$ k $NN local linear estimation of the conditional density and mode for functional spatial high dimensional data

Fatimah Alshahrani; Wahiba Bouabsa; Ibrahim M. Almanjahie; Mohammed Kadi Attouch; Fatimah Alshahrani; Wahiba Bouabsa; Ibrahim M. Almanjahie; Mohammed Kadi Attouch

doi:10.3934/math.2023809

AIMS Mathematics

2023, Volume 8, Issue 7: 15844-15875. doi: 10.3934/math.2023809

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Research article

$k$ NN local linear estimation of the conditional density and mode for functional spatial high dimensional data

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, Riyadh 11671, Saudi Arabia
2.
Laboratory of Statistics and Stochastic Processes, University of Djillali Liabes, BP 89, Sidi Bel Abbes 22000, Algeria
3.
Department of Mathematics, College of Science, King Khalid University, Abha 62223, Saudi Arabia

Received: 12 January 2023 Revised: 02 April 2023 Accepted: 11 April 2023 Published: 04 May 2023
MSC : 62H12, 62G07, 62G35, 62G20

Traditionally, regression problems are examined using univariate characteristics, including the scale function, marginal density, regression error, and regression function. When the correlation between the response and the predictor is reasonably straightforward, these qualities are helpful and instructive. Given the predictor, the response's conditional density provides more specific information regarding the relationship. This study aims to examine a nonparametric estimator of a scalar response variable's function of a density and mode, given a functional variable when the data are spatially dependent. The estimator is then derived and established by combining the local linear and the $k$ nearest neighbors methods. Next, the suggested estimator's uniform consistency in the number of neighbors (UNN) is proved. Finally, to demonstrate the efficacy and superiority of the acquired results, we applied our new estimator to simulated and real data and compared it to the existing competing estimator.

Keywords:

Citation: Fatimah Alshahrani, Wahiba Bouabsa, Ibrahim M. Almanjahie, Mohammed Kadi Attouch. $k$ NN local linear estimation of the conditional density and mode for functional spatial high dimensional data[J]. AIMS Mathematics, 2023, 8(7): 15844-15875. doi: 10.3934/math.2023809

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Abstract

1. Introduction

The notion of fractional derivative (FD) is more than three thousand years old, the role of fractional calculus has been increasing due to its application zone in various domains including biology, semiconductor industry, optical communication, energy quantization, quantum chemistry, wave propagation, protein folding and bending, condensed matter physics, solid state physics, nanotechnology and industry, laser propagation, nonlinear optics etc. the fractional differential equations (FDEs), have received a great deal of interest from scholars and researchers. Many mathematicians presented diverse types of FDs, such as that given in ^[1,2]. The most famous ones are Hadamard, Marchaud, Riemann-Liouville, Grunwald-Letnikov, Kober, Caputo, Riesz and Erdelyi.

Lately, a novel FD has been presented by Khalil et al. ^[3] and others ^[4,5,6], called conformable FD. Due to the significance of the exact solutions of NLSEs, plenty of mathematicians solved them with conformable derivative, who used the different methods like first integral method (FIM) ^[7,8], functional variable method (FVM) ^[7,9], trial equation method (TEM) ^[10], modified trial equation method (MTEM) ^[11], direct algebraic method (DAM) ^[12] and sine-Gordon expansion method (SGEM) ^[13] to find the exact solutions to NLSEs.

In ^[14], Younas et al. presented CTFMNLSE and studied the exact solutions of it by the generalized exponential rational function method. Also in ^[15,16], authors introduced some new solutions of CTFMNLSE via FIM, FVM, TEM and MTEM.

Motivated by the work done in ^[14,15,16], we consider the following CTFMNLSE:

$\begin{equation} \text{i}\mathbf{D}^{\alpha}_{\tau}\Psi+\sigma_{1} \Psi_{\mathfrak{X}\mathfrak{X}}+\sigma_{2} \vert \Psi \vert^{2}\Psi = \text{i}\delta_{1}\Psi_{\mathfrak{X}\mathfrak{X}\mathfrak{X}}+\text{i}\delta_{2} \Psi^{2}\Psi_{\mathfrak{X}}^{*}-\text{i}\delta_{3} \vert \Psi \vert^{2}\Psi_{\mathfrak{X}}+\delta_{4}\Psi \:\:\:\:\:\:\:\: 0 < \alpha\leq 1, \end{equation}$

(1.1)

where $\sigma_{1} = \dfrac{\mathcal{P}_{0}}{8\mathcal{K}_{0}^{2}(-3\cos(\Theta)+2)}$ , $\sigma_{2} = \dfrac{-\mathcal{P}_{0}\mathcal{K}_{0}^{2}}{2}$ , $\delta_{1} = \dfrac{\mathcal{P}_{0}\cos(\Theta)}{16\mathcal{K}_{0}^{3}(-5\cos^{2}(\Theta)-6)}$ , $\delta_{2} = \dfrac{\mathcal{P}_{0}\mathcal{K}_{0}\cos(\Theta)}{4}$ , $\delta_{3} = \dfrac{3\mathcal{P}_{0}\mathcal{K}_{0}}{2}, \delta_{4} = \mathcal{K}_{0}\vert \Psi \vert^{2}_{\mathfrak{X}} \bigg\vert_{\mathfrak{X} = 0}$ , $\mathcal{P}_{0}$ and $\mathcal{K}_{0}$ are the frequency and the wave number of the carrier wave, respectively, and the operator $\mathbf{D}^{\alpha}$ of order $\alpha$ , where $\alpha\in(0, 1]$ represents the conformable fractional derivative.

2. Preliminaries

In this section, we present some properties and definitions of the conformal derivative and other Preliminaries.

Definition 2.1. ^[3] Suppose $\Omega : (0, \infty) \rightarrow \mathbb{R}$ is a function. Therefore the conformal fractional derivative of $\Omega$ of order $\alpha$ is as follows

$\begin{equation} \mathbf{T}_{\alpha}(\tau) = \lim\limits_{\epsilon\rightarrow 0 } \dfrac{\Omega (\tau+\epsilon \tau^{1-\alpha})-\Omega(\tau)}{\epsilon} \end{equation}$

(2.1)

for all $0 < \alpha < 1, 0 < \tau$ .

Definition 2.2. ^[3] Suppose $\iota\geq 0$ and $\tau \geq \iota$ . Also, suppose $\Omega$ is a function defined on $(\iota, \tau]$ and $\alpha \in \mathbb{R}$ . Therefore, the $\alpha$ -fractional integral of $\Omega$ is defined by,

$\begin{equation} \mathbf{I}_{\iota}^{\alpha}\Omega(\tau) = \int_{\iota}^{\tau}\dfrac{\Omega(\varsigma)}{\varsigma^{1-\alpha}}d\varsigma, \end{equation}$

(2.2)

if the Riemann improper integral exists.

