Investigating bifurcation and Chaos in lossy electrical transmission line models with Hamiltonian dynamics

Abstract

This study on Lossy Nonlinear Electrical Transmission Line Models (LNETLM) highlights several novel contributions: (1) Introducing the modified $\left( \frac{G^{'}}{G^2}\right)$-expansion method applied to LNETLM with a beta derivative, providing precise soliton solutions previously undocumented in the field; (2) Through extensive computer simulations, uncovering diverse wave phenomena such as bright, solitary, and parabolic solitons, alongside oscillatory singular waves, advancing the characterization of wave dynamics within LNETLM; (3) Contrasting conformable, M-truncated derivatives with the beta derivative, revealing the unique advantages of the beta derivative approach in modeling electrical transmission line systems; (4) Transforming the LNETLM equation into a Hamiltonian system, analyzing phase portraits, Chaos, and bifurcation phenomena, enhancing understanding of system dynamics and stability; (5) Expanding the knowledge frontier in lossy electrical transmission line models, with implications for theoretical understanding and practical applications in signal transmission and communication systems. In conclusion, this research underscores the potential of pulse-like solitons for enhancing data transmission rates in telecommunication systems. The insights gained pave the way for advancements in telecommunication technology, promising improved efficiency and performance.

1 Introduction

Conducting guides, which form the basis of transmission lines, facilitate the propagation of electromagnetic signals. These lines are often used to transmit data from a system to external measuring devices [1]. However, when semiconductor devices are interfaced with these transmission lines, nonlinear effects start to play a significant role [2]. Lossy Nonlinear Electrical Transmission Line Models (LNETLM), are characterized by their nonlinearity, which are primarily due to the presence of varactors. These varactors undergo changes in capacitance when a voltage is applied. Furthermore, LNETLM display dispersion, a phenomenon that arises from their inherent structural periodicity. As a result, these lines functions as nonlinear dispersive media, facilitating the propagation of voltage waves. These waves, frequently referred to as cnoidal waves, travel in the form of electrical solitons [3]. LNETLM are widely utilized across various domains. LNETLM serve as an instrumental tool for investigating nonlinear excitations in nonlinear media. Moreover, LNETLM offer a unique approach to modulating the attributes of innovative systems [4, 5]. The research emphasis is on investigating both analytical and numerical nonlinear uses of signal-generating transistors driven by LNETLM. This has made them not only practical but also of substantial physical relevance, particularly in the context of exceptionally wideband scenarios [6,7,8]. The models of LNETLM and their soliton-like pulses hold a high degree of importance in the fields of electronic technology and telecommunications structures. This is chiefly owing to their extensive use in various aspects of communication innovation, including radar emissions, electromagnetic wave and wideband electronic instruments, antenna networks, and transient electromagnetics. This is primarily because of their widespread utilization across various domains of communication technology, such as microwave, radar broadcasts and wide-spectrum electronic devices, short-pulse electromagnetics, antenna grids.

LNETLM are particularly crucial in contexts where the amplification of pulses is vital for transmitting signals over considerable distances [6, 9, 10]. LNETLM are also well suited for transmitting data in high-speed digital circuits and amplifying short pulses [6], high-precision measurements and high-speed communication systems [11], and LNETLM are ideal for use in high-speed sampling oscilloscopes designed for microwave systems [12]. Moreover, LNETLM prove to be beneficial in the manipulation of microwave digital signals, the field of wireless communications, the operation of radar systems, and the functioning of sensor arrays [13]. In linear operational scenarios, LNETLM can serve as phase shifters in phased antenna arrays. The time delay in these systems can be controlled by applying a direct current bias to the variable reactance of Schottky diodes. This allows for precise management of signal propagation within the array [14]. Under conditions of high signal amplitude, LNETLM can function as an impulse compressor or act as a frequency multiplier. LNETLM has the capability to serve as either a pulse stretcher or to operate as a frequency enhancer. This highlights their versatility in different operational scenarios [3]. Experimental research has shown that LNETLM can be employed as electrical soliton-based oscillators. These devices autonomously produce soliton pulses amidst background noise, showcasing their distinct functionality in signal processing [15].

In recent times, a number of innovative generalizations of the nonlinear Schr$\ddot{o}$dinger equation (NSE) have been introduced to more accurately depict the propagation of optical pulses within optical fibers. In recent times, a range of novel extensions of the NSE have surfaced to enhance the understanding of optical burst propagation dynamics in optical waveguides. Recent progress in this field encompass various equations such as the cubic-quintic nonlinearity extension of the NSE [16,17,18], the cubic-quartic Fokas-Lenells equation [19], the Schr$\ddot{o}$dinger-Hirota equation [19], the chiral variant of the nonlinear Schr$\ddot{o}$dinger equation [20], the perturbed Chen-Lee-Liu equation [21], and the Sasa-Satsuma model [22]. Furthermore, these contemporary breakthroughs have established the foundation for the development of distributed NSE by employing similarity transformations that incorporate eigenmatrix representations. Scholars have effectively derived N-soliton solutions using specific distributions of eigenvalues and adjoint eigenvalues, employing the corresponding non-shimmering generalized Riemann-Hilbert problems, as described in the works of Ma [23, 24]. Moreover, a variety of techniques have been utilized to extract new isolated wave solutions and investigate their interplay dynamics. The aforementioned approaches encompass unified technique[25], the Hirota bilinear approach [26,27,28], Lie symmetry analysis[29,30,31,32], direct techniques [33, 34], the extended physics-informed neural network approach [35, 36], similarity transformations [37, 38], the square operator technique [39, 40], the Sardar sub-equation technique [41], a novel extended hyperbolic function technique [41], and the rational sine-cosine technique[42]. These methods have demonstrated their effectiveness in resolving alternative kinds of nonlinear PDEs, given the selection of suitable unknown functions. These solutions hold practical significance in examining key properties of nonlinear wave phenomena, extending beyond optical fibers to encompass areas like nonlinear optical phenomena, dynamics of ocean surfaces, plasma dynamics, and other relevant areas of research. An examination of the current body of literature indicates that a multitude of integration approaches have been employed within the LNETLM.

These methods have been instrumental in investigating the properties of solitary waves within fractionalized LNETLM, exploring the spatiotemporal behavior of pulse-like solitons propagating along these trajectories, including the modified Kudryshov technique and the sine-Gordon equation expansion technique, among others [43,44,45], the extended sinh-Gordon equation expansion technique [45], the $\left( \frac{G^{'}}{G}\right)$-expansion technique [46], the extended tanh function technique [46], the alternative $\left( \frac{G^{'}}{G}\right)$-expansion technique [47], the $\left( \frac{G^{'}}{G^2}\right)$-expansion technique[48], the new auxiliary equation method [49], the generalized auxiliary equation technique [56], the new Jacobi elliptic function expansion technique [51], the improved Bernoulli sub-equation function technique[52], the discrete tanh technique [53], the three-wave technique [54], the improved Sardar-sub equation tactic [55], and the extended direct algebraic tactic [56]. Nevertheless, it is essential to highlight that all the previously referenced investigations have been performed out utilizing the non-deficient traditional-integer LNETLM.

In 2023, Paul et al. [57] employs the $\left( \frac{G^{'}}{G^2}\right)$-expansion method to analyze a nonlinear beta derivative (BD) of LNETLM that is periodically loaded and features symmetric voltage-dependent capacitances. Very recently, Qi J M and colleagues [58, 59] employed the revised $\left( \frac{G^{'}}{G^2}\right)$-expansion method to explore the LNETLM. The modified $\left( \frac{G^{'}}{G^2}\right)$-expansion method offers a richer content and a more diverse form of solutions compared to the $\left( \frac{G^{'}}{G^2}\right)$-expansion method. Our research in this paper aims to broaden the study by utilizing the modified $\left( \frac{G^{'}}{G^2}\right)$-expansion technique on a more applicable and authentic LNETLM that integrates the BD. Hence, our research in this paper aims to broaden the investigation by utilizing the adapted $\left( \frac{G^{'}}{G^2}\right)$-expansion approach on a more pragmatic and authentic LNETLM including the BD. The aim of this expansion will delve into and scrutinize the dynamics of soliton-like pulses amid this innovative LNETLM.

Table 1 Terminology overview table

Fig. 1

Nonlinear electrical network schematic

Furthermore, the research investigates the impact of fractional derivatives (FD) on voltage pulses. At the same time, under specific conditions, this research extensively investigates aspects such as phase trajectories, bifurcation examination, responsiveness, and possible chaotic phenomena of LNETLM, which remain unexplored sufficiently investigated or explored in previous studies[57, 62]. Finally, this study will investigate the influence of losses on the generation of uniquely contoured voltage pulses in the framework, offering crucial realizations into their reactions under feasible conditions. By juxtaposing various existing research findings, this study implies a wealth of untapped potential. This research will facilitate a more profound insight into soliton dynamics in the circumstances of applied transmission lines models.

To prevent redundancy in the article, we have employed abbreviations for certain proprietary clauses, as outlined in Table 1. The remainder of this paper is structured as follows. In Sect. 1, we start by presenting the historical development of LNETLM. This is followed by Sect. 2, where we describe the model and the circuit’s equations. Sect. 3 delves into the mathematical formulation in detail, focusing on two main aspects: Subsec. 3.1 defines the BD and highlights its numerous advantageous characteristics, while Subsect. 3.2 elaborates on the modified $\left( \frac{G^{'}}{G^2}\right)$-expansion method. Moving on to Sect. 4, we present our findings and discussions, covering a range of topics. Subsection 4.1 details the application of the modified $\left( \frac{G^{'}}{G^2}\right)$-expansion method for seeking new exact solutions. Subsection 4.2 provides a visual interpretation of the derived solutions, while subsection 4.3 explores the effects of fractionality on these innovative solutions. Additional in Sect. 4, Subsect. 4.4 offers a comparative analysis of the BD compared to other FD, while Subsect. 4.5 discusses the impact of losses on bright solitary wave signals. Subsection 4.6 analyzes phase diagrams and bifurcation patterns, and Subsect. 4.7 conducts a sensitivity analysis regarding the initial values in LNETLM. Subsection 4.8 explores the occurrence of chaotic behavior. Moreover, we present a comparison between previously published results and the solutions obtained in our study in Sect. 4.9, showcasing the novelty of our generated solutions. Finally, Sect. 5 concludes the paper.

2 Description of the model and equation of the circuit

In Fig. 1, we have a nonlinear electrical network consisting of M cells. Each cell comprises a linear inductor with an inductance of L and a linear resistor with resistance $R_1$, both in series connection. In each cell, there is an additional linear dispersive element, which is a nonlinear capacitor with capacitance C, and a parallel-connected linear resistor with resistance $R_2$. The voltage-dependent capacitance c, affected by a steady voltage $U_0$, may be approximately represented by a multinomial of degree two in terms of the relationship between capacitance and voltage.

$$\begin{aligned} \begin{aligned}&c(U_0+U_m)=\frac{dQ_m}{dU_m}=c_0(1+2\alpha +3\beta U_m^2){,} \end{aligned} \end{aligned}$$

(1)

in this context, $c_0$ signifies the linear capacitance of the capacitor, whereas $\alpha$ and $\beta$ stand for nonlinear parameters. These coefficients are responsible for determining the electric charge $Q_m$ that is retained in the m th capacitor along the line. Additionally, $U_m$ refers to the voltage that is present across the m th capacitor. Additional information regarding Fig. 1 is available in reference [15]. Equation (1) is employed as a second-order bend proper for either semiconductor features or MOS varactor traits, reliant on the indications of $\alpha$ and $\beta$ [8]. In the case of the diode, $\alpha$ is negative and $\beta$ is positive. However, for the MOS varactor, this sign relationship is inverted [8, 15].

