New Complex Solutions to the Nonlinear Electrical Transmission Line Model

Mehmet Tahir Gulluoglu

doi:10.1515/phys-2019-0074

Article Open Access

New Complex Solutions to the Nonlinear Electrical Transmission Line Model

Mehmet Tahir Gulluoglu

Published/Copyright: December 31, 2019

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From the journal Open Physics Volume 17 Issue 1

Abstract

In this paper, with the help of an analytical approach, new complex singular and travelling dark solutionsto the nonlinear electrical transmission line are successfully constructed. 2D and 3Dfigures along with contour figures are plotted. Finally, at the end of manuscript, general conclusions about these novel findings, which differ from existing results, are given.

Keywords: Nonlinear electrical transmission line equation; Improved Bernoulli sub-equation function method; Complex solutions; Contour surfaces

PACS: 02.30.Jr; 02.30.Hq

1 Introduction

Energy is one of the most important requirementsof people from all over the world. “The demand for energy and natural resources is increasing rapidly in conjunction with rising population, industrialization, and urbanization as well as the growth in production and commercial opportunities resulting from globalization [1].” Furthermore, networks, which are related to energy, are another requirement for people. Therefore, they are an outstanding connection tool among people in the modern day. These two concepts,energy and networks, are used in many fields of science and real world problems arising in all aspect of daily life.Research conducted on these fields has attracted the attention of scientists from all over the world, and continues to be studied in the literature.

H. Reda et al. have investigated the wave propagation in pre-deformed periodic network materials based on large strains homogenization [2]. Wave emitting and propagation on neural networks have been investigated by J. Ma et al. in 2015 [3]. Z. Rostami et al. have studied a deepsearchfor propagating waves in a neural network using magnetic radiation [4]. Denys Dutykh and Jean-Guy Caputo have observedwave propagation on the wave dynamics on networks [5]. Atte Aalto and Jarmo Malinen have introduced that wave propagation on networks solvable (forward in time) and energy passive or conservative with the help of a theoretical approach [6]. Therefore, many novel models have been presented to the literature by many scientists [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32].

One such novel model, voltage wave propagation of electrical transmission lines, has been presented by E. Tala-Tebue as

(1) v t t − α v 2 t t + β v 3 t t − ϖ 0 2 δ 1 2 v x x − ϖ 0 2 δ 1 4 12 v x x x x − ω 0 2 δ 2 2 v y y − ω 0 2 δ 2 4 12 v y y y y = 0 ,

where α, β, ϖ₀, δ₁, ω₀, δ₂ are real, non-zero constants while v = v (x, y, t) is the voltage in the transmission lines [7, 8]. This nonlinear electrical transmission line model (NETLM) has been used to symbolize the wave propagation on the network system [7, 8].

This manucript is composed of the following three sections. In section 2, we present the Improved Bernoulli Sub-Equation function method (IBSEFM) in a detailed manner. We apply IBSEFM to find new complex travelling wave solution in section 3. In the last section of paper, we introduce a comprehensive conclusion.

2 General properties of IBSEFM

The general properties of IBSEFM are given as follows:

Step 1. The following nonlinear model in two variables and a dependent variable u can be considered:

(2) P v x , v y , v t , v x y t , ⋯ = 0 ,

and taking the travelling wave transformation

(3) v x , y , t = V ξ , ξ = k x + y − c t ,

in which k, c are real constants and non-zero. Substituting Eq. (3) in Eq. (2) yields a nonlinear ordinary differential equation (NLODE) as follows;

(4) N V , V ′ , V ″ , ⋯ = 0 ,

where V = V ξ , V ′ = d V d ξ , V ″ = d 2 V d ξ 2 , ⋯ .

Step 2. Taking the trial solution for Eq. (4) as follows [9, 10, 11, 12, 13]:

(5) V = ∑ i = 0 n a i F i ∑ j = 0 m b j F j = a 0 + a 1 F + a 2 F 2 + ⋯ + a n F n b 0 + b 1 F + b 2 F 2 + ⋯ + b m F m ,

and

(6) F ′ = p F + d F M ,

where F = F (ξ) is a Bernoulli differential polynomial and p ≠ 0, d ≠ 0, M ∈ − {0, 1, 2}. Putting Eqs. (5, 6) into Eq. (4) produces an equation of polynomial Ω (F)of F as follows:

(7) Ω F = ρ s F s + ⋯ + ρ 1 F + ρ 0 = 0.