Theorem 2.3. ^[3]Suppose $0 < \alpha \leq 1$ , and $\Omega$ and $\mho$ are $\alpha-$ differentiable at a point $\tau$ , therefore

(i) $\mathbf{T}_{\alpha}(\varpi_{1}\Omega+\varpi_{2} \mho) = \varpi_{1} \mathbf{T}_{\alpha}(\Omega)+\varpi_{2} \mathbf{T}_{\alpha}(\mho), \:\: \forall \varpi_{1}, \varpi_{2}\in \mathbb{R}$ .

(ii) $\mathbf{T}_{\alpha}(t^{\varpi}) = \varpi \tau^{\varpi-\alpha}, \:\:\forall \varpi\in \mathbb{R}$ .

(iii) ${}_{\tau}\mathbf{T}_{\alpha}(\Omega\mho) = \Omega \mathbf{T}_{\alpha}(\mho)+\mho\mathbf{T}_{\alpha}(\Omega)$ .

(iv) $\mathbf{T}_{\alpha}\bigg(\dfrac{\Omega}{\mho}\bigg) = \dfrac{\mho\mathbf{T}_{\alpha}(\Omega)-\Omega\mathbf{T}_{\alpha}(\mho)}{\mho^{2}}$ .

Furthermore, if $\Omega$ is differentiable, then $\mathbf{T}_{\alpha}(\Omega)(\tau) = \tau^{1-\alpha}\dfrac{d\Omega}{d\tau}$ .

Theorem 2.4. ^[3] Suppose $\Omega:(0, \infty)\rightarrow \mathbb{R}$ is a function s.t. $\Omega$ is differentiable and also $\alpha$ -differentiable. Suppose $\mho$ is a function defined in the range of $\Omega$ and also differentiable; therefore, one has the following rule

$\mathbf{T}_{\alpha}(\Omega \mathit{\text{o}} \: \mho) (\tau) = \tau^{1-\alpha} \mho(\tau)\Omega^{\prime} (\mho(\tau)).$

Remark 2.5. Let

$\begin{equation} \mathcal{Q}^{\prime}(\xi) = \text{Ln} (A)(\mathfrak{P}_{0}+\mathfrak{P}_{1}\mathcal{Q}(\xi)+\mathfrak{P}_{2}\mathcal{Q}^{2}(\xi)), \:\:\:\:\:\:A\neq 0, 1. \end{equation}$

(2.3)

The solutions of ODE (2.3) are:

(1) When $\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2} < 0$ and $\mathfrak{P}_{2}\neq 0,$

$\begin{equation*} \mathcal{Q}_{1}(\xi) = -\dfrac{\mathfrak{P}_{1}}{2\mathfrak{P}_{2}}+\dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{2\mathfrak{P}_{2}}\text{tan}_{A}\bigg( \dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{2}\xi \bigg), \end{equation*}$

$\begin{equation*} \mathcal{Q}_{2}(\xi) = -\dfrac{\mathfrak{P}_{1}}{2\mathfrak{P}_{2}}-\dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{2\mathfrak{P}_{2}}\text{cot}_{A}\bigg( \dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{2}\xi \bigg), \end{equation*}$

$\begin{eqnarray*} \mathcal{Q}_{3}(\xi)& = &-\dfrac{\mathfrak{P}_{1}}{2\mathfrak{P}_{2}}+\dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{2\mathfrak{P}_{2}}\text{tan}_{A}( \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}\xi )\\ &\pm &\notag\dfrac{\sqrt{-sr ( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{2\mathfrak{P}_{2}}\text{sec}_{A}( \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}\xi ), \end{eqnarray*}$

$\begin{eqnarray*} \mathcal{Q}_{4}(\xi)& = &-\dfrac{\mathfrak{P}_{1}}{2\mathfrak{P}_{2}}-\dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{2\mathfrak{P}_{2}}\text{cot}_{A}( \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}\xi )\\ &\pm &\notag\dfrac{\sqrt{-sr ( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{2\mathfrak{P}_{2}}\text{csc}_{A}( \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}\xi ), \end{eqnarray*}$

$\begin{eqnarray*} \mathcal{Q}_{5}(\xi)& = &-\dfrac{\mathfrak{P}_{1}}{2\mathfrak{P}_{2}}+\dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{4\mathfrak{P}_{2}}\text{tan}_{A}( \dfrac{ \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{4}\xi )\\ &- &\notag\dfrac{\sqrt{- ( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{4\mathfrak{P}_{2}}\text{cot}_{A}( \dfrac{ \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{4}\xi ), \end{eqnarray*}$

(2) When $\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2} > 0$ and $\mathfrak{P}_{2}\neq 0,$

$\begin{equation*} \mathcal{Q}_{6}(\xi) = -\dfrac{\mathfrak{P}_{1}}{2\mathfrak{P}_{2}}-\dfrac{\sqrt{\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{2\mathfrak{P}_{2}}\text{tanh}_{A}\bigg( \dfrac{\sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{2}\xi \bigg), \end{equation*}$

$\begin{equation*} \mathcal{Q}_{7}(\xi) = -\dfrac{\mathfrak{P}_{1}}{2\mathfrak{P}_{2}}-\dfrac{\sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{2\mathfrak{P}_{2}}\text{coth}_{A}\bigg( \dfrac{\sqrt{\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{2}\xi \bigg), \end{equation*}$

$\begin{eqnarray*} \mathcal{Q}_{8}(\xi)& = &-\dfrac{\mathfrak{P}_{1}}{2\mathfrak{P}_{2}}-\dfrac{\sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{2\mathfrak{P}_{2}}\text{tanh}_{A}( \sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}\xi )\\ &\pm &\notag\text{i}\dfrac{\sqrt{sr ( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{2\mathfrak{P}_{2}}\text{sech}_{A}( \sqrt{\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}\xi ), \end{eqnarray*}$

$\begin{eqnarray*} \mathcal{Q}_{9}(\xi)& = &-\dfrac{\mathfrak{P}_{1}}{2\mathfrak{P}_{2}}-\dfrac{\sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{2\mathfrak{P}_{2}}\text{coth}_{A}( \sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}\xi )\\ &\pm &\notag\dfrac{\sqrt{sr( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{2\mathfrak{P}_{2}}\text{csch}_{A}( \sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}\xi ), \end{eqnarray*}$

$\begin{eqnarray*} \mathcal{Q}_{10}(\xi)& = &-\dfrac{\mathfrak{P}_{1}}{2\mathfrak{P}_{2}}-\dfrac{\sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{4\mathfrak{P}_{2}}\text{tanh}_{A}( \dfrac{ \sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{4}\xi )\\ &- &\notag\dfrac{\sqrt{\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{4\mathfrak{P}_{2}}\text{coth}_{A}( \dfrac{ \sqrt{\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{4}\xi ), \end{eqnarray*}$

where generalized hyperbolic and triangular functions are given by

$\text{cosh}_{A}(\theta) = \dfrac{s A^{\theta}+r A^{-\theta}}{2}, \:\:\:\:\:\:\text{sinh}_{A}(\theta) = \dfrac{s A^{\theta}-r A^{-\theta}}{2},$