By applying the Kirchhoff current law at node m, which has a voltage $U_m$ relative to the ground, and enforcing the Kirchhoff voltage law across the pair of inductors linked to this node, we obtain the subsequent outcomes:

$$\begin{aligned} \left\{ \begin{aligned}&I_{m-1}-I_m=\frac{dQ_m}{dt},\\&U_{m-1}-U_m=L\frac{dL_{m-1}}{dt}+I_{m-1}R_1,\\&U_m-U_{m+1}=L\frac{dL_m}{dt}+I_mR_1. \end{aligned} \right. \end{aligned}$$

(2)

In a similar vein, when we take into account the voltage across the capacitor as $U_m+U_0$, we derive the subsequent result:

$$\begin{aligned} \begin{aligned}&U_m=R_2(I_{m-1}-I_m)+U_0. \end{aligned} \end{aligned}$$

(3)

We can restructure Eq. (2) as follows:

$$\begin{aligned} \left\{ \begin{aligned}&L\frac{dL_{m-1}}{dt}=U_{m-1}-U_m-I_{m-1}R_1,\\&L\frac{dL_m}{dt}=U_m-U_{m+1}-I_mR_1. \end{aligned} \right. \end{aligned}$$

(4)

Drawing from Eq. (2), we obtain the following:

$$\begin{aligned} \begin{aligned}&L\frac{d^2Q_m}{dt^2}=L\frac{dI_{m-1}}{dt}-L\frac{dI_m}{dt}. \end{aligned} \end{aligned}$$

(5)

By substituting Eq. (4) into Eq. (5), we arrive at the following result:

$$\begin{aligned} \begin{aligned}&L\dfrac{d^2Q_m}{dt^2}=U_{m+1}+U_{m-1}-2U_m+R_1(I_m-I_{m-1}). \end{aligned} \end{aligned}$$

(6)

By incorporating the values of $U_{m+1}$, $U_m$, and $U_{m-1}$ from Eq. (3) into Eq. (6), we establish a relationship between the voltages of the neighboring nodes on these LNETLM through a PDE as shown below:

$$\begin{aligned}{} & {} L\frac{d^{2}}{dt^{2}}[c_{0}(U_{m}+\alpha U_{m}^{2} ) ]+R_{1}c_{0}\frac{d}{dt} (U_{m}+\alpha U_{m}^{2} )+R_{2}c_{0} \nonumber \\{} & {} [-\frac{d}{dt} (U_{m-1}+\alpha U_{m-1}^{2}) +2\frac{d}{dt}(U_{m}+\alpha U_{m}^{2}) \nonumber \\{} & {} \qquad -\frac{d}{dt}({U_{m+1}}+U_{m+1}^{2}] \nonumber \\{} & {} \quad =U_{m-1}-2U_{m}+U_{m+1}. \end{aligned}$$

(7)

The expression on the portion to the right of Eq. (7) can be approximated by computing partial x-derivatives from the origin of the line. This estimation supposes that the distance amid two consecutive parts is $\lambda$ (represented as $x_m = m\lambda$). Suppose function U(x, t) that maintains continuity across the variables x and t, thus $U_m(t) = U(x, t)$ and $U_{m\pm 1(t)} = U(x\pm \lambda , t)$. By employing Taylor series expansions, we can derive an approximate continuous PDE,

$$\begin{aligned} \left\{ \begin{aligned}&U_m(t)\rightarrow U(x,t){,}\\&U_{m\pm 1}(t)=U(x\pm \lambda ,t)=U\pm \lambda \frac{\partial U}{\partial x}+\frac{\lambda ^2}{2!}\frac{\partial ^2U}{\partial x^2}\\&\pm \frac{\lambda ^3}{3!}\frac{\partial ^2U}{\partial x^3}+\frac{\lambda ^4}{4!}\frac{\partial ^4U}{\partial x^4}\pm \dots \end{aligned} \right. \end{aligned}$$

(8)

the task at hand is to compute the value of the right-hand side of Eq. (7),

$$\begin{aligned} \begin{aligned}&U_{m+1}-2U_m+U_{m-1}=\lambda ^2\frac{\partial ^2U}{\partial x^2}\pm \frac{\lambda ^4}{4!}\frac{\partial ^4U}{\partial x^4}. \end{aligned} \end{aligned}$$

(9)

We disregard terms of order higher than $\lambda ^4$, considering them to be negligible. We postulate that the temporal fluctuations of the local voltage U are minuscule in comparison to the steady background voltage $U_0$. As a result, we can infer that time derivatives of the magnitude of a small element $\tau$, along with the nonlinear voltage terms $\alpha U_2$, are within the range of $\tau ^2$. This leads us to the subsequent nonlinear partial differential equation for the disturbed voltage:

$$\begin{aligned}{} & {} R_{r2}C_0\lambda ^2\frac{\partial }{\partial t}\left( \frac{\partial ^2U}{\partial x^2}\right) +\lambda ^2\left( \frac{\partial ^2U}{\partial x^2}\right) -L_{1}C_0\frac{\partial ^2U}{\partial t^2} \nonumber \\{} & {} \quad +2R_{r1}C_0\alpha U\frac{\partial U}{\partial t}-R_{r1}C_0\frac{\partial U}{\partial t}=0, \end{aligned}$$

(10)

here $L_1 =L/\lambda ,R_{r1}=R_1/\lambda ,R_{r2}=R_2/\lambda ,\mathrm {~and~}C(U)=c(U)/\lambda$ are designated as the inductance, resistances, and capacitance per unit length, respectively [8]. Equation (10) characterizes the transmission of a solitary wave with envelope modulation along the LNETLM.

In this study, we employed fractional complex transform to analyze the equation of motion governing the LNETLM’s dynamics. This innovative approach presents advantages compared to traditional derivatives in characterizing circuit equations. Fractional-order derivatives are adept at capturing system memory, accurately portraying its history and dynamics, particularly beneficial for modeling complex systems with long-term memory. They also enhance the precision of fitting experimental data and capturing real-world system behavior, especially in scenarios where non-ideal or nonlinear behavior cannot be effectively described by integer-order derivatives. Presently, we examine Eq. (10) in the context of the BD, we get:

$$\begin{aligned}{} & {} R_{r2}C_{0}\lambda ^{2}\frac{A}{0}D_{t}^{\beta }\left( \frac{A}{0}D_{xx}^{2\beta }U\right) +\lambda ^{2}\frac{A}{0}D_{xx}^{2\beta }U-L_1C_{0}{}_{0}^{A}D_{tt}^{2\beta }U \nonumber \\{} & {} \quad +2R_{r1}C_{0}\alpha (U\frac{A}{0}D_{t}^{\beta }N) -R_{r1}C_0\frac{A}{0}D_{t}^{\beta }U=0. \end{aligned}$$

(11)

Here $\frac{A}{0}D_{x}^{\beta }U$ and $\frac{A}{0}D_{x}^{\beta }U$ represent the BD of U with respect to t and x; where $\beta$ ranges between 0 and 1, and U(x, t) is a function that is differentiable in terms of x and t. It is important to note that by substituting $\beta = 1$ into Eq. (11), the BD about LNETLM equation returns back to the LNETLM equation via the standard derivative, as depicted in Eq. (10). The introduction of the BD will be presented in the forthcoming chapter.

3 Mathematical formulation

3.1 The BD is defined and its multitude of beneficial characteristics are outlined

Fractional models, renowned for offering distinctive insights into the physical realm and its applications, exhibit variation subject to the precise interpretation of FD employed.

There are numerous definitions, including the Caputo derivative, Jumarie derivative, Conformable derivative (CD), M-truncated derivative (MTD), Atangana’s conformable derivative, among others [63,64,65,66]. These diverse definitions present a challenge for researchers in identifying the supreme precise one and comprehending the intrinsic constraints of these FD.

In-depth knowledge about the descriptions, characteristics, and constraints of diverse FD can be obtained from the exhaustive research conducted by Teodoro and colleagues [65]. In Tarasov’s work [66], it is suggested that a fractional operator should conform to the overarching Leibniz principle. This interpretation reveals that FD of non-integer orders, such as the Jumarie derivative and the local derivative, are fundamentally unable to satisfy the Leibniz rule. In a separate study, Cresson et al [67] engaged in a comprehensive exploration of various properties of Jumarie’s fractional derivative and the local fractional derivative, particularly in the real-world context. They also examined the chain rule assistant within the same analytical framework.

In light of the complexities in identifying the supreme precise FD, this research concentrates on elucidating the clarification and assorted properties of Atangana’s CD, also known as the BD. The notion of the BD and its attributes were initially put forth by Atangana et al. [68]. Later, Atangana and Alqahtani [69] illustrated that the BD can more accurately depict the proliferation of the river vision loss epidemiological model than the Caputo derivative. The composition of Atangana’s CD is totally based on the primary definition about derivative’s limit, and it is described as ensues according to [68]:

$$\begin{aligned}{} & {} \frac{A}{0}D_{t}^{\beta }(f(t))=\lim _{\Delta h\rightarrow 0}\frac{f(t+\Delta h(t+\frac{1}{\Gamma (\beta )})^{1-\beta })-f(t)}{\Delta h}, \nonumber \\{} & {} \quad \text {for all}\quad t>0, 0<\beta \le 1. \end{aligned}$$

(12)

Commonly known as the beta derivative (BD), this derivative has been utilized by numerous researchers over recent years. They have applied the definition and properties of the BD to fractional PDEs, deriving analytical solutions via a range of analytic methods[45, 70,71,72,73,74]. To leverage the definition and attributes of the BD in this investigation, we revisit certain advantageous characteristics outlined in reference [68], augmenting our understanding of its utility.

We posit $a, b \in {R}$, here R denotes real number set. Additionally, let $g(t) \ne 0$ and f(t) be pair of functions that exhibit $\beta$-differentiability at a positive number t, with $0<\beta \le 1$. The ensuing features are as follows:

• 1. $\frac{A}{0}D_t^\beta (af(t)+bg(t))=a\frac{A}{0}D_t^\beta f(t)+cD_t^\beta g(t),$ $arbitrary\,\, b, c\in R.$

• 2. $\frac{A}{0}D_{t}^{\beta }(C)=0$, here C denotes a constant.

• 3. $\frac{A}{0}D_t^\alpha (f(t)g(t))=g(t)\frac{A}{0}D_t^\alpha f(t)+ f(t)\frac{A}{0}D_t^\alpha g(t).$

• 4. $\frac{A}{0}D_t^\alpha (\frac{f(t)}{g(t)})=\frac{g(t)\frac{A}{0}D_t^\alpha f(t)+ f(t)\frac{A}{0}D_t^\alpha g(t)}{g(t)^2}.$

Considering $\Delta h = (t+\frac{1}{\Gamma _\beta })^{\beta -1}h$, where $h\rightarrow 0$ as $\Delta h \rightarrow 0$, we derive a valuable property: $\frac{A}{0} D_t^\beta f(t) = (t+\frac{1}{\Gamma \beta })^{\beta -1}\frac{df(t)}{dt}.$

3.2 The modified $\left( \frac{G^{'}}{G^2}\right)$-expansion method

Firstly, let’s assume a fractional nonlinear PDE with variables (x, t) is established as follows:

$$\begin{aligned}&Y\left( U \right. ,{U _x},{U _t},{U _{xt}}, {U^{\alpha } _{x}}, {U^{\alpha } _{t}}, {U^{2\alpha } _{xx}}, {U^{2\alpha } _{tt}}, {U^{2\alpha } _{xt}}\left. \cdots \right) = 0,\nonumber \\&\quad 0<\alpha \le 1{.} \end{aligned}$$

(13)

In this place, Y denotes an algebraic expression incorporating the uncertain function U(x, t) and its fractional derivative. Subsequently, we provide a detailed description of the recently formulated modified $\left( \frac{G^{'}}{G^2}\right)$-expansion method.

Process 1. We begin by substituting $U(x, t) = U(\xi )$, and $\xi =k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta })$ into Eq. (13). This converts Eq. (13) into the following ranked integer differential equation:

$$\begin{aligned} H({U},{U^{'}},{U^{''}},{U^{'''}}, \cdots ) = 0{.} \end{aligned}$$

(14)

In Eq. (14), we denote an algebraic expression composed of $U(\xi )$ and its derivatives as H.

Process 2. We consider the potential of representing the solution to Eq. (14) in the form of a polynomial in $\left( \frac{G^{'}}{G^2}\right)$ as follows:

$$\begin{aligned} {U}({\xi }) = a_0+\sum _{n=1}^m(a_n\left( \frac{G^{'}}{G^2}\right) ^n+b_n\left( \frac{G^{'}}{G^2}\right) ^{-n}). \end{aligned}$$

(15)

The phrase ${G} = {G}({\xi })$ in Eq. (15) fulfills the subsequent differential equation:

$$\begin{aligned} \left( \frac{G^{'}}{G^2}\right) '=\sigma +\mu \left( \frac{G^{'}}{G^2}\right) +\rho \left( \frac{G^{'}}{G^2}\right) ^2{.} \end{aligned}$$

(16)

With the undetermined constants $\mu$, $\sigma$, and $\rho$ present, we can ascertain the positive integer m by equating the nonlinear variables in Eq. (14) with the topmost-order derivatives.

Process 3. Substituting Eq. (15) into Eq. (14) allows us to reformulate it as a $\left( \frac{G^{'}}{G^2}\right)$ equation. Upon equating the coefficients of each algebraic expression in Eq. (14) to zero, we derive a system of algebraic equations involving ${a_{m}}$, $b_m$, and other variables such as ${\rho }$ and ${\mu }$.