We can obtain a relationship between n, m and M under the rules of the balance principle.

Step 3. Setting the coefficients of Ω (F) all equal to zero gives an algebraic system of equations;

(8) ρ i = 0 , i = 0 , ⋯ , s .

Solving this system, we obtain the values of a₀, a₁, · · · , a_n and b₀, b₁, · · · , b_m

Step 4. When we solve Eq. (6), we obtain the following two situations according to the values of p and d;

(9) F ξ = − d p + E e p M − 1 ξ 1 1 − M , p ≠ d ,

(10) F ξ = E − 1 + E + 1 tanh p 1 − M ξ 2 1 − tanh p 1 − M ξ 2 1 1 − M , p = d , E ∈ .

Substituting Eq. (5) into Eq. (4), we can find the polynomial of F. Considering all the coefficients of the same power of F to be zero gives an algebraic system of equations. By solving this system with the help of various computational programs, we can find some new values of parameters. This process gives many solutions to the model considered. For a better understanding of the solutions obtained in this manner, we plot 2D, 3D and contour graphs of results with the suitable values of ₀₂parameters.

3 Application of the IBSEFM

In this section, IBSEFM has been successfully considered to the NETLM to find additional novel complex solutions.

Example

Let’s consider the travelling wave transformation as following;

(11) v = v x , y , t = V ξ , ξ = k x + y − c t ,

where k, c are real constants and non-zero. Substituting Eq. (11) into Eq. (1) produces the following NLODE;

(12) c 2 − ω 0 2 δ 2 2 − ϖ 0 2 δ 1 2 k 2 V ″ − 2 α k 2 c 2 V V ′ ′ + 3 β k 2 c 2 V 2 V ′ ′ − ϖ 0 2 δ 1 4 + ω 0 2 δ 2 4 k 4 12 V 4 = 0.

Integrating twice and considering the zero for both constants, Eq. (12) can be rewritten as

(13) 12 c 2 − ϖ 0 2 δ 1 2 − ω 0 2 δ 2 2 V + 12 β c 2 V 3 − 12 α c 2 V 2 − k 2 ϖ 0 2 δ 1 4 + ω 0 2 δ 2 4 V ″ = 0

With the balance principle for V^′′ and V³, we obtain the relationship among n, m and M as

(14) M + m = n + 1.

Case 1: Choosing M = 3, n = 3 and m = 1, we can find V and its derivatives from Eq. (5) as follows:

(15) V = a 0 + a 1 F + a 2 F 2 + a 3 F 3 b 0 + b 1 F = Υ Ψ ,

(16) V ′ = Υ ′ Ψ − Υ Ψ ′ Ψ 2 ,

(17) V ″ = Υ ″ Ψ − Υ Ψ ′ Ψ 2 − Υ Ψ ′ ′ Ψ 2 − 2 Υ Ψ ′ 2 Ψ Ψ 4 , ⋮

where F^′ = pF + dF³, a₃ ≠ 0, b₁ ≠ 0, p ≠ 0, d ≠0. When we use Eq. (17) in Eq. (13), we obtain a system of algebraic equations. By solving this system of equations with the help of software programs, we find the coefficients, which give the complex solutions to the NETLM as follows.

Case 1a.

(18) a 0 = p a 2 d , a 1 = p a 2 b 1 d b 0 , a 3 = a 2 b 1 b 0 , c = 2 d 2 b 0 2 w 0 2 δ 1 2 + ω 0 2 δ 2 2 − p 2 β a 2 2 + 2 d 2 b 0 2 , α = 3 p β a 2 2 d b 0 , k = − i a 2 3 β w 0 2 δ 1 2 + δ 2 2 ω 0 2 p 2 β a 2 2 − 2 d 2 b 0 2 w 0 2 δ 1 4 + δ 2 4 ω 0 2 ,