$\text{coth}_{A}(\theta) = \dfrac{s A^{\theta}+r A^{-\theta}}{s \text{e}^{\theta}-r\text{e}^{-\theta}}, \:\:\:\:\:\: \text{tanh}_{A}(\theta) = \dfrac{s A^{\theta}-r A^{-\theta}}{s \text{e}^{\theta}+r\text{e}^{-\theta}},$

$\text{csch}_{A}(\theta) = \dfrac{2}{s A^{\theta}-r A^{-\theta}}, \:\:\:\:\:\:\text{sech}_{A}(\theta) = \dfrac{2}{s A^{\theta}+rA^{-\theta}},$

$\text{cos}_{A}(\theta) = \dfrac{s A^{\text{i}\theta}+r A^{-\text{i}\theta}}{2\text{i}}, \:\:\:\:\:\:\text{sin}_{A}(\theta) = \dfrac{s A^{\text{i}\theta}-r A^{-\text{i}\theta}}{2\text{i}} ,$

$\text{cot}_{A}(\theta) = \text{i}\dfrac{s A^{\text{i}\theta}+r A^{-\text{i}\theta}}{s A^{\text{i}\theta}-r A^{-\text{i}\theta}}, \:\:\:\:\:\:\text{tan}_{A} (\theta) = -\text{i}\dfrac{s A^{\text{i}\theta}-r A^{-\text{i}\theta}}{s A^{\text{i}\theta}+r A^{-\text{i}\theta}},$

$\text{csc}_{A} (\theta) = \dfrac{2\text{i}}{s A^{\text{i}\theta}-r A^{-\text{i}\theta}}, \:\:\:\:\:\:\:\:\:\text{sec}_{A}(\theta) = \dfrac{2}{s A^{\text{i}\theta}+r A^{-\text{i}\theta}}, \:\:\:\:\:$

where $\theta$ is an independent variable, $A\neq 0, 1,$ and $s$ and $r$ are arbitrary constants greater than zero and are called deformation parameters.

3. Methods and applications

In this section, we present the first step of the DAM and the SGEM, for finding analytical solutions of CTFMNLSE defined as (1.1). Suppose a CTFNLPDE,

$\begin{equation} \Gamma \bigg(\Phi , \Phi_{\tau}, \Phi_{\mathfrak{X}}, \mathbf{D}_{\tau}^{\alpha}\Phi , \mathbf{D}_{\mathfrak{X}}^{\beta}\Phi , \mathbf{D}_{\tau}^{2\alpha}, \mathbf{D}_{\mathfrak{X}}^{2\beta}, \ldots\bigg) = 0, \:\:\:\:\:\:0 < \alpha, \beta\leq 1, \end{equation}$

(3.1)

where $\Gamma$ and $\Phi$ are a polynomial and an unknown function in its arguments, respectively. Using a fractional travelling wave transformation

$\begin{equation} \Phi (\mathfrak{X}, \tau) = \Lambda(\xi), \:\:\:\:\:\:\:\: \xi = \mathfrak{X}-\dfrac{\mathcal{V}}{\alpha}\tau^{\alpha} \end{equation}$

(3.2)

where $\mathcal{V}$ is velocity and substituting (3.2) into (3.1), we have a NLODE given by

$\begin{equation} \Upsilon (\Lambda , \Lambda^{\prime}, \Lambda^{\prime\prime}, \Lambda^{\prime\prime\prime}, \ldots) = 0, \end{equation}$

(3.3)

in which ${}^{\prime}$ signifies the derivative with respect to $\xi$ .

Since $\Psi = \Psi(\mathfrak{X}, \tau)$ in (1.1) is a complex function, we begin with the following travelling wave assumption

$\begin{equation} \Psi(\mathfrak{X}, \tau) = \Lambda(\xi)\text{e}^{\text{i}\psi} \end{equation}$

(3.4)

where $\xi = \eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})$ and $\psi = -\mathcal{K}\mathfrak{X}+ \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta$ , and $\zeta, \mathcal{P}$ and $\mathcal{K}$ are parameters, represent the phase constant, frequency and wave number respectively. Substitute (3.4) into (1.1), we get real and imaginary parts as follows

$\begin{equation} \eta^{2}(\sigma_{1}-3\delta_{1}\mathcal{K})\Lambda^{\prime\prime}+(\sigma_{2}+(\delta_{2}+\delta_{3})\mathcal{K})\Lambda^{3}+(-\mathcal{P} -\sigma_{1}\mathcal{K}^{2}+\delta_{1}\mathcal{K}^{3}-\delta_{4})\Lambda = 0, \end{equation}$

(3.5)

and

$\begin{equation} (3\delta_{1}\mathcal{K}^{2}-\mathcal{V}-2\sigma_{1}\mathcal{K})\Lambda^{\prime}-\delta_{1}\eta^{2}\Lambda^{\prime\prime\prime} +(\delta_{3}-\delta_{2})\Lambda^{2}\Lambda^{\prime} = 0. \end{equation}$

(3.6)

Now integrating the imaginary part of the equation and taking constant equal to zero one may have

$\begin{equation} 3(3\delta_{1}\mathcal{K}^{2}-\mathcal{V}-2\sigma_{1}\mathcal{K})\Lambda - 3\delta_{1}\eta^{2}\Lambda^{\prime\prime}+(\delta_{3}-\delta_{2})\Lambda^{3} = 0. \end{equation}$

(3.7)

From (3.5) and (3.7), it can be followed that

$\begin{equation} \dfrac{\mathcal{K}^{3}\delta_{1}-\mathcal{P}-\sigma_{1}\mathcal{K}^{2}-\delta_{4}}{3(3\delta_{1}\mathcal{K}^{2}-\mathcal{V}-2\sigma_{1})\mathcal{K}} = \dfrac{\sigma_{1}-3\delta_{1}\mathcal{K}}{-3\delta_{1}} = \dfrac{\mathcal{K}\delta_{2}+\mathcal{K}\delta_{3}+\sigma_{2}}{\delta_{3}-\delta_{2}}. \end{equation}$

(3.8)

From above, it can be followed that

$\mathcal{V} = -\dfrac{\delta_{1}\mathcal{P}+\delta_{1}\delta_{4}+2\mathcal{K}(\sigma_{1}-2\delta_{1}\mathcal{K})^{2}}{\sigma_{1}-3\delta_{1}\mathcal{K}},$

$\mathcal{K} = \dfrac{\sigma_{1}(\delta_{2}-\delta_{3})-3\sigma_{2}\delta_{1}}{6\delta_{1}\delta_{2}}.$

Rewrite (3.5) into following form

$\begin{equation} \Lambda^{\prime\prime}+\lambda_{1}\Lambda^{3}-\lambda_{2}\Lambda = 0 \end{equation}$

(3.9)

$\begin{equation} \Lambda^{\prime\prime} = \lambda_{2}\Lambda -\lambda_{1}\Lambda^{3} \end{equation}$

(3.10)

where $\lambda_{1} = \dfrac{\sigma_{2}+(\delta_{2}+\delta_{3})\mathcal{K}}{\eta^{2}(\sigma_{1}-3\delta_{1}\mathcal{K})}$ and $\lambda_{2} = -\dfrac{-\mathcal{P}-\sigma_{1}\mathcal{K}^{2}+\delta_{1}\mathcal{K}^{3}-\delta_{4}}{\eta^{2}(\sigma_{1}-3\delta_{1}\mathcal{K})}$ .