Process 4. This collection of mathematical expressions involves determining precise values for the constants ${a_{m}}$, $a_0$, $b_1$, $b_m$, and so forth, as outlined in Process 3. Through an examination of the five prospective solutions outlined below Eq. (16), we ascertain the exact solution for the provided Eq. (13). At this location:

$$\begin{aligned} \left\{ \begin{aligned}&\text {Case 1:}\quad \text {While}\,\, \sigma \rho >0, \mu =0,\,\text {subsequently} \\&\left( \frac{G^{'}}{G^2}\right) (\xi )=\frac{\sqrt{\sigma \rho }}{\sigma } [\frac{C_1\cos \sqrt{\sigma \rho }\xi +C_2\sin \sqrt{\sigma \rho }\xi }{C_2\cos \sqrt{\sigma \rho }\xi -C_1\sin \sqrt{\sigma \rho }\xi }];\\&\text {Case 2:}\quad \text {When}\,\, \sigma \rho<0, \mu =0,\,\text {after that}\\&\left( \frac{G^{'}}{G^2}\right) (\xi )=-\frac{\sqrt{|\sigma \rho |}}{\sigma }\\&[\frac{C_1\sinh 2\sqrt{|\sigma \rho |}\xi +C_2\cosh 2\sqrt{|\sigma \rho |}\xi +C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}\xi +C_1\sinh 2\sqrt{|\sigma \rho |}\xi -C_2}];\\&\text {Case 3:}\quad \text {When}\,\,\sigma =0, \rho \ne 0,\mu =0,\,\text {after that}\\&\left( \frac{G^{'}}{G^2}\right) (\xi )=-\frac{C_1}{\rho (C_1\xi +C_2)};\\&\text {Case 4:}\quad \text {When}\,\, \mu \ne 0, \Delta \ge 0,\,\text {after that}\\&\left( \frac{G^{'}}{G^2}\right) (\xi )=-\frac{\mu }{2\rho }-[\frac{\sqrt{\Delta }(C_1\cosh \left( \frac{\sqrt{\Delta }}{2}\right) \xi +C_2\sinh \left( \frac{\sqrt{\Delta }}{2}\right) \xi )}{2\rho (C_2\cosh \left( \frac{\sqrt{\Delta }}{2}\right) \xi +C_1\sinh \left( \frac{\sqrt{\Delta }}{2}\right) \xi )}]; \\&\text {Case 5:}\quad \text {When}\,\, \mu \ne 0, \Delta <0, \,\text {after that}\\&\left( \frac{G^{'}}{G^2}\right) (\xi )=-\frac{\mu }{2\rho }-[\frac{\sqrt{-\Delta } (C_1\cos \left( \frac{\sqrt{-\Delta }}{2}\right) \xi -C_2\sin \left( \frac{\sqrt{-\Delta }}{2}\right) \xi )}{2\rho (C_2\cos \left( \frac{\sqrt{-\Delta }}{2}\right) \xi +C_1\sin \left( \frac{\sqrt{-\Delta }}{2}\right) \xi )}]{.}\\ \end{aligned} \right. \end{aligned}$$

(17)

Here $\Delta$ is defined as $\mu ^2-4\rho \sigma$, and $C_1, C_2$ are arbitrary constants.

Process 5. By applying the inverse transform to the expressions for the solutions $U(\xi )$, defined as $U(\xi ) = U(x, t) = U\left( k\left( \frac{1}{\beta }\left( x + \frac{1}{\Gamma (\beta )}\right) ^{\beta } - \frac{c}{\beta }\left( t + \frac{1}{\Gamma (\beta )}\right) ^{\beta }\right) \right)$, we can obtain all accurate solutions of the original Eq. (14).

4 Findings and discussion

4.1 Utilization of the modified $\left( \frac{G^{'}}{G^2}\right)$-expansion methodology

In this segment, we employ the $\left( \frac{G^{'}}{G^2}\right)$ -expansion technique to resolve the space-time fractional differential equation that dictates wave propagation in LNETLM. To formulate the analytical solutions using the $\left( \frac{G^{'}}{G^2}\right)$-expansion method, we consider the subsequent LNETLM equation with a BD:

$$\begin{aligned}{} & {} R_{r2}C_{0}\lambda ^{2}\frac{A}{0}D_{t}^{\beta }\left( \frac{A}{0}D_{xx}^{2\beta }U\right) +\lambda ^{2}\frac{A}{0}D_{xx}^{2\beta }U-L_1C_{0}{}_{0}^{A}D_{tt}^{2\beta }U\nonumber \\{} & {} \quad +2R_{r1}C_{0}\alpha (U\frac{A}{0}D_{t}^{\beta }U)-R_{r1}C_0\frac{A}{0}D_{t}^{\beta }U=0. \end{aligned}$$

(18)

Upon applying the transformation $U(x, t) = U(\xi )$ to Eq. (18), where $\xi =k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right)$, with c and k as nontrivial constants, subsequently integrating the resultant equation once with respect to $\xi$ and subsequently equating the constant term to zero. We get the subsequent ODE:

$$\begin{aligned} \begin{aligned}&R_{r2}C_0k^2\lambda ^2cU''+k(L_1C_0c^2-\lambda ^2)U'\\&\quad +R_{r1}C_0\alpha cU^2-R_{r1}C_0cU =0. \end{aligned} \end{aligned}$$

(19)

Consequently, Eq. (19) can be restructured in the following manner:

$$\begin{aligned} \begin{aligned}&PU''+TU'+QU^{2}-cU=0, \end{aligned} \end{aligned}$$

(20)

here, $P = \frac{R_{r2}}{R_{r1}}k^2\lambda ^2c, T = \frac{k(L_1C_0c^2-\lambda ^2)}{R_{r1}C_0} = \frac{kL_1}{R_{r1}}(c^2 -\lambda ^2\eta ^2_0)$, and $Q = \alpha c$. The parameter $\eta _0^2$ is defined as $\eta _0^2 = \frac{1}{L_0C_0}$, and the highest rate of movement of the traveling waves, $c_{max}$ is given by $c_{max} = \lambda \eta _0$(refer to [3]). Adhering to the precepts of the $(G^{'}/G^2)$-expansion approach, we hypothesize that the precise solution of Eq. (20) can be articulated as follows:

$$\begin{aligned} {U}({\xi }) = a_0+\sum _{n=1}^m(a_n\left( \frac{G^{'}}{G^2}\right) ^n+b_n\left( \frac{G^{'}}{G^2}\right) ^{-n}). \end{aligned}$$

(21)

We can easily find that $m = 2$ using the principle of homogeneous equilibrium. Consequently, we can establish the general solution of Eq. (19) as follows:

$$\begin{aligned} {U}({\xi })&= a_0+a_1\left( \frac{G^{'}}{G^2}\right) +a_2\left( \frac{G^{'}}{G^2}\right) ^2+b_1\left( \frac{G^{'}}{G^2}\right) ^{-1}\nonumber \\&\quad +b_2\left( \frac{G^{'}}{G^2}\right) ^{-2}, \end{aligned}$$

(22)

here the unknown constants $a_0, a_1, a_2, b_1, b_2$ will be found later.

By substituting Eq. (22) into Eq. (20) and applying Eq. (19), we gather the coefficients that share the same order of $\left( \frac{G^{'}}{G^2}\right)$. This process results in a set of nine nonlinear algebraic equations. Solving this algebraic system using Maple 2020 yields the following values for the unspecified constants:

By utilizing Eq. (16), and setting the indeterminate coefficients of identical terms in the $\left( \frac{G'}{G^2}\right)$ functions to zero, we derive the following equations:

$$\begin{aligned} \left\{ \begin{aligned}&P(a_1\mu \sigma +2 a_2 \sigma ^2+b_1 \mu \rho +2 b_2 \rho ^2)+T(a_1 \sigma -b_1 \rho )\\&\quad +Q (2 a_1 b_1+a_0^2+2 a_2 b_2)-c a_0=0,\\&P(a_1\mu ^2+2 a_2 \sigma \mu +2a_1 \sigma \rho +4 a_2 \mu \sigma )+T(a_1 \mu +2a_2 \sigma )\\&\quad +Q (2 a_1 a_0+2 a_2 b_1)-c a_1=0,\\&P(a_1\mu \rho +2 a_2 \sigma \rho +2a_1 \mu \rho +4 a_2 \mu ^2+6a_2\rho \sigma )\\&\quad +T(a_1 \rho +2a_2 \mu )+Q (2 a_2 a_0+a_1^2)-c a_2=0,\\&P(2a_1\rho ^2+4a_2\mu \rho +6a_2\rho \mu )+2Ta_2\rho +2Qa_1a_2 = 0,\\&6Pa_2\rho ^2+Qa_2^2= 0,\\&P(b_1\mu ^2+2b_2\mu \rho +2b_1\sigma \rho +4b_2\mu \rho )\\&\quad +Q(2a_1b_2+2a_0b_1)+T(-b_1\mu -2b_2\rho )-cb_1 = 0,\\&P(2b_1\sigma \mu +4b_2\mu ^2+b_1\mu \sigma +2b_1\rho \sigma +6b_2\sigma \rho )\\&\quad +T(-b_1\sigma -2b_2\mu )+Q(2a_0b_2+b_1^2)-cb_2= 0,\\&P(6b_2\sigma \mu +2b_1\sigma ^2+4b_2\mu \sigma )-2Tb_2\sigma +2Qb_1b_2= 0,\\&6Pb_2\sigma ^2+Qb_2^2= 0. \end{aligned} \right. \end{aligned}$$

(23)

The equations represented by Eq. (23) are computed using Maple 2020, which yields solutions for the undetermined coefficients as follows:

$$\begin{aligned} \mathbf{Set 1.}\,\,a_0&=-\frac{5 P^{2} \mu ^{3}+40 P^{2} \mu \rho \sigma +6 P T \,\mu ^{2}+12 P T \rho \sigma -5 P c \mu +T^{2} \mu -T c}{2 Q \left( 5 P \mu +T \right) },\nonumber \\ a_1&=-\frac{6\rho (5P\mu +T)}{5Q},\nonumber \\ b_1&=0, a_2=-\frac{6P\rho ^{2}}{Q},b_2=0,\nonumber \\ k&=\pm \frac{\sqrt{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) R_{r1} }}{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) \lambda },\nonumber \\ c&=\pm \frac{\lambda }{\sqrt{C_0 L_1}}{,}\nonumber \\ U_1(x,t)&=-\frac{5 P^{2} \mu ^{3}+40 P^{2} \mu \rho \sigma +6 P T \,\mu ^{2}+12 P T \rho \sigma -5 P c \mu +T^{2} \mu -T c}{2 Q \left( 5 P \mu +T \right) }-\frac{6\rho (5P\mu +T)}{5Q}\nonumber \\&\quad \left( -\frac{\mu }{2\rho }-\![\frac{\sqrt{\Delta }(C_1\cosh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x\!+\!\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ) \!+\!C_2\sinh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x\!+\!\frac{1}{\Gamma (\beta )})^{\beta }\!-\!\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}{2\rho (C_2\cosh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x\!+\!\frac{1}{\Gamma (\beta )})^{\beta }\!-\!\frac{c}{\beta }(t\!+\!\frac{1}{\Gamma (\beta )})^{\beta }\right) )\!+\!C_1\sinh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x\!+\!\frac{1}{\Gamma (\beta )})^{\beta }\!-\!\frac{c}{\beta }(t\!+\!\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}]\right) \!-\!\frac{6P\rho ^{2}}{Q}\nonumber \\&\quad \left( -\frac{\mu }{2\rho }-[\frac{\sqrt{\Delta }(C_1\cosh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ) +C_2\sinh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}{2\rho (C_2\cosh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}]\right) ^{2},\nonumber \\&\quad here, \Delta \ge 0, \mu \ne 0. \end{aligned}$$

(24)

$$\begin{aligned} \mathbf{Set 2.}\,\,a_0&=-\frac{5 P^{2} \mu ^{3}+40 P^{2} \mu \rho \sigma +6 P T \,\mu ^{2}+12 P T \rho \sigma -5 P c \mu +T^{2} \mu -T c}{2 Q \left( 5 P \mu +T \right) },\nonumber \\ a_1&=-\frac{6\rho (5P\mu +T)}{5Q},\nonumber \\ b_1&=0, a_2=-\frac{6P\rho ^{2}}{Q},b_2=0,\nonumber \\ k&=\pm \frac{\sqrt{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) R_{r1} }}{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) \lambda },\nonumber \\ c&=\pm \frac{\lambda }{\sqrt{C_0 L_1}},\nonumber \\ U_2(x,t)&=-\frac{5 P^{2} \mu ^{3}+40 P^{2} \mu \rho \sigma +6 P T \,\mu ^{2}+12 P T \rho \sigma -5 P c \mu +T^{2} \mu -T c}{2 Q \left( 5 P \mu +T \right) } \nonumber \\&\quad -\frac{6\rho (5P\mu +T)}{5Q}\nonumber \\&\quad \left( -\frac{\mu }{2\rho }-\left[ \frac{\sqrt{-\Delta } (C_1\cos \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta } -\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2\sin \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta } -\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}{2\rho (C_2\cos \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta } -\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ) +C_1\sin \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta } -\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}\right] \right) \nonumber \\&\quad -\frac{6P\rho ^{2}}{Q}\nonumber \\&\quad \left( -\frac{\mu }{2\rho }-\left[ \frac{\sqrt{-\Delta } (C_1\cos \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta } -\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2\sin \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}{2\rho (C_2\cos \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sin \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta } -\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}\right] \right) ^{2},\nonumber \\&\quad here, \Delta <0, \mu \ne 0.\end{aligned}$$