we have

(19) v 1 x , y , t = E a 2 p 2 e i 2 a 2 3 β w 0 2 δ 1 2 + δ 2 2 ω 0 2 p 2 β a 2 2 − 2 d 2 b 0 2 w 0 2 δ 1 4 + δ 2 4 ω 0 2 x + y − 2 t d 2 b 0 2 w 0 2 δ 1 2 + ω 0 2 δ 2 2 − p 2 β a 2 2 + 2 d 2 b 0 2 − b 0 d 2 + p d E b 0 e i 2 a 2 3 β w 0 2 δ 1 2 + δ 2 2 ω 0 2 p 2 β a 2 2 − 2 d 2 b 0 2 w 0 2 δ 1 4 + δ 2 4 ω 0 2 x + y − 2 t d 2 b 0 2 w 0 2 δ 1 2 + ω 0 2 δ 2 2 − p 2 β a 2 2 + 2 d 2 b 0 2 .

For a better understanding of the physical meaning of the solution with the help of the complex structure of Eq. (19) in Eq. (1), and with suitable values of parameters, 2D and 3D figures along with contour graphs may be observed in Figures 1, 2, 3, 4.

Figure 1

The 3D graphs of Eq. (19) for a₂ = 0.9, d = 0.2, b₀ = b₁ = 0.3, p = 0.25, δ₁ = δ₂ = 0.5, β = 0.9, E = 0.1, w₀ = 0.7, ω₀ = 0.8, y = 0.03, −2 < x < 2, −2 < t < 5.

Figure 2

The 2D graphs of Eq. (19) for a₂ = 0.9, d = 0.2, b₀ = b₁ = 0.3, p = 0.25, δ₁ = δ₂ = 0.5, β = 0.9, E = 0.1, w₀ = 0.7, ω₀ = 0.8, y = 0.03, t = 0.1, −2 < x < 2

Figure 3

The contour graphs of Eq. (19) for a₂ = 0.9, d = 0.2, b₀ = b₁ = 0.3, p = 0.25, δ₁ = δ₂ = 0.5, β = 0.9, E = 0.1, w₀ = 0.7, ω₀ = 0.8, y = 0.03, −10 < x < 10, −10 < t < 10.

Figure 4

The compilation of contour graphs of both side of Eq. (19) for a₂ = 0.9, d = 0.2, b₀ = b₁ = 0.3, p = 0.25, δ₁ = δ₂ = 0.5, β = 0.9, E = 0.1, w₀ = 0.7, ω₀ = 0.8, y = 0.03, −10 < x < 10, −10 < t < 10.

Case 1b. When

(20) a 0 = a 1 = 0 , a 3 = a 2 b 1 b 0 , c = − w 0 2 δ 1 2 3 + k 2 p 2 δ 1 2 + ω 0 2 δ 2 2 3 + k 2 p 2 δ 2 2 3 , α = − 3 d k 2 p b 0 w 0 2 δ 1 4 + δ 2 4 ω 0 2 a 2 w 0 2 δ 1 2 3 + k 2 p 2 δ 1 2 + a 2 ω 0 2 δ 2 2 3 + k 2 p 2 δ 2 2 , β = 2 d 2 k 2 b 0 2 w 0 2 δ 1 4 + δ 2 4 ω 0 2 a 2 2 w 0 2 δ 1 2 3 + k 2 p 2 δ 1 2 + a 2 2 ω 0 2 δ 2 2 3 + k 2 p 2 δ 2 2 ,

we find the dark solitary wave structure as following;

(21) v 2 x , y , t = p a 2 ( − d b 0 + p b 0 E − 1 − tanh − 2 k p x − 2 k p y − 2 k p t ϖ − 1 + tanh − 2 k p x − 2 k p y − 2 k p t ϖ ) − 1 ,

where ϖ = w 0 2 δ 1 2 3 + k 2 p 2 δ 1 2 + ω 0 2 δ 2 2 3 + k 2 p 2 δ 2 2 3 . With the suitable values of parameters, 2D and 3D figures along with contour graphs may be observed in Figures 5, 6.

