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There is no denying fact that harmonic crystals, cold plasma or liquids and compressible fluids are usually dependent of acoustic-gravity waves, acoustic waves, hydromagnetic waves, surface waves with long wavelength and few others. In this context, the exact solutions of the modified Camassa-Holm equation have been successfully constructed on the basis of comparative analysis of (G′ / G − 1 / G) and (1 / G′)-expansion methods. The (G′ / G − 1 / G) and (1 / G′)-expansion methods have been proved to be powerful and systematic tool for obtaining the analytical solutions of nonlinear partial differential equations so called modified Camassa-Holm equation. The solutions investigated via (G′ / G − 1 / G) and (1 / G′)-expansion methods have remarkably generated trigonometric, hyperbolic, complex hyperbolic and rational traveling wave solutions. For the sake of different traveling wave solutions, we depicted 3-dimensional, 2-dimensional and contour graphs subject to the specific values of the parameters involved in the governing equation. Two methods, which are important instruments in generating traveling wave solutions in mathematics, were compared both qualitatively and quantitatively. In addition, advantages and disadvantages of both methods are discussed and their advantages and disadvantages are revealed.

Computational and travelling wave solutions provide significant improvements in accuracy and characterize novelty of imposed techniques. In this context, computational and travelling wave solutions have been traced out for Tzitzéica and Dodd-Bullough-Mikhailov equations by means of (1/G′)-expansion method. The different types of solutions have constructed for Tzitzéica and Dodd-Bullough-Mikhailov equations in hyperbolic form. Moreover, solution function of Tzitzéica and Dodd-Bullough-Mikhailov equations has been derived in the format of logarithmic nature. Since both equations contain exponential terms so the solutions produced are expected to be in logarithmic form. Traveling wave solutions are presented in different formats from the solutions introduced in the literature. The reliability, effectiveness and applicability of the (1/G′)-expansion method produced hyperbolic type solutions. For the sake of physical significance, contour graphs, two dimensional and three dimensional graphs have been depicted for stationary wave. Such graphical illustration has been contrasted for stationary wave verses traveling wave solutions. Our graphical comparative analysis suggests that imposed method is reliable and powerful method for obtaining exact solutions of nonlinear evolution equations.

In this article, the Sinh–Gordon function method and sub-equation method are used to construct traveling wave solutions of modified equal width equation. Thanks to the proposed methods, trigonometric soliton, dark soliton, and complex hyperbolic solutions of the considered equation are obtained. Common aspects, differences, advantages, and disadvantages of both analytical methods are discussed. It has been shown that the traveling wave solutions produced by both analytical methods with different base equations have different properties. 2D, 3D, and contour graphics are offered for solutions obtained by choosing appropriate values of the parameters. To evaluate the feasibility and efficacy of these techniques, a nonlinear evolution equation was investigated, and with the help of symbolic calculation, these methods have been shown to be a powerful, reliable, and effective mathematical tool for the solution of nonlinear partial differential equations.

In this study, numerical solutions of the fractional Harry Dym equation are investigated. Linearization techniques are utilized for non-linear terms existing in the fractional Harry Dym equation. The error norms L2 and L∞ are computed. Stability of the finite difference method is studied with the aid of Von Neumann stabity analysis.

In this work, new traveling wave solutions are generated for the system that models nematicons in liquid crystals using the (1 ∕ G ′) -expansion method. In the equation system, nonlinearity is taken into account for Kerr law and Power law. Also, the existence of exact solutions under restriction conditions is guaranteed. We suggest that the solutions produced are of a different type than the solutions in the literature. Figures representing the intensity of the produced traveling wave solutions are presented. In addition, the simulation of the solitary wave is made for different values of the parameter that affects the inclination angle of the molecules in the nematicons mechanism in liquid crystals. How classical solitary wave behavior translates into triangular wave behavior is discussed. We believe this paper will provide an important perspective on the problems encountered in various application areas such as fluid dynamics, chemical engineering, chaos and complex networks in terms of investigating different mechanisms by taking into account nonlinearity factors. [ABSTRACT FROM AUTHOR]