(25)

$$\begin{aligned} \mathbf{Set 3.}\,\,a_0&=\frac{-25P^2\mu ^2-200P^2\rho \sigma -30PT\mu +25Pc+T^2}{50QP},\nonumber \\ a_1&=-\frac{6\rho (5P\mu +T)}{5Q}, b_1 =0,\nonumber \\ a_2&=-\frac{6P\rho ^{2}}{Q},b_2=0,\nonumber \\ k&=\pm \frac{\sqrt{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) R_{r1} }}{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) \lambda },\nonumber \\ c&=\pm \frac{\lambda }{\sqrt{C_0 L_1}},\nonumber \\ U_3(x,t)&=\frac{-25P^2\mu ^2-200P^2\rho \sigma -30PT\mu +25Pc+T^2}{50QP}\nonumber \\&\quad -\frac{6\rho (5P\mu +T)}{5Q}\nonumber \\&\quad \left( -\frac{\mu }{2\rho }-\left[ \frac{\sqrt{\Delta }(C_1\cosh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ) +C_2\sinh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}{2\rho (C_2\cosh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}\right] \right) \nonumber \\&\quad -\frac{6P\rho ^{2}}{Q}\nonumber \\&\quad \left( -\frac{\mu }{2\rho }-\left[ \frac{\sqrt{\Delta }(C_1\cosh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ) +C_2\sinh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}{2\rho (C_2\cosh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}\right] \right) ^{2},\nonumber \\&\quad here, \Delta \ge 0, \mu \ne 0. \end{aligned}$$

(26)

$$\begin{aligned} \mathbf{Set 4.}\,\,a_0&=\frac{-25P^2\mu ^2-200P^2\rho \sigma -30PT\mu +25Pc+T^2}{50QP},\nonumber \\ a_1&=-\frac{6\rho (5P\mu +T)}{5Q},b_1=0,\nonumber \\ a_2&=-\frac{6P\rho ^{2}}{Q},b_2=0,\nonumber \\ k&=\pm \frac{\sqrt{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) R_{r1} }}{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) \lambda },\nonumber \\ c&=\pm \frac{\lambda }{\sqrt{C_0 L_1}},\nonumber \\ U_4(x,t)&=\frac{-25P^2\mu ^2-200P^2\rho \sigma -30PT\mu +25Pc+T^2}{50QP}-\frac{6\rho (5P\mu +T)}{5Q}\nonumber \\&\quad \left( -\frac{\mu }{2\rho }-\left[ \frac{\sqrt{-\Delta } (C_1\cos \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2\sin \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}{2\rho (C_2\cos \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sin \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}\right] \right) \nonumber \\&\quad -\frac{6P\rho ^{2}}{Q}\nonumber \\&\quad \left( -\frac{\mu }{2\rho }-\left[ \frac{\sqrt{-\Delta } (C_1\cos \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2\sin (\frac{\sqrt{-\Delta }}{2})(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}{2\rho (C_2\cos (\frac{\sqrt{-\Delta }}{2})(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sin (\frac{\sqrt{-\Delta }}{2})(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta })))}\right] \right) ^{2},\nonumber \\&\quad here, \Delta <0, \mu \ne 0. \end{aligned}$$

(27)

$$\begin{aligned} \mathbf{Set 5.}\,\,a_0&=\frac{\frac{c}{2}\pm \frac{\sqrt{600 P^{2} \mu ^{2} \rho \sigma +1200 P^{2} \rho ^{2} \sigma ^{2}+720 P T \mu \rho \sigma +120 T^{2} \rho \sigma +25 c^{2}}}{10}}{Q},\nonumber \\ a_1&=-\frac{6\rho (5P\mu +T)}{5Q},b_1=0,\nonumber \\ a_2&=-\frac{6P\rho ^{2}}{Q},b_2=0,\nonumber \\ k&=\pm \frac{\sqrt{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) R_{r1} }}{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) \lambda },\nonumber \\ c&=\pm \frac{\lambda }{\sqrt{C_0 L_1}},\nonumber \\ U_5(x,t)&=\frac{\frac{c}{2}\pm \frac{\sqrt{M}}{10}}{Q}-\frac{6\rho (5P\mu +T)}{5Q}\nonumber \\&\quad \left( -\frac{\mu }{2\rho }-\left[ \frac{\sqrt{\Delta }(C_1\cosh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ) +C_2\sinh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}{2\rho (C_2\cosh (\frac{\sqrt{\Delta }}{2})(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_1\sinh \left( \frac{\sqrt{\Delta }}{2}\right) (k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta })))}\right] \right) \nonumber \\&\quad -\frac{6P\rho ^{2}}{Q}\nonumber \\&\quad \left( -\frac{\mu }{2\rho }-\left[ \frac{\sqrt{\Delta }(C_1\cosh (\frac{\sqrt{\Delta }}{2})(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta })) +C_2\sinh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}{2\rho (C_2\cosh \left( \frac{\sqrt{\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh \left( \frac{\sqrt{\Delta }}{2}\right) (k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta })))}\right] \right) ^{2},\nonumber \\&\quad here, \Delta \ge 0, \mu \ne 0, M=600 P^{2} \mu ^{2} \rho \sigma +1200 P^{2} \rho ^{2} \sigma ^{2}+720 P T \mu \rho \sigma +120 T^{2} \rho \sigma +25 c^{2}. \end{aligned}$$

(28)

$$\begin{aligned} \mathbf{Set 6.}\,\,a_0&=\frac{\frac{c}{2}\pm \frac{\sqrt{600 P^{2} \mu ^{2} \rho \sigma +1200 P^{2} \rho ^{2} \sigma ^{2}+720 P T \mu \rho \sigma +120 T^{2} \rho \sigma +25 c^{2}}}{10}}{Q},\nonumber \\ a_1&=-\frac{6\rho (5P\mu +T)}{5Q},b_1=0,\nonumber \\ a_2&=-\frac{6P\rho ^{2}}{Q},b_2=0,\nonumber \\ k&=\pm \frac{\sqrt{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) R_{r1} }}{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) \lambda },\nonumber \\ c&=\pm \frac{\lambda }{\sqrt{C_0 L_1}},\nonumber \\ U_6(x,t)&=\frac{\frac{c}{2}\pm \frac{\sqrt{M}}{10}}{Q}-\frac{6\rho (5P\mu +T)}{5Q}\nonumber \\&\left( -\frac{\mu }{2\rho }-\left[ \frac{\sqrt{-\Delta } (C_1\cos \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2\sin (\frac{\sqrt{-\Delta }}{2})(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta })))}{2\rho (C_2\cos \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sin (\frac{\sqrt{-\Delta }}{2})(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta })))}\right] \right) \nonumber \\&\quad -\frac{6P\rho ^{2}}{Q}\nonumber \\&\quad \left( -\frac{\mu }{2\rho }-\left[ \frac{\sqrt{-\Delta } (C_1\cos \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2\sin \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) ))}{2\rho (C_2\cos \left( \frac{\sqrt{-\Delta }}{2}\right) (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sin (\frac{\sqrt{-\Delta }}{2})(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta })))}\right] \right) ^{2},\nonumber \\&\quad here, \Delta <0, \mu \ne 0, M=600 P^{2} \mu ^{2} \rho \sigma +1200 P^{2} \rho ^{2} \sigma ^{2}+720 P T \mu \rho \sigma +120 T^{2} \rho \sigma +25 c^{2}.\end{aligned}$$

(29)

$$\begin{aligned} \mathbf{Set 7.}\,\,a_0&=-\frac{24 P T \rho \sigma -T c}{2 Q T},\nonumber \\ a_1&=-\frac{6 \rho T}{5 Q},b_1=\frac{6 \sigma T}{5 Q},\nonumber \\ a_2&=-\frac{6 P \rho ^{2}}{Q},b_2=-\frac{6 P \sigma ^{2}}{Q},\nonumber \\ k&=\frac{C_0 L_1 \,c^{2}-\lambda ^{2}}{80 C_0 R_{r2} c \,\lambda ^{2} \rho },\nonumber \\ c&=\pm \frac{\sqrt{-3 C_0\lambda \left( 25 C_0 R_{r1} R_{r2} -12 L_1 \pm 5 \sqrt{25 C_0^{2} R_{r1}^{2} R_2^{2}-24C_0 L_1 R_{r1} R_{r2}}\right) } }{6 C_0 L_1},\nonumber \\ U_7(x,t)&=-\frac{24 P T \rho \sigma -T c}{2 Q T}-\frac{6 \rho T}{5 Q}(\frac{\sqrt{\sigma \rho }}{\sigma }\nonumber \\&\quad \left[ \frac{C_1\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta } -\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}{C_2\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_1\sin \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))}\right] )\nonumber \\&\quad -\frac{6 P \rho ^{2}}{Q}\nonumber \\&\quad \left( \frac{\sqrt{\sigma \rho }}{\sigma }\left[ \frac{C_1\cos \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_2\sin \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))}{C_2\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_1\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}\right] \right) ^2\nonumber \\&\quad +\frac{6 \sigma T}{5 Q}\left( \frac{\sqrt{\sigma \rho }}{\sigma } \left[ \frac{C_1\cos \sqrt{\sigma \rho } (k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_2\sin \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))}{C_2\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_1\sin \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))}\right] \right) ^{-1}\nonumber \\&\quad -\frac{6 P \sigma ^{2}}{Q}\left( \frac{\sqrt{\sigma \rho }}{\sigma } \left[ \frac{C_1\cos \sqrt{\sigma \rho } (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\sin \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))}{C_2\cos \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))-C_1\sin \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))}\right] \right) ^{-2},\nonumber \\&here, \sigma \rho >0, \mu =0. \end{aligned}$$

(30)

$$\begin{aligned} \mathbf{Set 8.}\,\,a_0&=-\frac{24 P T \rho \sigma -T c}{2 Q T},\nonumber \\ a_1&=-\frac{6 \rho T}{5 Q}, b_1=\frac{6 \sigma T}{5 Q},\quad a_2=-\frac{6 P \rho ^{2}}{Q},\quad b_2=-\frac{6 P \sigma ^{2}}{Q},\nonumber \\ k&=\frac{C_0 L_1 \,c^{2}-\lambda ^{2}}{80 C_0 R_{r2} c \,\lambda ^{2} \rho },\quad c =\pm \frac{\sqrt{-3 C_0\lambda \left( 25 C_0 R_{r1} R_{r2} -12 L_1 \pm 5 \sqrt{25 C_0^{2} R_{r1}^{2} R_2^{2}-24C_0 L_1 R_{r1} R_{r2}}\right) } }{6 C_0 L_1},\nonumber \\ U_{8}(x,t)&=-\frac{24 P T \rho \sigma -T c}{2 Q T}-\frac{6 \rho T}{5 Q}\nonumber \\&\left( \frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\cosh 2\sqrt{|\sigma \rho |}(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2}\right) -\frac{6 P \rho ^{2}}{Q}\nonumber \\&\left( \frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_2\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2}\right) ^2+\frac{6 \sigma T}{5 Q}\nonumber \\&\quad \left( -\frac{\sqrt{|\sigma \rho |}}{\sigma } \left[ \frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2}\right] \right) ^{-1}\nonumber \\&\quad -\frac{6 P \sigma ^{2}}{Q}\nonumber \\&\quad \left( -\frac{\sqrt{|\sigma \rho |}}{\sigma } \left[ \frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2}\right] \right) ^{-2},\nonumber \\&here, \sigma \rho <0, \mu =0. \end{aligned}$$

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$$\begin{aligned} \mathbf{Set 9.}\,\,a_0&=\frac{\frac{5 c}{2}\pm \frac{\sqrt{-4800 P^{2} \rho ^{2} \sigma ^{2}+528 T^{2} \rho \sigma +25 c^{2}}}{2}}{5 Q},\nonumber \\ a_1&=-\frac{6 \rho T}{5 Q}, b_1=\frac{6 \sigma T}{5 Q},\nonumber \\ a_2&=-\frac{6 P \rho ^{2}}{Q},b_2=-\frac{6 P \sigma ^{2}}{Q},\quad k =\frac{C_0 L_1 \,c^{2}-\lambda ^{2}}{80 C_0 R_{r2} c \,\lambda ^{2} \rho },\nonumber \\ c&=\pm \frac{\sqrt{-3 C_0\lambda \left( 25 C_0 R_{r1} R_{r2} -12 L_1 \pm 5 \sqrt{25 C_0^{2} R_{r1}^{2} R_2^{2}-24C_0 L_1 R_{r1} R_{r2}}\right) } }{6 C_0 L_1},\nonumber \\ U_9(x,t)&=\frac{\frac{5 c}{2}\pm \frac{\sqrt{-4800 P^{2} \rho ^{2} \sigma ^{2}+528 T^{2} \rho \sigma +25 c^{2}}}{2}}{5 Q}\nonumber \\&\quad -\frac{6 \rho T}{5 Q}\left( \frac{\sqrt{\sigma \rho }}{\sigma } \left[ \frac{C_1\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\sin \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))}{C_2\cos \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))-C_1\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}\right] \right) -\frac{6 P \rho ^{2}}{Q}\nonumber \\&(\frac{\sqrt{\sigma \rho }}{\sigma }\left[ \frac{C_1\cos \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_2\sin \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))}{C_2\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_1\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}\right] )^2\nonumber \\&\quad +\frac{6 \sigma T}{5 Q}\left( \frac{\sqrt{\sigma \rho }}{\sigma } \left[ \frac{C_1\cos \sqrt{\sigma \rho } (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}{C_2\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_1\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}\right] \right) ^{-1}\nonumber \\&\quad -\frac{6 P \sigma ^{2}}{Q}\left( \frac{\sqrt{\sigma \rho }}{\sigma } \left[ \frac{C_1\cos \sqrt{\sigma \rho } (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\sin \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))}{C_2\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_1\sin \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))}\right] \right) ^{-2},\nonumber \\&here, \sigma \rho >0, \mu =0. \end{aligned}$$