In the next two subsections, we investigate the primary steps for detecting the exact solution of (3.5) by using DAM and the SGEM. Similarly we can find the exact solution of (3.7).

3.1. Direct algebraic method (DAM)

Firstly, CTFNLPDE (3.1) is reduced to NLODE (3.3) under the transformation (3.2). Secondly, let us consider that Eq (3.3) has a formal solution of the form

$\begin{equation} \Lambda (\xi) = \sum\limits_{j = 0}^{N}b_{j}\mathcal{Q}^{j}(\xi), \:\:\:\: b_{N}\neq 0, \end{equation}$

(3.11)

where $b_{j} (j = 0, \ldots, N)$ are constant coefficients to be detected later and $\mathcal{Q}(\xi)$ satisfies the ODE in the form (2.3). Now, we are able to determine the value $N$ in (3.11) by balancing the highest order derivative term and the highest order nonlinear term in (3.3). Substitute (3.11) along with its required derivatives into (3.3) and compares the coefficients of powers of $\mathcal{Q}(\xi)$ in the resultant equation for getting the set of algebraic equation. In the end, we {solve} the set of algebraic equations and put the results generated in (3.11) to obtain the exact solutions of (3.1).

Now, balancing the order of $\Lambda^{\prime\prime}$ and $\Lambda^{3}$ in (3.10), we get $N = 1.$ Therefore, Eq (3.11) is presented by

$\begin{equation} \Lambda(\xi) = b_{0}+b_{1}\mathcal{Q}(\xi). \end{equation}$

(3.12)

By substituting (3.12) into (3.10) and gathering all terms with the same order of $\mathcal{Q}(\xi)$ together, the left-hand side of (3.10) are converted into polynomial in $\mathcal{Q}(\xi)$ . {Putting} every coefficient of every polynomial to zero, we get a set of algebraic equations for $b_{0}$ and $b_{1}$ . Now, we have

$\begin{equation} \Lambda^{3} = b_{0}^{3}+3b_{0}^{2}b_{1}\mathcal{Q}+3b_{0}b_{1}^{2}\mathcal{Q}^{2}+b_{1}^{3}\mathcal{Q}^{3} \;\text{and}\;\Lambda^{\prime\prime} = b_{1} \mathcal{Q}^{\prime\prime}, \end{equation}$

(3.13)

where

$\begin{eqnarray} \mathcal{Q}^{\prime\prime} = \text{Ln}^{2}(A)(\mathfrak{P}_{0}+\mathfrak{P}_{1}\mathcal{Q}+\mathfrak{P}_{2}\mathcal{Q}^{2})[\mathfrak{P}_{1}+2\mathfrak{P}_{2}\mathcal{Q}]. \end{eqnarray}$

(3.14)

Coefficients of $\mathcal{Q}(\xi)$ as follows:

$\begin{eqnarray*} && \complement^{0} : \:\:\: b_{1}\text{Ln}^{2}(A)\mathfrak{P}_{0}\mathfrak{P}_{1}+\lambda_{1}b_{0}^{3}-\lambda_{2}b_{0} = 0\\ && \complement^{1} : \:\:\: b_{1}\text{Ln}^{2}(A) [\mathfrak{P}^{2}_{1}+2 \mathfrak{P}_{0}\mathfrak{P}_{2}] +3\lambda_{1}b_{0}^{2}b_{1}-\lambda_{2}b_{1} = 0 \\ && \complement^{2} : \:\:\: 3b_{1}\text{Ln}^{2}(A) \mathfrak{P}_{1}\mathfrak{P}_{2}+3\lambda_{1}b_{0}b_{1}^{2} = 0 \\ && \complement^{3} : \:\:\: 2b_{1}\text{Ln}^{2}(A)\mathfrak{P}_{2}^{2}+ \lambda_{1}b_{1}^{3} = 0. \end{eqnarray*}$

We earn the following values, by solving the above system of equations for $b_{0}$ and $b_{1}$ :

$\begin{equation} b_{0} = \pm\dfrac{\text{i}}{\sqrt{2\lambda_{1}}}\text{Ln}(A)\mathfrak{P}_{1}, \:\:\:\:\:\:\:\:\:\: b_{1} = \pm\text{i}\sqrt{\dfrac{2}{\lambda_{1}}}\text{Ln}(A)\mathfrak{P}_{2}. \end{equation}$

(3.15)

The solutions of (1.1) corresponding to (3.4), (3.12) and (3.15) are:

(1) When $\mathfrak{P}_{1}^{2}-4\mathfrak{P}_{0}\mathfrak{P}_{2} < 0,$ and $\mathfrak{P}_{2}\neq 0,$

$\begin{equation*} \Psi_{1, 2}(\mathfrak{X}, \tau) = (\pm\text{i})\text{ln}(A)\dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{2\lambda_{1}}} \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)} \text{tan}_{A}\bigg( \dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{2} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg), \end{equation*}$

$\begin{equation*} \Psi_{1, 3}(\mathfrak{X}, \tau) = (\pm\text{i})\text{ln}(A)\dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{2\lambda_{1}}}\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\text{cot}_{A}\bigg( \dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{2}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg), \end{equation*}$

$\begin{eqnarray*} \Psi_{1, 4}(\mathfrak{X}, \tau)& = &(\pm\text{i})\text{ln}(A)\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{2\lambda_{1}}}\text{tan}_{A}\bigg( \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\\ &\pm &\notag\dfrac{\sqrt{-sr ( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{2\lambda_{1}}}\text{sec}_{A}\bigg( \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\bigg], \end{eqnarray*}$

$\begin{eqnarray*} \Psi_{1, 5}(\mathfrak{X}, \tau)& = &(\pm\text{i})\text{ln}(A)\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[-\dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{2\lambda_{1}}}\text{cot}_{A}\bigg( \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\\ &\pm &\notag\dfrac{\sqrt{-sr ( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{2\lambda_{1}}}\text{csc}_{A}\bigg( \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg), \end{eqnarray*}$

$\begin{eqnarray*} \Psi_{1, 6}(\mathfrak{X}, \tau)& = &(\pm\text{i})\text{ln}(A)\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{8\lambda_{1}}}\text{tan}_{A}\bigg( \dfrac{ \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{4}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\\ &- &\notag\dfrac{\sqrt{- ( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{8\lambda_{1}}}\text{cot}_{A}\bigg( \dfrac{ \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{4}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\bigg], \end{eqnarray*}$

(2) When $\mathfrak{P}_{1}^{2}-4\mathfrak{P}_{0}\mathfrak{P}_{2} > 0,$ and $\mathfrak{P}_{2}\neq 0,$

