11], [12]proposed a more general model for describing pressure waves in a a gas and liquid bubbles mixture by taking into account of the heat transfer and the viscosity of liquid. The equation is known as Kudryashov-Sinelshchikov (KS) equation [16].…”