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$$\begin{aligned} \mathbf{Set 10.}\,\,a_0&=\frac{\frac{5 c}{2}\pm \frac{\sqrt{-4800 P^{2} \rho ^{2} \sigma ^{2}+528 T^{2} \rho \sigma +25 c^{2}}}{2}}{5 Q},\nonumber \\ a_1&=-\frac{6 \rho T}{5 Q},\quad b_1=\frac{6 \sigma T}{5 Q},\quad a_2=-\frac{6 P \rho ^{2}}{Q},b_2=-\frac{6 P \sigma ^{2}}{Q},\nonumber \\ k&=\frac{C_0 L_1 \,c^{2}-\lambda ^{2}}{80 C_0 R_{r2} c \,\lambda ^{2} \rho },\quad c =\pm \frac{\sqrt{-3 C_0\lambda \left( 25 C_0 R_{r1} R_{r2} -12 L_1 \pm 5 \sqrt{25 C_0^{2} R_{r1}^{2} R_2^{2}-24C_0 L_1 R_{r1} R_{r2}}\right) } }{6 C_0 L_1},\nonumber \\ U_{10}(x,t)&=\frac{\frac{5 c}{2}\pm \frac{\sqrt{-4800 P^{2} \rho ^{2} \sigma ^{2}+528 T^{2} \rho \sigma +25 c^{2}}}{2}}{5 Q}-\frac{6 \rho T}{5 Q}\nonumber \\&\left( \frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2}\right) -\frac{6 P \rho ^{2}}{Q}\nonumber \\&\left( \frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2}\right) ^2+\frac{6 \sigma T}{5 Q}\nonumber \\&\quad \left( -\frac{\sqrt{|\sigma \rho |}}{\sigma } \left[ \frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\cosh 2\sqrt{|\sigma \rho |}(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh 2\sqrt{|\sigma \rho |}(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))-C_2}\right] \right) ^{-1}\nonumber \\&\quad -\frac{6 P \sigma ^{2}}{Q}\nonumber \\&\quad \left( -\frac{\sqrt{|\sigma \rho |}}{\sigma } \left[ \frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\cosh 2\sqrt{|\sigma \rho |}(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh 2\sqrt{|\sigma \rho |}(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))-C_2}\right] \right) ^{-2},\nonumber \\&here, \sigma \rho <0, \mu =0. \end{aligned}$$

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$$\begin{aligned} \mathbf{Set 11.}\,\,a_0&=-\frac{200 P^{2} \rho \sigma -25 P c -T^{2}}{50 Q P},\quad a_1=-\frac{6 \rho T}{5 Q},\quad b_1=\frac{6 \sigma T}{5 Q},\quad a_2=-\frac{6 P \rho ^{2}}{Q},\nonumber \\ b_2&=-\frac{6 P \sigma ^{2}}{Q},\quad k =\frac{C_0 L_1 \,c^{2}-\lambda ^{2}}{80 C_0 R_{r2} c \,\lambda ^{2} \rho },\nonumber \\ c&=\pm \frac{\sqrt{-3 C_0\lambda \left( 25 C_0 R_{r1} R_{r2} -12 L_1 \pm 5 \sqrt{25 C_0^{2} R_{r1}^{2} R_2^{2}-24C_0 L_1 R_{r1} R_{r2}}\right) } }{6 C_0 L_1},\nonumber \\ U_{11}(x,t)&=-\frac{200 P^{2} \rho \sigma -25 P c -T^{2}}{50 Q P}-\frac{6 \rho T}{5 Q}\nonumber \\&\quad \left( \frac{\sqrt{\sigma \rho }}{\sigma } \left[ \frac{C_1\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}{C_2\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_1\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}\right] \right) -\frac{6 P \rho ^{2}}{Q}\nonumber \\&\left( \frac{\sqrt{\sigma \rho }}{\sigma }\left[ \frac{C_1\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}{C_2\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_1\sin \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))}\right] \right) ^2\nonumber \\&+\frac{6 \sigma T}{5 Q}\left( \frac{\sqrt{\sigma \rho }}{\sigma } \left[ \frac{C_1\cos \sqrt{\sigma \rho } (k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_2\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}{C_2\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_1\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}\right] \right) ^{-1}\nonumber \\&\quad -\frac{6 P \sigma ^{2}}{Q}\left( \frac{\sqrt{\sigma \rho }}{\sigma } \left[ \frac{C_1\cos \sqrt{\sigma \rho } (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}{C_2\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_1\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}\right] \right) ^{-2},\nonumber \\&here, \sigma \rho >0, \mu =0. \end{aligned}$$

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$$\begin{aligned} \mathbf{Set 12.}\,\,a_0&=-\frac{200 P^{2} \rho \sigma -25 P c -T^{2}}{50 Q P},\quad a_1=-\frac{6 \rho T}{5 Q},\quad b_1=\frac{6 \sigma T}{5 Q},\nonumber \\ a_2&=-\frac{6 P \rho ^{2}}{Q},\quad b_2=-\frac{6 P \sigma ^{2}}{Q},\quad k =\frac{C_0 L_1 \,c^{2}-\lambda ^{2}}{80 C_0 R_{r2} c \,\lambda ^{2} \rho },\nonumber \\ c&=\pm \frac{\sqrt{-3 C_0\lambda \left( 25 C_0 R_{r1} R_{r2} -12 L_1 \pm 5 \sqrt{25 C_0^{2} R_{r1}^{2} R_2^{2}-24C_0 L_1 R_{r1} R_{r2}}\right) } }{6 C_0 L_1},\nonumber \\ U_{12}(x,t)&=-\frac{200 P^{2} \rho \sigma -25 P c -T^{2}}{50 Q P}-\frac{6 \rho T}{5 Q}\nonumber \\&\left( \frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2}\right) -\frac{6 P \rho ^{2}}{Q}\nonumber \\&(\frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\cosh 2\sqrt{|\sigma \rho |}(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2})^2 +\frac{6 \sigma T}{5 Q}\nonumber \\&\quad \left( -\frac{\sqrt{|\sigma \rho |}}{\sigma } \left[ \frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2}\right] \right) ^{-1}\nonumber \\&-\frac{6 P \sigma ^{2}}{Q}\nonumber \\&\quad \left( -\frac{\sqrt{|\sigma \rho |}}{\sigma } \left[ \frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_2\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2}\right] \right) ^{-2},\nonumber \\&here, \sigma \rho <0, \mu =0.\end{aligned}$$

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$$\begin{aligned} \mathbf{Set 13.}\,\,a_0&=-\frac{150 P^{2} \rho \sigma -10 P T \rho -25 c P -T^{2}}{50 Q P},\quad a_1=-\frac{6 \rho T}{5 Q}, b_1=\frac{6 \sigma T}{5 Q},\nonumber \\ a_2&=-\frac{6 P \rho ^{2}}{Q}, b_2=-\frac{6 P \sigma ^{2}}{Q},\quad k =\frac{C_0 L_1 \,c^{2}-\lambda ^{2}}{80 C_0 R_{r2} c \,\lambda ^{2} \rho },\nonumber \\ c&=\pm \frac{\sqrt{-3 C_0\lambda \left( 25 C_0 R_{r1} R_{r2} -12 L_1 \pm 5 \sqrt{25 C_0^{2} R_{r1}^{2} R_2^{2}-24C_0 L_1 R_{r1} R_{r2}}\right) } }{6 C_0 L_1},\nonumber \\ U_{13}(x,t)&=-\frac{150 P^{2} \rho \sigma -10 P T \rho -25 c P -T^{2}}{50 Q P}-\frac{6 \rho T}{5 Q}\nonumber \\&\quad \left( \frac{\sqrt{\sigma \rho }}{\sigma } \left[ \frac{C_1\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}{C_2\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_1\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}\right] \right) \nonumber \\&\quad -\frac{6 P \rho ^{2}}{Q}\left( \frac{\sqrt{\sigma \rho }}{\sigma }\left[ \frac{C_1\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}{C_2\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_1\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}\right] \right) ^2\nonumber \\&\quad +\frac{6 \sigma T}{5 Q}\left( \frac{\sqrt{\sigma \rho }}{\sigma }\left[ \frac{C_1\cos \sqrt{\sigma \rho } (k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\sin \sqrt{\sigma \rho }(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))}{C_2\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_1\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}\right] \right) ^{-1}\nonumber \\&\quad -\frac{6 P \sigma ^{2}}{Q}\left( \frac{\sqrt{\sigma \rho }}{\sigma } \left[ \frac{C_1\cos \sqrt{\sigma \rho } (k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_2\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}{C_2\cos \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_1\sin \sqrt{\sigma \rho }(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )}\right] \right) ^{-2},\nonumber \\&here, \sigma \rho >0, \mu =0. \end{aligned}$$

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$$\begin{aligned} \mathbf{Set 14.}\,\,a_0&=-\frac{150 P^{2} \rho \sigma -10 P T \rho -25 c P -T^{2}}{50 Q P},\nonumber \\ a_1&=-\frac{6 \rho T}{5 Q}, b_1=\frac{6 \sigma T}{5 Q},\nonumber \\ a_2&=-\frac{6 P \rho ^{2}}{Q}, b_2=-\frac{6 P \sigma ^{2}}{Q},\nonumber \\ k&=\frac{C_0 L_1 \,c^{2}-\lambda ^{2}}{80 C_0 R_{r2} c \,\lambda ^{2} \rho },\nonumber \\ c&=\pm \frac{\sqrt{-3 C_0\lambda \left( 25 C_0 R_{r1} R_{r2} -12 L_1 \pm 5 \sqrt{25 C_0^{2} R_{r1}^{2} R_2^{2}-24C_0 L_1 R_{r1} R_{r2}}\right) } }{6 C_0 L_1},\nonumber \\ U_{14}(x,t)&=-\frac{150 P^{2} \rho \sigma -10 P T \rho -25 c P -T^{2}}{50 Q P}-\frac{6 \rho T}{5 Q}\nonumber \\&\left( \frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2}\right) -\frac{6 P \rho ^{2}}{Q}\nonumber \\&\left( \frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_2\cosh 2\sqrt{|\sigma \rho |}(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_1\sinh 2\sqrt{|\sigma \rho |}(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))-C_2}\right) ^2 +\frac{6 \sigma T}{5 Q}\nonumber \\&\quad \left( -\frac{\sqrt{|\sigma \rho |}}{\sigma } \left[ \frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2}\right] \right) ^{-1}\nonumber \\&\quad -\frac{6 P \sigma ^{2}}{Q}\nonumber \\&\quad \left( -\frac{\sqrt{|\sigma \rho |}}{\sigma } \left[ \frac{C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_2\cosh 2\sqrt{|\sigma \rho |}(k(\frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }))+C_2}{C_1\cosh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )+C_1\sinh 2\sqrt{|\sigma \rho |}(k\left( \frac{1}{\beta }(x+\frac{1}{\Gamma (\beta )})^{\beta }-\frac{c}{\beta }(t+\frac{1}{\Gamma (\beta )})^{\beta }\right) )-C_2}\right] \right) ^{-2},\nonumber \\&here, \sigma \rho <0, \mu =0. \end{aligned}$$

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4.2 Visual interpretation of the derived solutions

Within this portion, we immediate and examine the new derived calculated solutions, represented by $U_1, U_2, U_3$, along with their graphical illustrations. The main reason for selecting these three functions $U_1, U_2, U_3$ as representatives is that, in the modified $\left( \frac{G^{'}}{G^2}\right)$-expansion method, solutions obtained when $\mu \ne 0$ are not discovered in the $\left( \frac{G^{'}}{G^2}\right)$-expansion method[57]. Therefore, in both the graphical illustrations in this section and the subsequent chapters, we focus on studying the newly discovered solutions when $\mu \ne 0$. Comparing with literature [57], investigating various physical properties of these new solutions is the main focus of this paper. To offer a visual understanding of these calculated solutions’ form and dynamics, we furnish graphical illustrations of select proxy soliton solutions, as depicted in Figs. 2–5. These illustrations depict a variety of soliton types, encompassing bright, solitary, and parabolic solitons, oscillatory singular waves of various types, in that order. The visual representations offer precious insights into the characteristics and behaviors of these soliton solutions across various parameter settings. Through the examination of these illustrations, we can discern the propagation and interaction of solitons within the electrical network delineated by the LNETLM incorporating a BD. Moreover, the physical elucidations of the BD are instrumental in understanding the fundamental processes driving the behavior of the solitons.