In this study, the coupled Higgs system, which is a special case of the coupled Higgs field equation, which is effective in energy transport in the sub‐particles of the atom, is discussed. With the help of the modified generalized exponential rational function method, which is an important instrument in obtaining traveling wave solutions, both the propagating wave solutions and general form solutions of coupled Higgs system are presented. These solutions are examined under some restrictive conditions as they are presented. It is argued that these solutions differ from the literature. The advantages and disadvantages of the method discussed in the conclusion and discussion section are discussed. In addition, the changes in the behavior of the traveling wave solution are discussed by giving physical meaning to some constants in the traveling wave solutions produced by the method. The effects on the traveling wave solution are analyzed for different values of wave number, wave velocity, and wave frequency, which have physically important meanings. In addition, the behaviors caused by these effects are supported with the help of simulation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate solutions to the (3 + 1) -dimensional B-type Kadomtsev–Petviashvili–Boussinesq (B-KPB) equation, which has strong nonlinearity feature by using sub-equation and the modified (1 / G ′) -expansion methods. Using these analytical methods, dark soliton, trigonometric oscillation, singular soliton, algebraic and hyperbolic type oscillating traveling wave solutions are presented. By giving special values to the parameters in the obtained traveling wave solutions, 3D, 2D and contour graphs representing the solitary waves are drawn. The advantages and disadvantages of both analytical methods are presented in comparison with each other. In addition, the changes of the solutions of the B-KPB equation according to the velocity parameter and the refraction phenomenon are supported by simulation. In addition, the solutions of the equation discussed are first discussed in detail, physical interpretations. Secondly, the geometric interpretations of the traveling wave solutions are analyzed and it is shown that the graphs of the solutions correspond to flat hyper-surfaces. It is argued that these hyper-surfaces can be leveled on a hyperplane without distortion. [ABSTRACT FROM AUTHOR]

Copyright of Adiyaman University Journal of Science & Technology / Adıyaman Üniversitesi Fen Bilimleri Dergisi is the property of Adiyaman University, Institute of Science / Adiyaman Universitesi Fen Bilimleri Enstitusu and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this study, the modified (1 / G ′) -expansion method and modified sub-equation method have been successfully applied to the fractional Benjamin–Ono equation that models the internal solitary wave event in the ocean or atmosphere. With both analytical methods, dark soliton, singular soliton, mixed dark-singular soliton, trigonometric, rational, hyperbolic, complex hyperbolic, complex type traveling wave solutions have been produced. In these applications, we consider the conformable operator to which the chain rule is applied. Special values were given to the constants in the solution while drawing graphs representing the stationary wave. By making changes of these constants at certain intervals, the refraction dynamics and physical interpretations of the obtained internal solitary waves were included. These physical comments were supported by simulation with 3D, 2D and contour graphics. These two analytical methods used to obtain analytical solutions of the fractional Benjamin–Ono equation have been analyzed in detail by comparing their respective states. By using symbolic calculation, these methods have been shown to be the powerful and reliable mathematical tools for the solution of fractional nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

Copyright of Adiyaman University Journal of Science & Technology / Adıyaman Üniversitesi Fen Bilimleri Dergisi is the property of Adiyaman University, Institute of Science / Adiyaman Universitesi Fen Bilimleri Enstitusu and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, sub equation and ð1=G'Þ expansion methods are proposed to construct exact solutions of a nonlinear longitudinal wave equation (LWE) in a magneto-electro-elastic circular rod. The proposed methods have been used to construct hyperbolic, rational, dark soliton and trigonometric solutions of the LWE in the magnetoelectro-elastic circular rod. Arbitrary values are given to the parameters in the solutions obtained. 3D, 2D and contour graphs are presented with the help of a computer package program. Solutions attained by symbolic calculations revealed that these methods are effective, reliable and simple mathematical tool for finding solutions of nonlinear evolution equations arising in physics and nonlinear dynamics. [ABSTRACT FROM AUTHOR]

