Sound & Vibration |

DOI: 10.32604/sv.2021.014157

ARTICLE

Modeling of Dark Solitons for Nonlinear Longitudinal Wave Equation in a Magneto-Electro-Elastic Circular Rod

1Department of Computer Engineering, Faculty of Engineering, Ardahan University, Ardahan, Turkey

2Department of Mathematics, Faculty of Science, Firat University, Elazig, Turkey

3Department of Mathematics, Istanbul Commerce University, Istanbul, Turkey

4Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan

*Corresponding Author: Hijaz Ahmad. Email: hijaz555@gmail.com

Received: 03 September 2020; Accepted: 28 September 2020

Abstract: In this paper, sub equation and

Keywords: (1/G’)-expansion method; sub equation method; exact solution; traveling wave solution; nonlinear evolution equations

Nonlinear evolution equations (NLEEs) are used in various fields such as biological sciences, plasma physics, quantum mechanics, fluid dynamics and engineering. Many methods have been used to obtain solutions of NLEEs from past to present.

In particular,

We know that solitons have an important place in wave theory. There are many solitons that offer a mathematical perspective to many physical phenomena. Some of these are dark soliton, bright soliton, peaked solitary, topological soliton, non-topological soliton, singular soliton and so on. The mathematical expressions of these solitons appear as a solution of NLEEs.

It is very difficult to obtain the analytical solution of NLEEs. However, with the help of classical wave transformation, traveling wave solutions can be obtained by converting to ordinary differential equations. Traveling wave solutions, which have an important place in wave theory and contain many physical events, are important for mathematics. Different types of traveling wave solutions are available with different methods. Some of these methods are auxiliary equation method [1],

Let’s take the LWE in a magneto-electro-elastic circular rod [39]

where

The real-world physical response of a magneto-electro-elastic circular rod LWE is the combination of piezomagnetic and piezoelectric BaTiO3 [39]. The solutions offered especially for those working in this field are important. It will become more important with the physical meaning of the constants in the solution.

In this study, some researchers have examined the physically precious LWE. In the study of Iqbal et al. wave solutions have been obtained with extended auxiliary equation mapping and extended direct algebraic mapping methods [39]. Ilhan et al. have provided solutions including complex, hyperbolic and trigonometric functions with sine-Gordon expansion method [40]. Baskonus et al. have been presented topological, non-topological and singular soliton solutions using the extended sinh-Gordon equation expansion method [41]. Also, Baskonus et al. have obtained hyperbolic, complex and complex hyperbolic function solutions with the modified exp expansion function method [42]. In their study, Yang et al. achieved solitary wave solutions that peaked using direct integration with the boundary condition and symmetry condition [43]. Younis et al. have been presented dark, bright and singular solitons solutions with the solitary wave ansatz method [44].

In this study, we will present different solutions from the solutions presented in the literature. In particular, we offer a different solution than the dark solitons that Younis et al. present in their work.

Consider the sub-equation method for the solving NLEEs. Regard the NLEEs as

Applying the wave transmutation

Eq. (2) converts into ODE

where w is arbitrary constant. In the form it is supposed that Eq. (4) has a solution

in here

where

In Eq. (4), if we apply the Eqs. (6) and (5), we attained the new polynomial with respect

Using this analytical method, trigonometric provides solutions of hyperbolic and algebraic type. These solutions are in Eq. (7) formats. Especially our tanh solution contains dark soliton feature [45]. This method is a reliable, effective and powerful analytical method in obtaining the analytical solution of many differential equations.

Consider a general form of NLEEs,

Let

The solution of Eq. (9) is assumed to have the form

where

where

where

After calculating the n balancing term, the structure of the solution function of the assumed Eq. (10) emerges. The necessary derivatives of this solution are taken and replaced in the Eq. (9), and after some algebraic operations, a polynomial that accepts the expression

4 Application of Sub-Equation Method

If we apply the transform in the Eq. (3) to the Eq. (1), we find

or

If we take the integral twice according to

In Eq. (15), we get

If the equation given by (16) is placed in the Eq. (15) and the necessary arrangements are made, we can write the following equation system:

Case 1: If

Substituting values (18) into (16), we can also present the dark soliton for Eq. (1) using the classical wave transformation inverse, that is, using the

Case II: If

Substituting values (20) into (16), we can also present the singular for Eq. (1) using the classical wave transformation inverse, that is, using the

Case III: If

Substituting values (22) into (16), we attain trigonometric soliton for Eq. (1)

Case IV: If

Substituting values (24) into (16), we attain trigonometric soliton for Eq. (1)

Case V: If

Substituting values (26) into (16), we attain rational traveling wave solution for Eq. (1)

5 Application of (1/G’)-Expansion Method

Considering Eq. (15), we get balancing term

Replacing Eq. (28) into Eq. (15) and the coefficients of the algebraic Eq. (1) are equal to zero, can find the following algebraic equation systems:

Case 1.

Replacing values Eq. (30) into Eq. (28) and we have the following hyperbolic type solutions for Eq. (1):

In this study, the LWE of a magneto-electro-elastic circular rod has been successfully produced using two different analytical methods. These solutions are emphasized to be hyperbolic, rational, dark soliton and trigonometric type traveling wave solutions. Generating the solutions of this equation is mathematically valuable as much in terms of physical meaning. The u solutions presented in this article represent the electrostatic potential of the magneto-electro-elastic circular rod. Also, using this potential, a different physical perspective can be presented. If we represent the pressure of this physical event with P, the P pressure can be calculated for different analytical solutions as follows:

The potential u calculated here and the P pressure magneto-electro-elastic circular rod can offer different interpretations and different perspectives on the physical event [46–49].

The figures presented in this work are the graphs of solitons representing the standing wave. Fig. 1 represents the dark soliton, Fig. 2 represents the singular soliton, Figs. 3 and 4 represent the trigonometric solitons, Fig. 5 represents the rational soliton, and Fig. 6 represents the graphics of hyperbolic type solitons.

In this study, we have achieved hyperbolic, rational, dark soliton and trigonometric traveling wave solutions for the LWE in a magneto-electro-elastic circular rod using sub equation method and

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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