Fig. 2

Bright shape solitons in three-dimensional representation: the outcome of the BD on $|U_1(t, x; \beta )|$ is illustrated by 3D plots for numerous $\beta$ data points: a $\beta = 0.3$, b $\beta = 0.6$, and c $\beta = 0.9$. These plots are in the framework of of the LNETLM model with BDs. Additionally, and d presents 2D wave shapes of a–c post propagation time of $t = 5\mu s$. We use the next set of adjustable parameters: $\alpha = 0.25, C_0 = 8.7pF, \lambda = 2\,m, C_1= 0, C_2 = 1, \rho = 1, \mu =5.2, R_{r1} = 0.8\Omega , R_{r2} = 1\Omega , L_1 = 2nH$, and $\sigma = 2$

Fig. 3

Evolution of solitary waves with parameter $\beta$: the influence of the BD on $|U_2(x, t; \beta )|$ is depicted via 3D plots for a range of $\beta$ data points: a $\beta = 0.3$, b $\beta =0.6$, and c $\beta =0.9$. These visualizations are contextualized within the framework of the LNETLM that incorporates BD. In addition, 2D waveforms of a–c are presented in d after a propagation duration of $t = 5\mu s$. The next set of adjustable parameters is employed: $\alpha = 0.25, C_0 = 8.7pF, \lambda = 2\,m, C_1= 1, C_2 = 0, \rho = 1, \mu =-1, R_{r1} = 0.8\Omega , R_{r2} = 1\Omega , L_1 = 2nH$, and $\sigma = 4$

Figure 2: The three-dimensional diagrams of the bright shape solitons, represented by $|U_1(x, t; \beta )|$, are depicted in Fig. 2a–c. A clear observation from 2D cross-section diagrams about Fig. 2d that the gradually decreasing degree of opening with increasing fraction $\beta$. To generate the desired figure from the $|U_1(x, t; \beta )|$, we establish the rebellion data points at $R_{r1} = 0.8\Omega$ and $R_{r2} = 1\Omega$, while maintaining a constant linear inductance of $L_1 = 2nH$. We also set the added independent factors as follows: $\alpha = 0.25, C_0 = 8.7pF, \lambda = 2\,m, C_1= 0, C_2 = 1, \rho = 1, \mu =5.2$, and $\sigma = 2$.

Figure 3: Three-dimensional diagrams showcasing the evolution of a solitary with the escalating parameter $\beta$ in the interval of (0, 1) are presented in Fig. 3a–c. As the value of $\beta$ escalates, there is a corresponding gradual amplification in the amplitude of the solitary wave. This is evident from the behavior of the absolute value of $|U_2(x, t;\beta )|$. The associated two-dimensional cross-section diagrams, captured at $t=5\mu s$, are depicted in Fig. 2d. In order to construct the desired figure from the $|U_2(x, t; \beta )|$, we array the rebellion data points at $R_{r1} = 0.8\Omega$ and $R_{r2} = 1\Omega$, while keeping a steady linear inductance of $L_1 = 2nH$. The extra unrestricted elements are determined as follows: $\alpha = 0.25, C_0 = 8.7pF, \lambda = 2\,m, C_1= 1, C_2 = 0, \rho = 1, \mu =-1$, and $\sigma = 4$.

Fig. 4

Parabolic propagation of wave patterns with parameter $\beta$: the influence of BD on $|U_3(x, t; \beta )|$ is demonstrated using 3D plots for a range of $\beta$ values: a $\beta =0.3$, b $\beta =0.6$, and c $\beta =0.9$. These graphical representations are interpreted in the scope of the LNETLM, which integrates BDs. Moreover, 2D waveforms of a–c are showcased in d following a propagation period of $t = 5\mu s$. The subsequent set of adjustable parameters is utilized: $\alpha = 0.25, C_0 = 8.7pF, \lambda = 2\,m, C_1= 1, C_2 = -180, \rho = 1, \mu =4.4, R_{r1} = 0.8\Omega , R_{r2} = 1\Omega , L_1 = 2nH$, and $\sigma = 2$

Fig. 5

Transition to singular waveform with parameter $\beta$: the outcome of the BD on the $|U_5(x, t; \beta )|$ is demonstrated by 3D plots for a spectrum of $\beta$ data points: a $\beta =0.3$, b $\beta =0.6$, and c $\beta =0.9$. These graphical representations are interpreted within the scope of the LNETLM, which integrates BDs. Moreover, 2D waveforms of a–c are showcased in d following a propagation period of $t = 5\mu s$. The subsequent set of free parameters is utilized: $\alpha = 0.25, C_0 = 8.7pF, \lambda = 2\,m, C_1= 1, C_2 = -188, \rho = 1, \mu =7, R_{r1} = 0.8\Omega , R_{r2} = 1\Omega , L_1 = 2nH$, and $\sigma = 2$

Fig. 6

Evolution of soliton amplitude with fractional Parameter $\beta$: the impact of fractional behavior on $|U_1(t, x; \beta )|$ with specified parameters: $\alpha = 0.25, C_0 = 8.7pF, \lambda = 2\,m, C_1= 0, C_2 = -1, \rho = 1, \mu =5.2, R_{r1} = 0.8\Omega , R_{r2} = 1\Omega , L_1 = 2nH$, and $\sigma = 2$ is examined: a 3D plot is presented and b the alteration of the surface topography across the t-axis in relation to varying fractional values is depicted

Fig. 7

Impact of fractional parameters on soliton oscillation amplitude: a study is conducted on how fractionality impacts $|U_5(x, t; \beta )|$ using the next parameters: $\alpha = 0.25, C_0 = 8.7pF, \lambda = 2\,m, C_1= -110, C_2 = 1, \rho = 1, \mu =8, R_{r1} = 0.8\Omega , R_{r2} = 1\Omega , L_1 = 2nH$, and $\sigma = 2$: a 3D plot is presented and b the alteration of the surface topography across the t-axis in relation to varying fractional data points are depicted

Fig. 8

Fractional wave solutions comparison using different fractional derivatives: BD analysis of the obtained solutions, specifically$|U_1(x, t; \beta )|$ and $|U_2(x, t; \beta )|$, in relation to the CD and MTD solutions is performed. Figure a displays the 2D cross-section diagram of the comparative study of $|U_1(x, t; \beta )|$ when $\beta =0.4$. Figure b portrays the two-dimensional cross-section of the comparative investigation of$|U_2(x, t; \beta )|$ when $\beta =0.4$. The following parameters are employed: $C_0 = 8.7pF, \alpha = 0.25, \lambda = 2\,m, C_1= 1, C_2 = -180, \rho = 1, R_{r1} = 0.8\Omega , R_{r2} = 1\Omega , L_1 = 2nH, \sigma = 2$. As for the parameter $\mu$, it is 4.4 in Fig. (a) and 2.5 in Fig. (b)

Figure 4: The wave patterns exhibit a parabolic propagation towards the right with the incremental rise of the fractional parameter $\beta$ within the interval (0,1), as depicted in Fig. 4a–c. This trend is discernible from the absolute value of $|U_3(x, t; \beta )|$. In addition, Fig. 4d showcase the associated two-dimensional cross-section diagrams at $t=20s$. To generate the desired image from the computed $|U_3(t, x; \beta )|$, we establish the rebellion data points at $R_{r1} = 0.8\Omega$ and $R_{r2} = 1\Omega$, maintaining a constant linear inductance of $L_1 = 2nH$. The remaining adjustable factors are designated as follows: $\alpha = 0.25, C_0 = 8.7pF, \lambda = 2\,m, C_1= 1, C_2 = -180, \rho = 1, \mu =4.4$, and $\sigma =2$.

Figure 5: The wave patterns display oscillatory waves transitioning gradually toward a singular waveform with the incremental rise of the fractional parameter $\beta$ within the interval (0,1), as depicted in Fig. 5a–c. This trend is depicted from the absolute value of $|U_5(x, t; \beta )|$. Moreover, 2D waveforms of Fig. 5a–c are showcased in Fig. 5d following a propagation period of $t = 5\mu s$. The subsequent set of free parameters is utilized: $\alpha = 0.25, C_0 = 8.7pF, \lambda = 2\,m, C_1= 1, C_2 = -188, \rho = 1, \mu =7, R_{r1} = 0.8, R_{r2} = 1, L_1 = 2nH$, and $\sigma = 2$.

The pictorial examination of the investigated soliton solutions suggests that the employed approach can proficiently address our question, taking advantage of mathematical symbolic computing software similar to Maple. This strategy facilitates the efficient management of complex calculations and considerably diminishes computation time.

4.3 Effects of fractionality about the derived solutions

The distinct impact of fractionality on the got solution $|U_1(x, t; \beta )|$ is further illustrated in Fig. 6a–b, where the amplitude of soliton oscillations initially increases with the increment of the fractional parameter $\beta$, followed by a decrease. However, the influence of fractionality on $|U_5(x, t; \beta )|$ depicted in Fig. 7a–b exhibits the opposite trend: the amplitude of soliton oscillations decreases initially with the increment of the fractional parameter $\beta$, followed by an increase.

The above behaviors not only highlight the intricate influence of fractional parameters on soliton characteristics but also suggests a fascinating complexity in the dynamics governed by fractional models. It implies that fractional parameters play a pivotal role not just in modulating soliton amplitudes and shapes, but also in shaping the overall evolution and stability of soliton systems. Moreover, this insight opens avenues for deeper exploration into the interplay between fractionality and nonlinear phenomena, offering potential applications in various fields ranging from optics and fluid dynamics to quantum mechanics and beyond. Thus, understanding and harnessing the rich dynamics driven by fractional models can lead to breakthroughs in both theoretical understanding and practical applications of soliton physics.

Fig. 9

Impact of parameter variations on bright pulses in LNETLM with BD: the impact of losses on bright pulses in the LNETLM with BD, as provided by the $|U_1(t, x; \beta =0.9)|$ with $\alpha = 0.25, C_0 = 8.7pF, \lambda = 2, C_1= 1, C_2 = -180, \rho = 1, \mu =4.4, R_{r1} = 0.8\Omega , R_{r2} = 1\Omega , L_1 = 2nH$, and $\sigma = 2$, is examined: a The fluctuation of $L_1$ is considered when $R_{r1} = 0.8\Omega , R_{r2} = 1\Omega$, b The change in $R_{r1}$is observed when $L_1 = 2nH, R_{r2} = 1\Omega$, and c The variation of $R_{r2}$ is studied when $L_1 = 2nH, R_{r1} = 0.8\Omega$

Fig. 10

Impact of parameter variations on nonlinear transmission lines: the effects of LNETLM with BD, utilizing solution $|U_1(x, t; \beta =0.9)|$, are examined with particular focus on a $L_1$ is inductance, b $R_{r1}$ is resistance, c $R_{r2}$ is resistance, all within the framework of bright-shaped voltage pulses

4.4 A comparative analysis of the BD in relation to other FD.

In this research, we conduct a comparative examination between the BD, a form of FD, and other types of FD. The aforementioned serve to transform a fractional PDE into an ODE, thereby simplifying the analysis. Our focus is on three distinct fractional derivatives: the CD, the MTD, and the BD. Given particular limitations of the variable U(x, t) can be portrayed as $U(\xi )$ using various transformations: for the CD, $\xi = k(\frac{x^\beta }{\beta } -c\frac{t^\beta }{\beta })$; for the MTD, $\xi = k\Gamma (\tau +1)(\frac{x^\beta }{\beta } -c\frac{t^\beta }{\beta })$; and for the BD, $\xi = k\left( \frac{1}{\beta }(x + \frac{1}{\Gamma (\beta )})^\beta -\frac{c}{\beta }(t + \frac{1}{\Gamma {\beta }})^\beta \right)$. These transformations are employed to convert the fractional PDE described in Eq. (18) to an ODE shown in Eq. (19), under specific conditions.