$\begin{equation*} \Psi_{1, 7}(\mathfrak{X}, \tau) = (\pm\text{i})\text{ln}(A)\dfrac{\sqrt{\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{\sqrt{2\lambda_{1}}}\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\text{tanh}_{A}\bigg( \dfrac{\sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{2}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg), \end{equation*}$

$\begin{equation*} \Psi_{1, 8}(\mathfrak{X}, \tau) = (\pm\text{i})\text{ln}(A)\dfrac{\sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{\sqrt{2\lambda_{1}}}\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\text{coth}_{A}\bigg( \dfrac{\sqrt{\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{2}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg), \end{equation*}$

$\begin{eqnarray*} \Psi_{1, 9}(\mathfrak{X}, \tau)& = &(\pm\text{i})\text{ln}(A)\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[-\dfrac{\sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{\sqrt{2\lambda_{1}}}\text{tanh}_{A}\bigg( \sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)\bigg)\\ &\pm &\notag\text{i}\dfrac{\sqrt{sr ( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{2\lambda_{1}}}\text{sech}_{A}\bigg( \sqrt{\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\bigg], \end{eqnarray*}$

$\begin{eqnarray*} \Psi_{1, 10}(\mathfrak{X}, \tau)& = &\pm\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[-\dfrac{\sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{\sqrt{2\lambda_{1}}}\text{coth}_{A}\bigg( \sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\\ &\pm &\notag\dfrac{\sqrt{sr ( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{2\lambda_{1}}}\text{csch}_{A}\bigg( \sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\bigg], \end{eqnarray*}$

$\begin{eqnarray*} \Psi_{1, 11}(\mathfrak{X}, \tau)& = &(\pm\text{i})\text{ln}(A)\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\dfrac{\sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{\sqrt{8\lambda_{1}}}\text{tanh}_{A}\bigg( \dfrac{ \sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{4}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)\bigg)\\ &+ &\notag\dfrac{\sqrt{\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{\sqrt{8\lambda_{1}}}\text{coth}_{A}\bigg( \dfrac{ \sqrt{\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{4}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\bigg], \end{eqnarray*}$

where $\xi_{0}$ is an arbitrary constant. For more details see ^[17,18].

3.2. Sine-Gordon expansion method (SGEM)

Consider the sine-Gordon equation from ^[19,20]

$\begin{equation} \Psi_{\mathfrak{X}\mathfrak{X}}-\Psi_{\tau\tau} = m^{2}\sin(\Psi), \end{equation}$

(3.16)

where $\Psi = \Psi (\mathfrak{X}, \tau)$ and $m$ is a constant. To solve the equation through sine-Gordon expansion algorithm, first we use the transformation $\Psi (\mathfrak{X}, \tau) = \Lambda(\xi)$ where $\xi = \mu (\mathfrak{X}-c\tau)$ which reduce (3.16) to the following NLODE:

$\begin{equation} \Lambda^{\prime\prime} = \dfrac{m^{2}}{\mu^{2}(1-c^{2})}\sin(\Lambda). \end{equation}$

(3.17)

After that, we multiply $\Lambda^{\prime}$ on both sides of (3.17) and integrate it once which gives

$\begin{equation} \bigg[ \bigg( \dfrac{\Lambda}{2} \bigg) \bigg]^{2} = \dfrac{m^{2}}{\mu^{2}(1-c^{2})}\sin^{2}\bigg( \dfrac{\Lambda}{2} \bigg) +k, \end{equation}$

(3.18)

in which $k$ is an integration constant. Therefore by putting $k = 0, \: \Lambda_{2} = w(\xi),$ and $\dfrac{m^{2}}{\mu^{2}(1-c^{2})} = a^{2}$ in (3.18), we get

$\begin{equation} w^{\prime} = a\sin (w), \end{equation}$

(3.19)

which by setting $a = 1$ in (3.19), we have

$\begin{equation} w^{\prime} = \sin (w). \end{equation}$

(3.20)

Equation (3.20) is a simplified form of the sine-Gordon Eq (3.16). Thus, it has the following solutions:

$\begin{equation} \sin(w) = \text{sech}(\xi), \:\:\:\:\:\:\:\cos(w) = \text{tanh}(\xi), \end{equation}$

(3.21)

and

$\begin{equation} \sin(w) = \text{i} \text{csch}(\xi), \:\:\:\:\:\:\:\cos(w) = \text{coth}(\xi). \end{equation}$

(3.22)

Here, firstly, CTFNLPDE (3.1) is reduced to NLODE (3.3) under the transformation (3.2). Secondly, we apply the following transformation

$\begin{equation} \Lambda (w) = \sum\limits_{j = 1}^{N}\cos^{j-1}(w)[ B_{j} \sin(w)+A_{j}\cos(w)]+A_{0}. \end{equation}$

(3.23)

It is supposed that the solution $\Lambda (\xi)$ of the nonlinear (3.3) along with (3.21) and (3.22) can be demonstrated as

$\begin{equation} \Lambda (\xi) = \sum\limits_{j = 1}^{N}\text{tanh}^{j-1}(\xi)[ B_{j} \text{sech}(\xi)+A_{j}\text{tanh}(\xi)]+A_{0}. \end{equation}$

(3.24)

and

$\begin{equation} \Lambda (\xi) = \sum\limits_{j = 1}^{N}\text{coth}^{j-1}(\xi)[ B_{j} \text{csch}(\xi)+A_{j}\text{coth}(\xi)]+A_{0}. \end{equation}$

(3.25)

After detecting the value of $N$ by means of using the homogeneous balance principle, substituting its value into (3.23) and setting the result into the reduced ODE (3.20) give a nonlinear algebraic system. Equating the coefficients of $\sin^{j}(w)$ and $\cos^{j}(w)$ equal to zero and solving the acquired system yield the values of $A_{j}$ and $B_{j}$ . Finally, after substituting the values of $A_{j}$ and $B_{j}$ into (3.24) and (3.25), we are able to earn the solitary wave solutions for (3.1).

