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The analytical solution of the longitudinal wave equation in the MEE circular rod is analyzed by the powerful sine-Gordon expansion method. Sine - Gordon expansion is based on the well-known wave transformation and sine - Gordon equation. In the longitudinal wave equation in mathematical physics, the transverse Poisson MEE circular rod is caused by the dispersion. Some solutions with complex, hyperbolic and trigonometric functions have been successfully implemented. Numerical simulations of all solutions are given by selecting the appropriate parameter values. The physical meaning of the analytical solution explaining some practical physical problems is given. [ABSTRACT FROM AUTHOR]

The present study focuses on the acquisition of traveling wave solutions associated with the conformable time-fractional Zoomeron equation through the utilization of the modified exponential function method (MEFM). The solution functions derived from mathematical computations encompass hyperbolic, trigonometric, and rational functions. Various graphical representations, such as 2D, 3D, contour graphs, and density graphs, are utilized to visually depict the distinct features of the solution functions derived from the determination of suitable parameters. [ABSTRACT FROM AUTHOR]

In this paper, we investigate and use the new modified exp ( − Ω ( ξ ) ) -expansion method, (MEM). We apply the new MEM to nonlinear long–short-wave interaction systems (NLSWIS). Among our findings are sets of solutions including, but not limited to, new hyperbolic, complex, and dark soliton solutions. Not only is MEM shown to be highly adaptable for partial differential equations with strong nonlinearities, but also, it turns out to be highly efficient, despite its ease. [ABSTRACT FROM AUTHOR]

In this research, we seek exact solutions of generalized Gardner equation. We make use of extended trial equation method to attain exact solutions of generalized Gardner equation. Firstly, we find some exact solutions such as soliton solutions, rational, Jacobi elliptic and hyperbolic function solutions of generalized Gardner equation by the help of extended trial equation method. Then, for some parameters, we draw two and three dimensional graphics of imaginary and real values of some exact solutions that we acquired by using this method. [ABSTRACT FROM AUTHOR]

In this study, we apply the modified exp ( - Ω ( ξ ) ) - expansion function method to the new coupled Konno-Oono equation. We obtain some new travelling wave solutions such as complex function, hyperbolic function and rational function solutions. We observe that all travelling wave solutions have been verified the new coupled Konno-Oono equation by using Wolfram Mathematica 9. Then, we plot the two- and three-dimensional surfaces for all travelling wave solutions obtained in this paper by the same computer program. [ABSTRACT FROM AUTHOR]

In this paper, we apply the sine-Gordon expansion method which is one of the powerful methods to the generalized-Zakharov equation with complex structure. This algorithm yields new complex hyperbolic function solutions to the generalized-Zakharov equation with complex structure. Wolfram Mathematica 9 has been used throughout the paper for plotting two- and three-dimensional surface of travelling wave solutions obtained. [ABSTRACT FROM AUTHOR]

In this paper, exact solutions of the new Hamiltonian amplitude equation and Fokas-Lenells equation are successfully obtained. The extended trial equation method (ETEM) and generalized Kudryashov method (GKM) are applied to find several exact solutions of the new Hamiltonian amplitude equation and Fokas-Lenells equation. Primarily, we seek some exact solutions of the new Hamiltonian amplitude equation and Fokas-Lenells equation by using ETEM. Then, we research dark soliton solutions of the new Hamiltonian amplitude equation and Fokas-Lenells equation by using GKM. Lastly, according to the values of some parameters, we draw two and three dimensional graphics of imaginary and real values of certain solutions found by utilizing both methods. [ABSTRACT FROM AUTHOR]

In this study, we investigate some new analytical solutions to the (1 + 1)-dimensional nonlinear Dispersive Modified Benjamin-Bona-Mahony equation and the (2 + 1)-dimensional cubic Klein-Gordon equation by using the generalized Kudryashov method. After we submitted the general properties of the generalized Kudryashov method in Section 2, we applied this method to these problems to obtain some new analytical solutions, such as rational function solutions, exponential function solutions and hyperbolic function solutions in Section 3. Afterwards, we draw two- and three-dimensional surfaces of analytical solutions by using Wolfram Mathematica 9. [ABSTRACT FROM AUTHOR]

In this paper, we seek exact solutions of generalized Zakharov system. We use extended trial equation method to obtain exact solutions of this system. Consequently, we obtain some exact solutions including soliton solutions, rational, Jacobi elliptic and hyperbolic function solutions of this system by using extended trial equation method. [ABSTRACT FROM PUBLISHER]

In this paper, modified exp(-Ω(ξ))-expansion function method has been handled for finding exact solutions of positive Gardner-KP equation. Hyperbolic function solutions and dark soliton solution of positive Gardner-KP equation have been obtained by means of this method. Moreover, by the help of Mathematica 9, certain graphical notations were given to clarify the action of these solutions. [ABSTRACT FROM AUTHOR]