In this study, the (1/G') -expansion method was implemented to solve the Benney-Luke (BL) equation. Exact solutions of the BL equation were obtained via this method. The solutions obtained from the BL equation were in hyperbolic form. 3D, 2D and contour graphs of obtained solutions are presented. Results show that the (1/G') - expansion method provides an efficient and straightforward mathematical instrument for finding solutions of nonlinear evolution equations (NLEEs). [ABSTRACT FROM AUTHOR]

The purpose of this article is obtaining the exact solutions for (3+1)-dimensional Jimbo- Miwa Equation (3+1DJME). The (1/G')-expansion method which is an effective method in solving nonlinear evolution equations (NLEEs) is used. Then, 3D, contour and 2D graphics are presented by giving special values to the constants in the solutions obtained. These graphics are a special solution of the (3+1DJME) and represent a stationary wave of the equation. Ready computer package program is used to obtain the solutions and graphics presented in this study. [ABSTRACT FROM AUTHOR]

The traveling wave solutions of the combined Korteweg de Vries-modified Korteweg de Vries (cKdV-mKdV) equation and a complexly coupled KdV (CcKdV) equation are obtained by using the auto-Bäcklund Transformation Method (aBTM). To numerically approximate the exact solutions, the Finite Difference Method (FDM) is used. In addition, these exact traveling wave solutions and numerical solutions are compared by illustrating the tables and figures. Via the Fourier–von Neumann stability analysis, the stability of the FDM with the cKdV–mKdV equation is analyzed. The L 2 and L ∞ norm errors are given for the numerical solutions. The 2D and 3D figures of the obtained solutions to these equations are plotted. [ABSTRACT FROM AUTHOR]

In this study, by using the finite difference method (FDM for short) and operators, the discretized Cahn-Allen equation is obtained. New initial condition for the Cahn-Allen equation is introduced, considering the analytical solution given in Application of the modified exponential function method to the Cahn-Allen equation, AIP Conference Proceedings 1798, 020033 [1]. It is shown that the FDM is stable for the usage of the Fourier-Von Neumann technique. Accuracy of the method is analyzed in terms of the errors in L2 and L∞. Furthermore, the FDM is treated in order to obtain the numerical results and to construct a table including numerical and exact solutions as well as absolute measuring error. A comparison between the numerical and the exact solutions is supported with two and three dimensional graphics via Wolfram Mathematica 11. [ABSTRACT FROM AUTHOR]

In this study, we investigate the nonlinear time-fractional Korteweg–de Vries (KdV) equation by using the (1/G ′ )-expansion method and the finite forward difference method. We first obtain the exact wave solutions of the nonlinear time-fractional KdV equation. In addition, we used the finite-forward difference method to obtain numerical solutions in this equations. When these solutions are obtained, the indexed forms of both Caputo and conformable derivatives are used. By using indexing technique, it is shown that the numerical results of the nonlinear time-fractional KdV equation approaches the exact solution. The two- and three-dimensional surfaces of the obtained analytical solutions are plotted. The von Neumann stability analysis of the used numerical scheme with the studied equation is carried out. The L2 and L ∞ error norms are computed. The exact solutions and numerical approximations are compared by supporting with graphical plots and tables. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to find an exact solution of the traveling wave equation of a nonlinear time fractional Burgers' equation using the expansion method and the Cole-Hopf transformation. For this purpose, a nonlinear time fractional Burgers' equation with the initial conditions considered. The finite difference method (FDM for short) which is based on the Caputo formula is used and some fractional differentials are introduced. The Burgers' equation is linearized by using the Cole-Hopf transformation for a stability of the FDM. It shows that the FDM is stable for the usage of the Fourier-Von Neumann technique. Accuracy of the method is analyzed in terms of the errors in L2 and L∞. All of obtained results are discussed with an example of the Burgers' equation including numerical solutions for different situations of the fractional order and the behavior of potentials u is investigated with graphically. All the obtained numerical results in this study are presented in tables. We used the Mathematica software package in performing this numerical study. [ABSTRACT FROM AUTHOR]