Within the scope of a comparative analysis, the study entails the assessment and juxtaposition of the values of transformation variables, while acknowledging their inherent correlation with the fractional parameter. The potency of the results derived from this research is gauged against those unearthed via the $\left( \frac{G^{'}}{G^2}\right)$-expansion method. This section introduces two solutions, scrutinized from the standpoint of the definitions of the CD, the MTD, and the BD. The solutions, as outlined by Eqs. (25),(26) and (36), and expressed in terms of the CD, MTD, and BD, are presented as follows:

$$\begin{aligned} U_1(x,t){} & {} =-\frac{Z}{2 Q \left( 5 P \mu +T \right) }-\frac{6\rho (5P\mu +T)}{5Q}\left( -\frac{\mu }{2\rho }\right. \nonumber \\{} & {} \quad \left. -\left[ \frac{\sqrt{\Delta }(C_1\cosh (\frac{\sqrt{\Delta }}{2})(\xi ) +C_2\sinh \left( \frac{\sqrt{\Delta }}{2}\right) (\xi ))}{2\rho (C_2\cosh (\frac{\sqrt{\Delta }}{2})(\xi )+C_1\sinh \left( \frac{\sqrt{\Delta }}{2}\right) (\xi ))}\right] \right) \nonumber \\{} & {} \quad -\frac{6P\rho ^{2}}{Q}(-\frac{\mu }{2\rho } \nonumber \\{} & {} \quad -\left[ \frac{\sqrt{\Delta }(C_1\cosh (\frac{\sqrt{\Delta }}{2})(\xi ) +C_2\sinh \left( \frac{\sqrt{\Delta }}{2})(\xi )\right) }{2\rho (C_2\cosh (\frac{\sqrt{\Delta }}{2})(\xi )+C_1\sinh (\frac{\sqrt{\Delta }}{2})(\xi ))}\right] )^{2},\nonumber \\{} & {} here, \nonumber \\ Z{} & {} =5 P^{2} \mu ^{3}+40 P^{2} \mu \rho \sigma +6 P T \,\mu ^{2} \nonumber \\{} & {} \quad +12 P T \rho \sigma -5 P c \mu +T^{2} \mu -T c. \end{aligned}$$

(38)

here for the CD, $\xi = \frac{\sqrt{R_{r2} (\mu ^{2}-4 \rho \sigma ) R_{r1} }}{R_{r2} (\mu ^{2}-4 \rho \sigma ) \lambda \beta }(x^\beta -\frac{\lambda }{\sqrt{C_0 L_1}}t^\beta )$. For the MTD, $\xi = \frac{\sqrt{R_{r2} (\mu ^{2}-4 \rho \sigma ) R_{r1} }}{R_{r2} (\mu ^{2}-4 \rho \sigma ) \lambda \beta }\Gamma (\tau +1)(x^\beta - \frac{\lambda }{\sqrt{C_0 L_1}}t^\beta )$; and for the BD, $\xi = \frac{\sqrt{R_{r2} (\mu ^{2}-4 \rho \sigma ) R_{r1} }}{R_{r2} (\mu ^{2}-4 \rho \sigma ) \lambda }(\frac{1}{\beta }(x + \frac{1}{\Gamma (\beta )})^\beta - \frac{\lambda }{\beta \sqrt{C_0 L_1}}(t + \frac{1}{\Gamma {\beta }})^\beta )$.

$$\begin{aligned}{} & {} U_2(x,t)=-\frac{K}{2 Q \left( 5 P \mu +T \right) }-\frac{6\rho (5P\mu +T)}{5Q}(-\frac{\mu }{2\rho } \nonumber \\{} & {} \quad - [\frac{\sqrt{-\Delta }(C_1\cos (\frac{\sqrt{-\Delta }}{2})(\xi ) -C_2\sin (\frac{\sqrt{-\Delta }}{2})(\xi ))}{2\rho (C_2\cos (\frac{\sqrt{-\Delta }}{2})(\xi )+C_1\sin (\frac{\sqrt{-\Delta }}{2})(\xi ))}])\nonumber \\{} & {} \quad -\frac{6P\rho ^{2}}{Q}(-\frac{\mu }{2\rho } \nonumber \\{} & {} \quad -[\frac{\sqrt{-\Delta }(C_1\cos (\frac{\sqrt{-\Delta }}{2})(\xi ) -C_2\sin (\frac{\sqrt{-\Delta }}{2})(\xi ))}{2\rho (C_2\cos (\frac{\sqrt{-\Delta }}{2})(\xi )+C_1\sin (\frac{\sqrt{-\Delta }}{2})(\xi ))}])^{2},\nonumber \\{} & {} here,\nonumber \\{} & {} \quad K=5 P^{2} \mu ^{3}+40 P^{2} \mu \rho \sigma +6 P T \,\mu ^{2}\nonumber \\{} & {} \quad +12 P T \rho \sigma -5 P c \mu +T^{2} \mu -T c. \end{aligned}$$

(39)

For the conformable derivative, we define $\xi$ as $\xi = \frac{\sqrt{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) R_{r1} }}{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) \lambda \beta }(x^\beta -\frac{\lambda }{\sqrt{C_0 L_1}}t^\beta )$. In the case of the M-truncated derivative, $\xi$ is given by $\xi = \frac{\sqrt{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) R_{r1} }}{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) \lambda \beta }\Gamma (\tau +1)(x^\beta - \frac{\lambda }{\sqrt{C_0 L_1}}t^\beta )$. Lastly, for the BD, we express $\xi$ as

$\xi = \frac{\sqrt{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) R_{r1} }}{R_{r2} \left( \mu ^{2}-4 \rho \sigma \right) \lambda }\left( \frac{1}{\beta }(x + \frac{1}{\Gamma (\beta )})^\beta - \frac{\lambda }{\sqrt{C_0 L_1}}\frac{1}{\beta }(t + \frac{1}{\Gamma {\beta }})^\beta \right)$.

By varying the values of the unconstrained inputs, we visually illustrated the influence of fractionality on the two distinct analytical solutions using the three diverse FD methods noted previously. To confirm the exactness of our investigated solutions, we conducted an analogous evaluation of three fractional wave solutions, as detailed in Eqs. (38), (39), in connection with the CD, MTD, and BD, as shown in Fig. 8a–b. From these visual representations, it is evident that, among the various derivatives considered, the CD and MTD closely resemble each other, while significantly differing from the BD (Fig. 6a–b). It is noteworthy that choosing the BD as the form of the solutions is rather unique, as evidenced by this research. Upon thorough examination of the results, it can be confidently asserted that the investigated solutions, rooted in the CD framework, are representative compared to those obtained through the application of the BD methodology.

4.5 Effects of losses on bright solitary wave signals

      The losses in the LNETLM are mainly affected by the inductance $L_1 (L_1 =L/\lambda )$ and resistances $R_{r1}(R_{r1} = R_1/\lambda )$ and $R_{r2} (R_{r2} = R_2/\lambda )$ (refer to Fig. 1). As previously noted, the solution $U_1(t, x; \beta = 0.9)$ exhibits a bright soliton pulse with $\alpha = 0.25$.

Figure 9a shows how the voltage pulse fluctuates $U_1( t=5\mu s, x=1\,m; \beta = 0.9)$ with increasing inductance $(L_1)$ without changing any other factors, keeping everything else the same. These parameters include $\alpha = 0.25, C_0 = 8.7pF, \lambda = 2\,m, C_1= 0, C_2 = 1, \rho = 1, \mu =5.2$, and $\sigma = 2$. The research substantiates that raising the inductance $(L_1)$ in the circuit results in a decrease in voltage. The inductive voltage guides the line electricity flow by 90 degrees, indicating it reaches its peak a quarter cycle ahead of the electricity flow.

In the context of Fig. 9b, the illustration highlights how the increase in resistance $(R_{r1})$ results in a decrease in voltage, ascribed to heightened phase voltage losses along the wire. This consequence is directly linked to the heightened resistance prompting increased energy consumption and heat generation within the wire, thereby inducing a voltage decrease throughout its length. The remaining parameters are set as$\alpha = 0.25, C_0 = 8.7pF, \lambda = 2\,m, C_1= 0, C_2 = 1, \rho = 1, \mu =5.2$, and $\sigma = 2$.

Figure 9c depicts the correlation amid the voltage pulse (VP) and the resistance $(R_{r2})$, here maintaining all other parameters constant in $U_1(x=1m, t=5\mu s; \beta = 0.9)$. The other values are specified as follows: $\alpha = 0.25, C_0 = 8.7pF, \lambda = 2\,m, C_1= 0, C_2 = 1, \rho = 1, \mu =5.2$, and $\sigma = 2$. From the data shown in Fig. 7c, it is evident that boosting the resistivity $(R_{r2}$ leads to a corresponding spike in voltage. This can be ascribed to the fact that an amplified resistance leads to a decrease in bypass power dissipation, which indicates the power lost in the alternate routes about the circuit. With a higher resistivity, lesser current flows through the alternative paths, allowing a larger part to pass via the primary load. As a result, less energy is lost, enabling greater ability to be accessible for driving the load, thereby causing a rise in both the load voltage and the total circuit voltage. The analysis made from the illustration underlines that an enhancement in resistance $(R_{r2})$ leads to mitigated shunt current losses, consequently culminating in a voltage surge.

Figure 10a–c illustrate the effects of the independent variables $L_1, R_{r1}, R_{r2}$ on the NLTLs through $|U_1(x, t; \beta )|$. These graphs were generated using a range of values for these variables. As shown in Fig. 8a, the effect of inductance on the shape of the VP is substantial. Increasing $(L_1)$ results in a narrower amplitude and lower amplitude. Figure 10b demonstrates how the dispersion of the VP decreases with a rise in resistance $(R_{r1})$. A higher $(R_{r1})$ causes the oscillations to shift to the right, while the amplitude width decreases and the amplitude enhancement. Conversely, Fig. 10c shows how the scattering of the VP decreases with an strengthen in resistance $(R_{r2})$. A higher $(R_{r2})$ leads to the fluctuations shifting to the left, the amplitude breadth expanding, and the amplitude decreasing.

4.6 Phase diagrams and bifurcation patterns

Here we focus on one circumstance where $c^2 =\lambda ^2\eta ^2_0$. In this case, the parameter $T=0$ and the first order term of Eq. (20)

$$\begin{aligned} \begin{aligned}&PU''+QU^{2}-cU=0. \end{aligned} \end{aligned}$$

(40)

Adjustments in the system’s parameters lead to the manifestation of bifurcation phenomena, profoundly influencing its dynamical responses. The forthcoming discussions will investigate Eq. (40), analyzing it within the context of a two-dimensional dynamical system.

By denoting $V = U^{'}$, we can reformulate Eq. (40) into the framework of dynamics governed by Hamilton equation.

$$\begin{aligned} \left\{ \begin{aligned}&\dot{U}=V,\\&\dot{V}=\frac{1}{P}(cU-QU^2). \end{aligned} \right. \end{aligned}$$

(41)

The Hamiltonian for system (41) is given by:

$$\begin{aligned} H(U,V)=\frac{V^2}{2}-\frac{c}{2P}U^2+\frac{Q}{3}U^3. \end{aligned}$$

(42)

Utilizing vector field analysis in planar dynamics, we can explore the phase trajectories of system (41). The phase portrait of (41) is subject to change based on the chosen set of nonzero parameters Q and c. In fact, system (41) has two equilibrium points:

$$\begin{aligned} M_1=(\frac{c}{Q}, 0),\ \ M_2=(0, 0). \end{aligned}$$

(43)

To determine the Jacobian of system (41), one can employ the following calculation method:

$$\begin{aligned} J(U, V)=\left| \begin{matrix} 0 &{} 1 \\ \frac{1}{P}(c-2QU) &{} 0 \\ \end{matrix} \right| =-\frac{1}{P}(c-2QU). \end{aligned}$$

(44)

By examining the equilibrium point $(M_i)$ under three different conditions - when $J(U, V)<0$, it is characterized as a saddle point; when $J(U, V)>0$, it exhibits a center behavior; and when $J(U, V)=0$, it transforms into a cuspidal point. To assess the phase portrait view of system (41), we investigate the bifurcation behavior by exploring various parameter settings.Therefore, we demonstrate the influence of bifurcation by illustrating the phase portraits of the system under the subsequent parameter configurations.

Configuring the system parameters with specific values, such as case 1: $Q=1$ and $c=1$, results in the equilibrium point $M_2$ being classified as a center, $M_1$ as saddle, which can be visualized in Fig. 11a; case 2: $Q=-1$ and $c=1$, results in the equilibrium point $M_1$ being classified as a center, $M_2$ as saddle, which can be visualized in Fig. 12a; case 3: $Q=-1$ and $c=-1$, results in the equilibrium point $M_2$ being classified as a center, $M_1$ as saddle, which can be visualized in Fig. 13a; case 4: $Q=1$ and $c=-1$, results in the equilibrium point $M_1$ being classified as a center, $M_2$ as saddles, which can be visualized in Fig. 14a.

Fig. 11

Phase portraits and 3D phase portraits of case 1: the phase portraits and 3d phase portraits of case 1 are exhibited in Figure a and Figure b, respectively. The star symbol indicates the equilibrium points

Fig. 12

Phase portraits and 3D phase portraits of case 2: the phase portraits and 3d phase portraits of case 2 are exhibited in Figure a and Figure b, respectively. The star symbol indicates the equilibrium points

Fig. 13

Phase portraits and 3D phase portraits of case 3: the phase portraits and 3d phase portraits of case 3 are exhibited in Figure a and Figure b, respectively. The star symbol indicates the equilibrium points

Fig. 14

Phase portraits and 3D phase portraits of case 4: the phase portraits and 3d phase portraits of case 4 are exhibited in Figure a and Figure b, respectively. The star symbol indicates the equilibrium points

4.7 Sensitive analysis in regards to the initial value

To examine the sensitivity characteristics of Eq. (41) concerning the initial value, we explore previously mentioned case 1 and 2. Results are presented in Fig. 15a and b for when the system settings c and Q are configured in response to case 1. And results are presented in Fig. 16a and b for when the system settings c and Q are configured in response to case 2. Phase diagrams depicted in Figs. 15b–16b showcase the system (41) with two disparate initial configurations. In contrast, the time series of the system (41) are presented in Fig. 15a–16a, the blue curve depicts the system behavior when (U, V) equals (0.25, 0.1) and $(-0.25,0.1)$, whereas the red curve illustrates the system dynamics at $(U,V)=(0.5,0.25)$ and $(-0.5,0.25)$.