In this article, we employed the powerful sine-Gordon expansion method in obtaining analytical solutions of the Benjamin-Bona-Mahony equation. We obtain some new solutions with the hyperbolic function structures. Benjamin-Bona-Mahony equation has a wide range of applications in modelling long surface gravity waves of small amplitude. We also plot the 2- and 3-dimensional graphics of all analytical solutions obtained in this paper. On the other hand, we analyze the finite difference method and operators, we obtain discretize equation using the finite difference operators. We consider one of the analytical solutions to the Benjamin-Bona-Mahony equation with the new initial condition. We observe that finite difference method is stable when Fourier-Von Neumann technique is used. We also analyze the accuracy of the finite difference method with terms of the errors $$L_{2}$$ and $$L_{\infty }$$ . We use the finite difference method in obtaining the numerical solutions of the Benjamin-Bona-Mahony equation. We compare the numerical results and the exact solution that are obtained in this paper, we support this comparison with the graphic plot. We perform all the computations and graphics plot in this study with the help of Wolfram Mathematica 9. [ABSTRACT FROM AUTHOR]

In this paper, we seek for the new soliton solutions to the coupled nonlinear Maccari's system by using sine-Gordon expansion method. Sine-Gordon expansion method is based on sine-Gordon Eq. (2.1) and a travelling wave transformation. We obtain new complex hyperbolic functions solutions to this system with help of Wolfram Mathematica 9, we also plot 3-dimensional and 2-dimensional graphics to some obtained solution using the same program in Wolfram Mathematica 9. Finally, we submit a comprehensive conclusions. [ABSTRACT FROM AUTHOR]

In this study, we consider the t a n ( F ( ξ ) 2 ) -expansion methodfor the traveling wave solutions of the potential KdV equation (pKdV)and the (3 + 1)-dimensional shallow water wave equation (SWWE). We find some traveling wave solutions such as trigonometric function, hyperbolic function, exponential function solutions. Then, we showthe two and three dimensional surfaces for some traveling wave solutions obtained in this study by the Mathematica. [ABSTRACT FROM AUTHOR]

Modelling of wave motion and propagation characteristics of waves plays an important role in coastal, ocean and maritime engineering, and for most geographical areas, waves are the major source of environmental actions on beaches or on man-made fixed or floating structures. The travelling wave solutions are described by the hyperbolic functions, the trigonometric functions and the rational functions. In this paper, exact travelling wave solutions of Hirota equation and Hirota–Maccari system are successfully surveyed. Extended trial equation method and generalized Kudryashov method are applied to gain some exact travelling wave solutions of Hirota equation and Hirota–Maccari system. Firstly, we find some exact travelling wave solutions of Hirota equation and Hirota–Maccari system with the aid of extended trial equation method. Secondly, we obtain dark soliton solutions of Hirota equation and Hirota–Maccari system by means of generalized Kudryashov method. Finally, for certain parameter values, we plot two and three dimensional graphics of imaginary and real values of some solutions that we obtained by using these methods. [ABSTRACT FROM AUTHOR]

In this work, we survey exact solutions of Sasa–Satsuma equation (SSE). We utilize extended trial equation method (ETEM) and generalized Kudryashov method to acquire exact solutions of SSE. First of all, we gain some exact solutions such as soliton solutions, rational, Jacobi elliptic, and hyperbolic function solutions of SSE by means of ETEM. Furthermore, we procure dark soliton solution of this equation by the help of generalized Kudryashov method. Lastly, for certain parameter values, we draw two- and three-dimensional graphics of imaginary and real values of some exact solutions that we achieved using these methods. [ABSTRACT FROM PUBLISHER]

In this study, some solitary wave solutions of the Wu-Zhang system are analyzed using the modified expansion function method and the sine–Gordon expansion method. Solitary wave solutions of this non-linear mathematical model consisting of hyperbolic and trigonometric function structures are get. Two and three dimensional, density graphics of the solitary solutions of the mathematical models are drawn by choosing the appropriate parameters. It was seen that all solution functions found as a result provide the mathematical model. In this study, Wolfram Mathematica software program was used for all mathematical calculations. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the (2 + 1) -dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation. Based on bilinear method, the multiple rogue wave solutions and the novel multiple soliton solutions are constructed by giving some specific activation functions in the considered model. By means of symbolic computation, these analytical solutions and corresponding rogue waves are obtained, via Maple 18. By utilizing improved tan (Φ (ρ) ∕ 2) -expansion technique the series of novel exact solutions in terms of rational, periodic and hyperbolic functions for the fractional cases are derived. Also, the semi-inverse variational principle is offered to get the solitary solutions. We construct the exact lump and rogue wave solutions, by solving the under-determined nonlinear system of algebraic equations for the specified parameters. Via various three-dimensional plots, curve plots, density plots and contour plots, dynamical characteristics of these waves are represented. [ABSTRACT FROM AUTHOR]

This study focuses on acquiring new complex hyperbolic and trigonometric function solutions to the generalized conformable fractional Gardner equation by using two distinct integration schemes, namely: the modified Kudryashov method and the hyperbolic function method with the help of symbolic computation package. We generate new closed-form solitary wave solutions to the governing model. Implemented schemes are pragmatically effective for solving various types of nonlinear partial differential equations with integer or fractional order arising in various fields of applied sciences. [ABSTRACT FROM AUTHOR]