We used the balancing technique to Eq (3.10) by considering the highest derivative $\Lambda^{\prime\prime}$ and the highest power nonlinear term $\Lambda^{3}$ , which the value of $N$ is gained as $N+2 = 3N$ or $N = 1$ . Thus, we have the following equations

$\begin{equation} \Lambda(w) = B_{1}\sin(w)+A_{1}\cos(w)+A_{0}, \end{equation}$

(3.26)

$\begin{equation} \Lambda^{\prime}(w) = B_{1}\cos(w)\sin(w)-A_{1}\sin^{2}(w), \end{equation}$

(3.27)

and

$\begin{equation} \Lambda^{\prime\prime}(w) = B_{1}[ \cos^{2}(w)\sin(w)-\sin^{3}(w)]-2A_{1}\sin^{2}(w)\cos(w). \end{equation}$

(3.28)

Also we have

$\begin{eqnarray} \Lambda^{3}& = & B_{1}^{3}\sin^{3}(w)+A_{1}^{3}\cos^{3}(w)\\ &+& 3A_{0}B_{1}^{2}\sin^{2}(w)+3A_{1}^{2}A_{0}\cos^{2}(w)\\ &+& 3A_{0}^{2}B_{1}\sin(w)+3 A_{0}^{2}A_{1}\cos(w)\\ &+& 3A_{1}B_{1}^{2}\sin^{2}(w)\cos(w) +3A_{1}^{2}B_{1}\cos^{2}(w)\sin(w)\\ &+& 6A_{0}A_{1}B_{1}\sin(w)\cos(w)\\ &+& A_{0}^{3}. \end{eqnarray}$

(3.29)

By substituting (3.26)–(3.29) into (3.10), we obtain the following nonlinear algebraic system:

$\begin{eqnarray*} && \sin^{3}(w):\:\:\: \:\:\: -B_{1}+\lambda_{1}B_{1}^{3} = 0, \\ && \cos^{3}(w):\:\:\:\:\:\: \lambda_{1}A_{1}^{3} = 0, \\ && \sin^{2}(w):\:\:\: \:\:\: 3\lambda_{1}A_{0}B_{1}^{2} = 0, \\ && \cos^{2}(w)\:\:\: \:\:\: 3\lambda_{1}A_{1}A_{0} = 0, \\ &&\sin(w):\:\:\: \:\:\: 3\lambda_{1}A_{0}^{2}B_{1}-\lambda_{2}B_{1} = 0\\ &&\cos(w):\:\:\:\: \:\:\: 3\lambda_{1}A_{0}^{2}A_{1}-\lambda_{2}A_{1} = 0, \\ &&\sin^{2}(w)\cos(w):\:\:\: \:\:\: -2A_{1}+3\lambda_{1}A_{1}B_{1}^{2} = 0, \\ &&\cos^{2}(w)\sin(w):\:\:\: \:\:\: B_{1}+3\lambda_{1}A_{1}^{2}B_{1} = 0, \\ &&\sin(w)\cos(w):\:\:\:\:\:\: 6\lambda_{1}A_{0}A_{1}B_{1} = 0, \\ &&\sin^{0}(w)\cos^{0}(w): \:\:\:\:\:\: \lambda_{1}A_{0}^{3}-\lambda_{2}A_{0} = 0. \end{eqnarray*}$

Using (3.21) and (3.26), we earn the following traveling wave solutions:

Case 1: $A_{0} = \pm \sqrt{\dfrac{\lambda_{2}}{\lambda_{1}}}\: \:or \:\:\pm \sqrt{\dfrac{\lambda_{2}}{3\lambda_{1}}}\:\: or \:\: 0, \: A_{1} = \pm \sqrt{-\dfrac{1}{3\lambda_{1}}}, \: B_{1} = 0.$

$\begin{equation} \Psi_{2, 1} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\pm \sqrt{-\dfrac{1}{3\lambda_{1}}} \text{tanh} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg], \end{equation}$

(3.30)

$\begin{equation} \Psi_{2, 2} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\pm \sqrt{-\dfrac{1}{3\lambda_{1}}} \text{coth} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg]. \end{equation}$

(3.31)

Case 2: $A_{0} = \pm \sqrt{\dfrac{\lambda_{2}}{\lambda_{1}}}\: \:or \:\:\pm \sqrt{\dfrac{\lambda_{2}}{3\lambda_{1}}}\:\: or \:\: 0, \: A_{1} = 0, \: B_{1} = \pm \sqrt{\dfrac{2}{3\lambda_{1}}}.$

$\begin{equation} \Psi_{2, 3} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\pm \sqrt{\dfrac{2}{3\lambda_{1}}} \text{sech} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg], \end{equation}$

(3.32)

$\begin{equation} \Psi_{2, 4} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\pm\text{i} \sqrt{\dfrac{2}{3\lambda_{1}}} \text{csch} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg]. \end{equation}$

(3.33)

Case 3: $A_{0} = \pm \sqrt{\dfrac{\lambda_{2}}{\lambda_{1}}}\: \:or \:\:\pm \sqrt{\dfrac{\lambda_{2}}{3\lambda_{1}}}\:\: or \:\: 0, \: A_{1} = 0, \: B_{1} = \pm \sqrt{\dfrac{1}{\lambda_{1}}}.$

$\begin{equation} \Psi_{2, 5} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\pm \sqrt{\dfrac{1}{\lambda_{1}}} \text{sech} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg], \end{equation}$

(3.34)

$\begin{equation} \Psi_{2, 6} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\pm\text{i} \sqrt{\dfrac{1}{\lambda_{1}}}\text{csch} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg]. \end{equation}$

(3.35)

Case 4: $A_{0} = \pm \sqrt{\dfrac{\lambda_{2}}{\lambda_{1}}}\: \:or \:\:\pm \sqrt{\dfrac{\lambda_{2}}{3\lambda_{1}}}\:\: or \:\: 0, \: A_{1} = \pm \sqrt{-\dfrac{1}{3\lambda_{1}}}, \: B_{1} = \pm \sqrt{\dfrac{1}{\lambda_{1}}}.$

$\begin{equation} \Psi_{2, 7} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[ \pm \sqrt{\dfrac{1}{\lambda_{1}}} \text{sech} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)\pm \sqrt{-\dfrac{1}{3\lambda_{1}}}\text{tanh} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg], \end{equation}$

(3.36)

$\begin{equation} \Psi_{2, 8} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[ \pm \text{i}\sqrt{\dfrac{1}{\lambda_{1}}} \text{csch} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)\pm \sqrt{-\dfrac{1}{3\lambda_{1}}}\text{coth} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg]. \end{equation}$

(3.37)

Case 5: $A_{0} = \pm \sqrt{\dfrac{\lambda_{2}}{\lambda_{1}}}\: \:or \:\:\pm \sqrt{\dfrac{\lambda_{2}}{3\lambda_{1}}}\:\: or \:\: 0, \: A_{1} = \pm \sqrt{-\dfrac{1}{3\lambda_{1}}}, \: B_{1} = \pm \sqrt{\dfrac{2}{3\lambda_{1}}}.$

$\begin{equation} \Psi_{2, 9} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[ \pm \sqrt{\dfrac{2}{3\lambda_{1}}} \text{sech} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)\pm \sqrt{-\dfrac{1}{3\lambda_{1}}}\text{tanh} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg], \end{equation}$

(3.38)

$\begin{equation} \Psi_{2, 10} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[ \pm \text{i}\sqrt{\dfrac{2}{3\lambda_{1}}} \text{csch} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)\pm \sqrt{-\dfrac{1}{3\lambda_{1}}}\text{coth} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg], \end{equation}$

(3.39)

where $\xi_{0}$ is an arbitrary constant. For more details, see ^[21,22].