We examine the orbits generated from different initial conditions to observe the influence of sensitivity. This phenomenon becomes evident when examining the contrast between the blue and red curves depicted in Figs. 15a, b and 16a, b. The outcomes unequivocally show that a significant modification to the initial condition exerts a profound influence regarding the behavior of the system, assuming all other parameters remain unchanged.

Subsequently, our research delved into how parameter variations influence the system. This scrutiny is evidenced by comparing Fig. 15a, b with Fig. 16a, b. The results uncover that the dynamic system exhibits considerable sensitivity to changes in parameters, demonstrating how minor adjustments to these values can markedly affect the behavior of the system.

Fig. 15

Sensitivity analysis of initial conditions in system: we utilize figure a to demonstrate the time series and figure b to showcase the phase plot of case 1, enabling us to perform sensitivity analysis

Fig. 16

Sensitivity analysis of initial conditions in system: we utilize figure a to demonstrate the time series and figure b to showcase the phase plot of case 2, enabling us to perform sensitivity analysis

4.8 Chaotic analysis

Understanding the dynamic behaviors of system (20) holds significant importance. This section thoroughly explores the response of Eq. (41) to perturbations induced by noise and critically examines its demonstration of chaotic dynamics. The analysis utilizes the following model,

$$\begin{aligned} \left\{ \begin{aligned}&\dot{U}=V,\\&\dot{V}=\frac{1}{P}(cU-QU^2)+f\sin (\omega _0 t). \end{aligned} \right. \end{aligned}$$

(45)

To conduct the chaotic analysis, we will evaluate the influence of noise sensitivity from two perspectives. Our particular attention will be on examining the parameters f representing the amplitude and $\omega _0$ representing the frequency. By varying these parameters, we aim to ascertain their impact on the system’s chaotic behavior.

Initially, we investigate the effect of f on Eq. (41), while maintaining a constant frequency. The simulation outcomes are visualized in Fig. 17a and b, providing understading of the system’s response to variations in amplitude parameter.

Next, we explore influence of the frequency $\omega _0$ on Eq. (41), while holding amplitude f constant. Outcomes are illustrated in Fig. 18a and b, providing insights into the system’s behavior as the frequency parameter varies.

Using the same experimental settings as in cases 1 and 2, we do initialization at (0.25, 0.2) for case 1. In Fig. 17a and b, we explore the system’s response by examining the behavior under a constant frequency $\omega _0=\pi$ and a zero amplitude. The outcome depicted with blue line, illustrate the time evolution and phase trajectories. Subsequently, we investigate the cases of $f=0.2$ (indicated by the red curve) and $f=0.4$ (represented by the yellow curve). The presence of chaotic dynamics in the system becomes apparent as we observe a distinct difference between the behavior depicted by the blue line and the rest of the lines, influenced by noise. This means, the system exhibits significant sensitivity to changes in amplitude.

Fig. 17

Chaotic analysis: in case 1, we examine the chaotic dynamics of the system. Figure a depicts time series, Figure b showcases the phase plot

To assess the influence of the frequency $\omega _0$ on the dynamic system, we display the findings in Fig. 18a and b. Firstly, by setting $f=0$, the simulation results of the time series and phase orbits are plotted as the blue curve. Then, we consider the case where $f=0.2$ and $\omega _0=\pi$, represented by the red curve. Furthermore, we examine the scenario with $f=0.2$ and $\omega _0=2\pi$, depicted as the yellow curve. The disparity observed between the blue line and the other lines indicates the existence of chaotic dynamics in the system. Moreover, the dynamic system demonstrates significant sensitivity to frequency parameter.

Fig. 18

Chaotic analysis: in case 1, we analyze the chaotic behavior of the system. Figure a depicts time series, Figure b showcases the phase plot

5 Comparison

In this study, we applied the modified $\left( \frac{G^{'}}{G^2}\right)$-expansion method to analyze LNETLM, uncovering a multitude of new exact solutions not previously documented in relevant literature [57].

In a prior work by Paul G C et al. [57] in 2023, they also utilized the ($\frac{G'}{G^2}$)-expansion method to explore exact solutions of lossy LNETLM. However, our investigation, represented by equation (16), introduces a different formulation: $(\frac{G'}{G^2})'=\sigma +\mu (\frac{G'}{G^2})+\rho (\frac{G'}{G^2})^2$. This equation significantly deviates from equation (18) ($(\frac{G'}{G^2})'=u+\lambda (\frac{G'}{G^2})^2$) in [57]. Notably, when $\mu \ne 0$, equation (16) extends the findings of [57].

Comparison between our study and [57] reveals novel solutions, denoted as $u_1, u_2, u_3, u_4, u_5, u_6$, obtained specifically when $\mu \ne 0$ in our analysis. Our approach is characterized by a more intricate computational method compared to the methodology employed in [57].

When the maximum velocity of propagating waves $c_{\text {max}}$ meets the condition $c_{\text {max}} = \lambda \eta _0$, this paper advances to convert the equations into a Hamiltonian system in subsequent sections, investigating the phase portrait of the LNETLM. Moreover, Chaos and bifurcation phenomena are scrutinized using these phase trajectories. These analytical approaches have not been explored in prior literature [57, 60,61,62].

Table 2 Contrasting our findings with those documented in earlier studies

Furthermore, we have included Table 2, which offers a comprehensive comparison between the solutions obtained in this study and those published in previous research. This table emphasizes the distinctive and innovative characteristics of our obtained solutions, showcasing the novelty of our findings.

In Table 2, the symbol “$\surd$” signifies the presence of a specific feature or characteristic, while “$\times$” denotes its absence. “AFEL” stands for the “fraction effect analysis on LNETLM”, and “AEDFDFEL” denotes the “effect analysis of different fractional derivatives on LNETLM”.

6 The accuracy analysis of results

       Due to the structure of the network equation, which includes high-order derivatives, it is necessary to substitute the obtained solution back into the original equation for accuracy analysis. Here, we introduce an accuracy analysis function, and if the solution is substituted into this accuracy analysis function, the closer the function value approaches zero, the more accurate the solution of this equation will be. However, due to space limitations in the article, we have only selected a few representative solutions to analyze the values of the accuracy function. In this paper, we randomly selected $U_3(x, t)$ and $U_5(x, t)$ from the newly discovered solutions. In practice, we have performed similar operations on other solutions as well, although they are not individually listed in the paper. The next Tables 3 and tab:4 clearly reflect the high accuracy and precision of the solutions we obtained.

We define the accuracy analysis function of this structure (AA) in Eq. (19) as follows:

$$\begin{aligned} \begin{aligned} AA&= R_{r2}C_0k^2\lambda ^2cU''+k(L_1C_0c^2-\lambda ^2)U'\\&\quad +R_{r1}C_0\alpha cU^2-R_{r1}C_0cU. \end{aligned} \end{aligned}$$

(46)

Table 3 AA ($C_1=0,C_2=1, R_{r1}=0.8, R_{r2}=1, L_1=2, C_0=8.7$) for the solution $U_{3}(x, t)$ in Eq. (26)

Table 4 AA ($C_1=0,C_2=1, R_{r1}=0.8, R_{r2}=1, L_1=2, C_0=8.7$) for the solution $U_{5}(x, t)$ in Eq. (28)

7 Conclusion and Future outlook

       This paper explores the application of the modified $(\frac{G'}{G^2})$-expansion method to LNETLM with a BD, a novel approach not previously investigated for deriving analytical solutions. We primarily investigate the LNETLM system from the following five aspects:

• (1) Employing the modified $\left( \frac{G^{'}}{G^2}\right)$-expansion method on LNETLM with a beta derivative marks a groundbreaking approach for deriving analytical solutions. This introduces a novel perspective, presenting previously unrecorded precise soliton solutions for LNETLM.

• (2) Through computer simulations, this study unveils various wave phenomena, including bright, solitary, and parabolic solitons, along with oscillatory singular waves. These findings deepen our understanding of soliton behavior within the model, enriching the characterization of wave dynamics in LNETLM.

• (3) The comparison between conformable derivative and M-truncated derivative solutions, juxtaposed with the beta derivative solution, illuminates the unique aspects of employing the beta derivative methodology. This comparative analysis provides insights into the advantages and limitations of different derivative frameworks in modeling electrical transmission line systems.

• (4) Transforming the LNETLM equation into a Hamiltonian system and exploring the phase portrait of LNETLM, this study offers a comprehensive analysis of the system’s dynamic behavior. Investigating how phase trajectories change under various parameter settings, coupled with examining Chaos and bifurcation phenomena, enhances our understanding of system dynamics and stability.

• (5) The analyses presented in this study, previously unexplored in the literature, push the knowledge frontier within the domain of lossy electrical transmission line models. This contributes to advancing theoretical understanding and potential practical applications in areas such as signal transmission and communication systems.

We uncovered numerous precise solutions for nonlinear electrical transmission line models, previously undocumented in relevant literature. These solutions, including $U_1(x, t)-U_6(x, t)$, were revealed through computer simulations, illustrating various wave phenomena such as bright, solitary, and parabolic solitons, as well as oscillatory singular waves. By adjusting free parameters and fractional derivative orders, we generated a plethora of charts, providing valuable perspectives into the behavior of solitons in the model.

An examination of these visual representations reveals a strong similarity between the conformable derivative and M-truncated derivative solutions, which stand in stark contrast to the BD solution. Notably, opting for the BD as the solution form is a distinctive choice, as evidenced by this study. A comprehensive examination of the results confirms that solutions derived within the conformable derivative framework are more representative compared to those obtained using the BD methodology.

The practical implications of this study: The presence of Chaos in signal transmission may have far-reaching effects, impacting the stability and reliability of systems. Controlling chaotic phenomena becomes essential to optimize system performance, especially in applications such as transient effects in electrical systems, nonlinear transmission lines, wireless communication, and telecommunication systems. Through further research on controlling Chaos, we can enhance the predictability and stability of these systems. The inherent fractal structure in the LNETLM possesses unique properties like self-similarity and scale invariance, which can significantly influence various aspects of system behavior. Understanding these fractal characteristics is crucial for accurately characterizing signal transmission, waveform patterns, and ensuring system stability. This understanding opens up new avenues for practical applications in fields like wireless communication, where optimized waveform patterns are critically important for effective transmission. Sensitivity analysis has become a key tool for evaluating the impact of parameter variations on system behavior. This analytical method is essential for determining optimal parameter ranges, optimizing system design, and predicting the stability and reliability of system responses. In practical applications, such as adjusting parameters of obtained analytic solutions in LNETLM to appropriate values, sensitivity analysis allows us to customize wave characteristics to meet specific requirements, addressing the stringent demands of real-world applications or comparing theoretical results with experimental data. This capability has significant potential in enhancing the performance and reliability of various systems, including those in the telecommunications and electrical engineering fields. It is important to mention that the width and oscillations of soliton-like pulses can be controlled by propagation velocity and relevant parameters of the method and model, and the soliton-like pulses studied in this paper can also be used to explain voltage wave propagation behavior in physical systems with memory.

Overall, the research outcomes provide some contributions to the field by offering novel analytical solutions, revealing intricate wave phenomena, comparing derivative methodologies. By tuning the parameters in the got accurate solutions for LNETLM, the model can proficiently shape wave characteristics and generate desired wave profiles for comparison with real-world results. The results highlight the significant influence of fractional data points on wave behaviors and offer insights into the implicit Hamiltonian dynamics of the system, enhancing our comprehension of system dynamics in LNETLM and making some contributions to the field.

In future studies, we consider expanding the polynomial in equation (22) to a rational expression:

$$\begin{aligned} {{U}({\xi }) = \frac{a_0+a_1\left( \frac{G^{'}}{G^2}\right) +a_2\left( \frac{G^{'}}{G^2}\right) ^2+b_1\left( \frac{G^{'}}{G^2}\right) ^{-1}+b_2\left( \frac{G^{'}}{G^2}\right) ^{-2}}{c_0+c_1\left( \frac{G^{'}}{G^2}\right) +c_2\left( \frac{G^{'}}{G^2}\right) ^2+d_1\left( \frac{G^{'}}{G^2}\right) ^{-1}+d_2\left( \frac{G^{'}}{G^2}\right) ^{-2}},} \end{aligned}$$

(47)

here the unknown constants $a_0, a_1, a_2, b_1, b_2, c_0, c_1, c_2, d_1, d_2$ will be found later. Due to the more complex form of Eq. (47) compared to Eq. (22), if we substitute the rational form of $U(\xi )$ into the original equation, we should be able to obtain more accurate solutions, which implies the possibility of discovering more soliton solutions. Indeed, such calculations will undoubtedly be more complex, but this is a direction we are prepared to explore fearlessly in the future!