Figure 5

The 2D and 3D graphs of Eq. (21) for a₂ = 1, d = 2, b₀ = 4, b₁ = 3, p = 5, k = 0.9, δ₁ = δ₂ = 0.3, E = 2, w₀ = 0.7, ω₀ = 0.8, y = 0.03, −2 < x < 2, −2 < t < 2, and t = 0.85 for 2D surfaces.

Figure 6

The contour graphs of Eq. (21) for a₂ = 1, d = 2, b₀ = 4, b₁ = 3, p = 5, k = 0.9, δ₁ = δ₂ = 0.3, E = 2, w₀ = 0.7, ω₀ = 0.8, y = 0.03, −2 < x < 2, −2 < t < 2.

Case 1c. If we consider

(22) a 0 = 3 b 0 c 2 − w 0 2 δ 1 2 − ω 0 2 δ 2 2 c 2 α , a 1 = 3 b 1 c 2 − w 0 2 δ 1 2 − ω 0 2 δ 2 2 c 2 α , a 3 = a 2 b 1 b 0 , p = 3 d b 0 c 2 − w 0 2 δ 1 2 − ω 0 2 δ 2 2 c 2 α a 2 , k = − i c 2 α a 2 3 d 2 b 0 2 c 2 − w 0 2 δ 1 2 − ω 0 2 δ 2 2 w 0 2 δ 1 4 + δ 2 4 ω 0 2 , β = 2 c 2 α 2 9 c 2 − w 0 2 δ 1 2 − ω 0 2 δ 2 2 ,

we obtain another new complex dark solitary wave structure as

(23) v 3 x , y , t = 3 α − 3 w 0 2 δ 1 2 + ω 0 2 δ 2 2 α c 2 + − 1 + tanh 2 i ϖ x + y − c t − α c 2 3 ζ − 1 + tanh 2 i ϖ x + y − c t + E b 0 a 2 − 1 − tanh 2 i ϖ x + y − c t ,

where ζ = c 2 − w 0 2 δ 1 2 − ω 0 2 δ 2 2 , ϖ = 3 d b 0 ζ κ , κ = − d 2 b 0 2 w 0 2 δ 1 4 + δ 2 4 ω 0 2 For suitable values of parameters, 2D and 3D figures along with contour graphs the complex structure Eq. (23) with Eq. (1) may be observed in Figures 7, 8, 9, 10.

Figure 7

The 3D graphs of Eq. (23) for a₂ = 0.1, d = −0.2, b₀ = b₁ = 0.3, c = 0.12, δ₁ = δ₂ = 0.5, α = 12, E = 0.1, w₀ = 0.7, ω₀ = 0.8, y = 0.03, −2 < x < 2, −5 < t < 5.

Figure 8

The 2D graphs of Eq. (23) for a₂ = 0.1, d = −0.2, b₀ = b₁ = 0.3, c = 0.12, δ₁ = δ₂ = 0.5, α = 12, E = 0.1, w₀ = 0.7, ω₀ = 0.8, y = 0.03, t = 5, −6 < x < 6.

Figure 9

The contour graphs of Eq. (23) for a₂ = 0.1, d = −0.2, b₀ = b₁ = 0.3, c = 0.12, δ₁ = δ₂ = 0.5, α = 12, E = 0.1, w₀ = 0.7, ω₀ = 0.8, y = 0.03, −180 < x < 180, −180 < t < 180.

Figure 10

The compilation of contour graphs of both sides of Eq. (23) for a₂ = 0.1, d = −0.2, b₀ = b₁ = 0.3, c = 0.12, δ₁ = δ₂ = 0.5, α = 12, E = 0.1, w₀ = 0.7, ω₀ = 0.8, y = 0.03, −180 < x < 180, −180 < t < 180.