4. Comparing the real and imaginary part of solutions of CTFMNLSE defined in (1.1)

Let $\delta_{1} = \delta_{3} = \eta = s = r = \mathfrak{P}_{0} = \mathfrak{P}_{1} = \mathfrak{P}_{2} = 1, \: \mathcal{P} = \delta_{2} = \delta_{4} = \sigma_{2} = A = 2, \: \zeta = \mathcal{V} = 0.5, \: \mathcal{K} = 0.25, \: \sigma_{1} = -1, \: \alpha = 0.90, \: \xi_{0} = A_{0} = 0.$ Therefore, we have $\lambda_{1} = -1.57143$ and $\lambda_{2} = -2.24107.$ We now present numerical results in tables and charts.

and show the 3D with the both real and imaginary parts of the solution $\Psi_{1, 1}$ and $\Psi_{1, 3}$ for different values of $\mathfrak{X}$ and $\tau$ and also, Figure 1 (b), (f) and , show the 3D and 2D with the both real and imaginary part of the solution $\Psi_{1, 1}$ and $\Psi_{1, 3}$ for fixed $\mathfrak{X}$ and different values of $\tau$ through DAM, using the above values. Now, and show the 3D with the both real and imaginary parts of the solution $\Psi_{2, 1}$ and $\Psi_{2, 3}$ for different values of $\mathfrak{X}$ and $\tau$ and also and show the 3D with the both real and imaginary part of the solution $\Psi_{2, 1}$ and $\Psi_{2, 3}$ for fixed $\mathfrak{X}$ and different values of $\tau$ through SGEM, using the above values.

Figure 1. The figures (a) and (e) show the 3D with the both real and imaginary parts of the solution

$\Psi_{1, 1}$ and

$\Psi_{1, 3}$ for different values of

$\mathfrak{X}$ and

$\tau$ and also the figures (b), (f) and (c), (d) show 3D and 2D with the both real and imaginary parts of the solution

$\Psi_{1, 1}$ and

$\Psi_{1, 3}$ for fixed

$\mathfrak{X}$ and different values of

$\tau$ through DAM, under the values presented in Section 4.

DownLoad: Full-Size Img PowerPoint

Figure 2. The figures (a) and (c) show the 3D with the both real and imaginary parts of the solution

$\Psi_{2, 1}$ and

$\Psi_{2, 3}$ for different values of

$\mathfrak{X}$ and

$\tau$ . The figures (b) and (d) show 3D with the both real and imaginary parts of the solution

$\Psi_{2, 1}$ and

$\Psi_{2, 3}$ for fixed

$\mathfrak{X}$ and different values of

$\tau$ through SGEM, under the values presented in Section 4.

DownLoad: Full-Size Img PowerPoint

Moreover, displays the 3D with the real and imaginary part of solution $\Psi_{2, 1}$ for fixed $\mathfrak{X}$ and different values of $\alpha \:( = 0.50-0.09),$ obtained via SGEM.

Figure 3. The 3D with the real and imaginary parts of solution

$\Psi_{2, 1}$ for fixed

$\mathfrak{X}$ and different values of

$\alpha \:( = 0.50-0.09),$ obtained via SGEM.

DownLoad: Full-Size Img PowerPoint

shows the 3D with the differences between the real and also the imaginary part of solutions $\Psi_{1, 1}$ and $\Psi_{2, 1}$ , and also shows the 3D with the differences between the real and also the imaginary part of solutions $\Psi_{1, 3}$ and $\Psi_{2, 3}$ , for fixed $\mathfrak{X}$ and fixed $\alpha$ . Note that, these differences are minor in a wide range of domains. which implies that both methods leads to similar results except for some values.

Figure 4. (a) shows the 3D with the differences between the real and also the imaginary part of solutions

$\Psi_{1, 1}$ and

$\Psi_{2, 1}$ , and also (b) shows the 3D with the differences between the real and also the imaginary part of the solutions

$\Psi_{1, 3}$ and

$\Psi_{2, 3}$ , for fixed

$\mathfrak{X}$ and fixed

$\alpha$ .

DownLoad: Full-Size Img PowerPoint

Tables 1–4 present the numerical results of the solutions of CTFMNLSE (1.1) obtained by DAM and SGEM with several point sources trough arbitrary.

Table 1. The real part of exact solutions of CTFMNLSE (1.1) obtained by DAM, with several point sources trough arbitrary.

$\mathfrak{X}$	$\tau$	$\Psi_{1, 1}^{\Re}(\mathfrak{X}, \tau)$	$\Psi_{1, 2}^{\Re}(\mathfrak{X}, \tau)$	$\Psi_{1, 3}^{\Re}(\mathfrak{X}, \tau)$	$\Psi_{1, 4}^{\Re}(\mathfrak{X}, \tau)$
	0.012	0.00058	-0.00058	587.19354	-587.19354
0.012	0.037	-0.00550	0.00550	-55.91444	55.91444
	0.062	-0.01056	0.01056	-26.29075	26.29075
	0.012	0.00933	-0.00933	36.46445	-36.46445
0.037	0.037	0.00284	-0.00284	109.34814	-109.34814
	0.062	-0.00266	0.00266	-105.25186	105.25186
	0.012	0.01816	-0.01816	18.87874	-18.87874
0.062	0.037	0.01124	-0.01124	27.81915	-27.81915
	0.062	0.00531	-0.00531	53.34247	-53.34247

| Show Table

DownLoad: CSV

Table 2. The imaginary part of exact solutions of CTFMNLSE (1.1) obtained by DAM, with several point sources trough arbitrary.

$\mathfrak{X}$	$\tau$	$\Psi_{1, 1}^{\Im}(\mathfrak{X}, \tau)$	$\Psi_{1, 2}^{\Im}(\mathfrak{X}, \tau)$	$\Psi_{1, 3}^{\Im}(\mathfrak{X}, \tau)$	$\Psi_{1, 4}^{\Im}(\mathfrak{X}, \tau)$
	0.012	0.00034	-0.00034	350.78189	-350.78189
0.012	0.037	-0.00385	0.00385	-39.18969	39.18969
	0.062	-0.00852	0.00852	-21.21368	21.21368
	0.012	0.00549	-0.00549	21.47530	-21.47530
0.037	0.037	0.00196	-0.00196	75.62594	-75.62594
	0.062	-0.00212	0.00212	-83.84577	83.84577
	0.012	0.01054	-0.01054	10.96007	-10.96007
0.062	0.037	0.00767	-0.00767	18.98398	-18.98398
	0.062	0.00417	-0.00417	41.95143	-41.95143

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Table 3. The real part of exact solutions of CTFMNLSE (1.1) obtained by SGEM, with several point sources trough arbitrary.

