Abstract
This research paper focuses on the application of the tanh function method to find the soliton solutions of the (2+1)-dimensional nonlinear electrical transmission line model. Materials used to form a transmitting line are very important to transmit electric charge. In this sense, we find some new voltage behaviors such as dark, trigonometric, and complex function solutions. Choosing some suitable values of parameters, we present some various surfaces of results obtained in this paper. These results play an important role in telecommunications lines used to stand for wave propagations.
1. Introduction
Nonlinear partial differential equations (NPDEs) are recently used to investigate the meanings of physical problems such as fluid dynamics, mathematics, physics and quantum field theory, and nonlinear fiber optics. Systems of NPDEs are also considered as a main tool to investigate in chemical and biological experiments. Some methods such as index the extended tanh method [1, 2], the sine cosine method [3], the inverse scattering transform method [4], finite difference method [5], tanh and extended tanh methods [6–8], Jacobi elliptic function expansion method [9], modified expansion method [10], generalized tanh method [11, 12], sine-Gordon expansion method [13], extended mean value theorem [14], and interval-valued fuzzy topsis method [15] to obtain various solutions of such NPDEs have been presented in the literatures. Cordero et al. have observed the stability analysis of the fourth-order iterative method in [16]. The (3+1) Dimensional Boiti-Leon-Manna-Pempinelli equation has been deeply studied in [17]. Using time scale calculus, discrete normal vector field approximation has been presented in [18]. A Handy Technique has been handled in [19]. Classical Boussinesq equations have been immensely studied in [20]. Traveling wave solutions of nonlinear Klein Gordon equation were observed in [21] and so on [15, 17, 18, 22–43].
The contents of this paper are as follows. Section 2 presents the general properties of the tanh function method (TFM) [44]. This TFM has been proposed as a strong and creditable method for finding the solutions of the nonlinear models. Section 3 introduces some new complex dark, trigonometric, and hyperbolic soliton solutions to the nonlinear electrical transmission line. To express physical properties in terms of mathematical dynamics, such as wave propagation of electrical transmission lines, has been presented by equation defined as [9]
where is used to explain the tightness throughout the electrical line and and are interpreted like the promulgation distance. is the period, and and are constants with nonzero. is the space between two proximate sections in during longitudinally flank, while is the space between two proximate sections in the transversal flank [9]. With the help of some computational programs, we are able to plot in terms of 2D, 3D, and contour surfaces of the results theoretically found. Finally, the main conclusions are given in the last section of the paper.
2. The Tanh Function Method
In this part of the paper, we present the general properties of the tanh function method in a detailed manner [45–48]. where is a polynomial in the dependent variable . Considering the traveling wave transformation as , , we obtain the following nonlinear ordinary differential equation with is a polynomial of . Now, finding the traveling wave solutions to Eq. (2) is equivalent to obtain the solution to reduced ordinary differential Eq. (3), and it can be introduced a new independent variable defined as
We can find the following for some derivations as
As the last step, we present the tanh series as being where is a positive integers. The values of , generally, with the help of the balance principle can be determined.
3. Mathematical Analysis
In this part of the paper, we find some new complex dark, trigonometric, and hyperbolic function solutions of Eq. (1) by using TFM. First of all, we consider the traveling wave transformation defined as where , are nonzero constants or complex-valued parameters. When considering Eq. (7) into Eq. (1), we find
Integrating Eq. (8) twice with regard to , setting the constants of integrations to zero yields
According to the general properties of TFM, it can be considered as
Substituting Eqs. (5), (10) into Eq. (9) gives
Using the balance rule, can be found as which result in
Substituting Eq. (13) into Eq. (11) by getting necessary derivations presents
After some calculations, it can be obtained as follows:
Y0: ,
Y1: ,
Y2: ,
Y3: .
Solving this system yields the following cases:
Case 1. When we get the following complex trigonometric function solution in which , , , , and are constant and nonzero. Choosing suitable values of coefficients in Eq. (16), we can observe some (Figures 1 and 2).
Case 2. Choosing as we get another new complex dark function solution with strain conditions are , , and also are real constants and nonzero or complex-valued parameters. Considering some values of parameters under the strain conditions, different wave patterns can be observed from (Figures 3 and 4) for Eq. (18).
Case 3. Selecting we get conjugate new complex dark function solution as with strain conditions are , , and also are real constants and nonzero or complex-valued parameters. 3D, 2D, and contour surfaces of Eq. (20) can be also seen (Figures 5–7) with the strain conditions.
Case 4. Choosing as produces the following dark soliton solution in which and strain conditions are are real constants and nonzero. 3D, 2D, and contour surfaces of Eq. (22) can be also observed (Figures 8 and 9).
Case 5. Taking as produces another complex dark traveling wave solution to the governing model with strain conditions and are real constants and nonzero. 3D, 2D, and contour surfaces of Eq. (24) can be also observed in (Figures 10–12).
Case 6. Once we select other coefficients given as results in another complex dark traveling wave solution to the Eq. (1) with strain conditions and are real constants and nonzero. Various surfaces of Eq. (26) with the considering suitable values of parameters can be also presented as (Figures 13–15).
4. Conclusions
In this manuscript, TFM being one of the powerful techniques has been successfully used to Eq. (1). Many new trigonometric, complex, and hyperbolic function solutions have been extracted, afterwards. The conditions which guarantee the existence of the valid solutions to this model are also given in a detailed manner. Considering the strain conditions of coefficients of results, various simulations have been also plotted by using some computational programs. These solutions are of various physical properties of the electrical transmission line. For example, the tanh function arises in gravitational potential as a dark structure [49]. Hence, it is estimated that the solution of is of such physical property. From (Figures 1–15), it can be also seen that the results simulate estimated wave behaviors. TFM used in this paper can be considered to solve other nonlinear problems arising in the theory of solitons and other areas of nonlinear science [50–56].
Data Availability
This work is not based on any data.
Conflicts of Interest
The authors declare that there is no conflict of interest.
Acknowledgments
This paper has been partially supported by Fundación Séneca (Spain), grant 20783/PI/18, and Ministerio de Ciencia, Innovación y Universidades (Spain), grant PGC2018-097198-B-I00. Secondly, the first author extends his special thanks to Harran University because this projected work was partially (not financial) supported by Harran University with the project HUBAP ID: 20124.