Case 1d. Taking

(24) a 0 = a 1 = 0 , a 2 = a 3 b 0 b 1 , p = 3 d b 1 − c 2 + w θ 2 δ 1 2 + δ 2 2 ω θ 2 c 2 α a 3 k = − i c 2 α a 3 − 3 d 2 b 1 2 c 2 − w 0 2 δ 1 2 − ω 0 2 δ 2 2 w 0 2 δ 1 4 + δ 2 4 ω 0 2 , β = 2 c 2 α 2 9 c 2 − w 0 2 δ 1 2 − ω 0 2 δ 2 2 ,

we find another new complex wave structure as

(25) v 4 x , y , t = a 3 3 c 2 − 3 w 0 2 δ 1 2 − 3 ω 0 2 δ 2 2 E b 1 3 c 2 − 3 w 0 2 δ 1 2 − 3 ω 0 2 δ 2 2 e − 2 i d b 1 3 c 2 − w 0 2 δ 1 2 − ω 0 2 δ 2 2 d 2 b 1 2 − w 0 2 δ 1 4 − δ 2 4 ω 0 2 x + y − c t + c 2 α a 3 .

2D and 3D figures along with contour graphs for the complex structure Eq. (25) in Eq. (1) may be seen in Figures 11, 12, 13, 14.

Figure 11

The 3D graphs of Eq. (25) for a₃ = 0.1, d = −0.2, b₀ = b₁ = 0.3, c = 0.25, δ₁ = δ₂ = 0.5, α = 0.9, E = 0.1, w₀ = 0.7, ω₀ = 0.8, y = 0.03, −2 < x < 2, −5 < t < 5.

Figure 12

The 2D graphs of Eq. (25) for a₃ = 0.1, d = −0.2, b₀ = b₁ = 0.3, c = 0.25, δ₁ = δ₂ = 0.5, α = 0.9, E = 0.1, w₀ = 0.7, ω₀ = 0.8, y = 0.03, t = 0.5, −6 < x < 6.

Figure 13

The contour graphs of Eq. (25) for a₃ = 0.1, d = −0.2, b₀ = b₁ = 0.3, δ₁ = δ₂ = 0.5, c = 0.25, α = 0.9, E = 0.1, w₀ = 0.7, ω₀ = 0.8, y = 0.03, −180 < x < 180, −180 < t < 180.

Figure 14

The compilation of contour graphs of both sides of Eq. (25) for a₃ = 0.1, d = −0.2, b₀ = b₁ = 0.3, δ₁ = δ₂ = 0.5, c = 0.25, α = 0.9, E = 0.1, w₀ = 0.7, ω₀ = 0.8, y = 0.03, −180 < x < 180, −180 < t < 180.

4 Conclusions

In this paper, complex dark travelling wave structures that are solutions to Eq. (1) have been extracted by using IB-SEFM. Many entirely new solutions such as exponential, rational, dark and the complex exponential have been successfully obtained. It has been observed that all solutions found in this paper have satisfied the nonlinear electrical transmission line model. Choosing suitable values for parameters, better understanding of the physical meanings of the solutions found, and the three- and two-dimensional graphs and contour simulations drawn via computational programs have all been employed.

The perspective view of the solutions Eqs. (19, 21, 23, 25) can be viewed from the 3D, 2D graphs in Figures 1, 5, 7, 11 and Figures 2, 8, 12, respectively. The contour patterns can be also viewed from Figures 3, 6, 9, 13, separately for the imaginary and real part of the solutions. As an alternative and new perspective to the 3D graph, contour surfaces give more detail on the model. Moreover, the contour patterns of combinations of imaginary and real part of the obtained solutions can be seen in Figures 4, 10, 14. When we consider these contour surfaces, it can be observed that the voltage is of high points among intervals in figures as a physical meanings.

Comparing results produced in this paper with the paper published in [7, 8], it can also be seen that Eq. (21) is similar hyperbolic type. Furthermore, it can be observed that the solutions Eqs. (19, 23, 25 are entirely new complex dark solutionsto the nonlinear electrical transmission line model. The calculations show that this method is a reliable and efficient scheme that yields many complex results to the other nonlinear models. To the best of our knowledge, the application of IBSEFM to Eq. (1) has been not submitted to literature in advance.

Acknowledgement

This study was supported by Harran University Scientific Research Projects Department with BAP project code: 16213. In calculations of system, Matlab 2016b version of HPC in Harran University has been used.

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Received: 2019-09-11

Accepted: 2019-10-10

Published Online: 2019-12-31

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https://doi.org/10.1515/phys-2019-0074

Keywords for this article

Nonlinear electrical transmission line equation; Improved Bernoulli sub-equation function method; Complex solutions; Contour surfaces

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