$\mathfrak{X}$	$\tau$	$\Psi_{2, 1}^{\Re}(\mathfrak{X}, \tau)$	$\Psi_{2, 2}^{\Re}(\mathfrak{X}, \tau)$	$\Psi_{2, 3}^{\Re}(\mathfrak{X}, \tau)$	$\Psi_{2, 4}^{\Re}(\mathfrak{X}, \tau)$
	0.012	0.00065	-0.00065	241.48348	-241.48348
0.012	0.037	-0.00624	0.00624	-22.75660	22.75660
	0.062	-0.01197	0.01197	-10.70930	10.70930
	0.012	0.01055	-0.01055	14.88481	-14.88481
0.037	0.037	0.00320	-0.00320	44.79798	-44.79798
	0.062	-0.00304	0.00304	-42.62661	42.62661
	0.012	0.02052	-0.02052	7.71125	-7.71125
0.062	0.037	0.01270	-0.01270	11.36482	-11.36482
	0.062	0.00599	-0.00599	21.83731	-21.83731

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Table 4. The imaginary part of exact solutions of CTFMNLSE (1.1) obtained by SGEM, with several point sources trough arbitrary.

$\mathfrak{X}$	$\tau$	$\Psi_{2, 1}^{\Im}(\mathfrak{X}, \tau)$	$\Psi_{2, 2}^{\Im}(\mathfrak{X}, \tau)$	$\Psi_{2, 3}^{\Im}(\mathfrak{X}, \tau)$	$\Psi_{2, 4}^{\Im}(\mathfrak{X}, \tau)$
	0.012	0.00038	-0.00038	144.25913	-144.25913
0.012	0.037	-0.00437	0.00437	-15.94980	15.94980
	0.062	-0.00965	0.00965	-8.64120	8.64120
	0.012	0.00621	-0.00621	8.76623	-8.76623
0.037	0.037	0.00221	-0.00221	30.98259	-30.98259
	0.062	-0.00241	0.00241	-33.95722	33.95722
	0.012	0.01191	-0.01191	4.47677	-4.47677
0.062	0.037	0.00866	-0.00866	7.75543	-7.75543
	0.062	0.00471	-0.00471	17.17404	-17.17404

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In Table 5, based on –, separately, we calculate the differences between solutions $\Psi_{1, 1}, \Psi_{1, 3}, \Psi_{2, 1},$ and $\Psi_{2, 3},$ represented by $\Delta \Psi_{1, 1}, \Delta \Psi_{2, 1}, \Delta \Psi_{2, 1},$ and $\Delta \Psi_{2, 3},$ for fixed $\mathfrak{X} = 0.012$ and different values of $\tau$ . Note that $\Re$ and $\Im$ show the real and imaginary part of solutions. As we can observe, for fixed $\mathfrak{X}$ by changing the value of $\tau$ , both DAM and SGEM result major changes for solutions $\Psi_{1, 3}$ and $\Psi_{2, 3},$ and here we are not dealing with an advantageous result. Nevertheless, SGEM results more minor changes than DAM.

Table 5. According to Tables 1-4, separately, we calculate the differences between solutions

$\Psi_{1, 1}, \Psi_{1, 3}, \Psi_{2, 1},$ and

$\Psi_{2, 3},$ represented by

$\Delta \Psi_{1, 1}, \Delta \Psi_{2, 1}, \Delta \Psi_{2, 1},$ and

$\Delta \Psi_{2, 3},$ for fixed

$\mathfrak{X} = 0.012$ and different values of

$\tau$ . Note that

$\Re$ and

$\Im$ show the real and imaginary part of solutions.

$\mathfrak{X}=0.012$
$\tau=0.012, 0.037, 0.062$
DAM				SGEM
$\Delta\Psi_{1, 1}^{\Re}$	$\Delta\Psi_{1, 1}^{\Im}$	$\Delta\Psi_{1, 3}^{\Re}$	$\Delta\Psi_{1, 3}^{\Im}$	$\Delta\Psi_{2, 1}^{\Re}$	$\Delta\Psi_{2, 1}^{\Im}$	$\Delta\Psi_{2, 3}^{\Re}$	$\Delta\Psi_{2, 3}^{\Im}$
0.00608	0.00419	643.10798	389.97158	0.00689	0.00475	264.24008	160.20893
0.01114	0.00886	613.48429	771.99557	0.01262	0.01003	252.19278	152.90033
0.00506	0.00467	29.62369	17.97601	0.00573	0.00528	12.04730	7.30860

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Tables 6–9 propose the real and imaginary part of exact solutions of CTFMNLSE (1.1) obtained by six different methods: FIM, FVM, TEM, MTEM ^[15,16] and also DAM and SGEM, with several point sources trough arbitrary. For some values the results obtained through DAM and SGEM, are near to the results obtained in four other methods.

Table 6. The real part of exact solutions of CTFMNLSE (1.1) obtained by 6 different methods, with several point sources trough arbitrary.

$\mathfrak{X}$	$\tau$	$\overbrace{\Psi_{1, 2}^{\Re}(\mathfrak{X}, \tau)}^{\text{FIM}}$	$\overbrace{\Psi_{1, 2}^{\Re}(\mathfrak{X}, \tau)}^{\text{FVM}}$	$\overbrace{\Psi_{1, 2}^{\Re}(\mathfrak{X}, \tau)}^{\text{TEM}}$	$\overbrace{\Psi_{1, 2}^{\Re}(\mathfrak{X}, \tau)}^{\text{MTEM}}$	$\overbrace{\Psi_{1, 2}^{\Re}(\mathfrak{X}, \tau)}^{\text{DAM}}$	$\overbrace{\Psi_{1, 2}^{\Re}(\mathfrak{X}, \tau)}^{\text{SGEM}}$
0.012	0.012	$\mp$ 0.00099	$\mp$ 0.86524	$\pm$ 1.44911	$\mp$ 0.00083	$\pm$ 0.00057	$\pm$ 0.00064
	0.037	$\pm$ 0.01209	$\mp$ 0.96849	$\pm$ 1.38186	$\pm$ 0.00941	$\mp$ 0.00549	$\mp$ 0.00623
	0.062	$\pm$ 0.02666	$\mp$ 1.06066	$\pm$ 1.31204	$\pm$ 0.02081	$\mp$ 0.01055	$\mp$ 0.01196
	0.087	$\pm$ 0.04264	$\mp$ 1.14551	$\pm$ 1.23880	$\pm$ 0.03333	$\mp$ 0.01482	$\mp$ 0.01679
0.037	0.012	$\mp$ 0.01689	$\mp$ 0.85415	$\pm$ 1.45334	$\mp$ 0.01342	$\pm$ 0.00933	$\pm$ 0.01055
	0.037	$\mp$ 0.00583	$\mp$ 0.95628	$\pm$ 1.38820	$\mp$ 0.00478	$\pm$ 0.00283	$\pm$ 0.00319
	0.062	$\pm$ 0.00691	$\mp$ 1.04758	$\pm$ 1.32017	$\pm$ 0.00518	$\mp$ 0.00266	$\mp$ 0.00303
	0.087	$\pm$ 0.02124	$\mp$ 1.13177	$\pm$ 1.24849	$\pm$ 0.01639	$\mp$ 0.00738	$\mp$ 0.00838
0.062	0.012	$\mp$ 0.03247	$\mp$ 0.84187	$\pm$ 1.45547	$\mp$ 0.02575	$\pm$ 0.01815	$\pm$ 0.02052
	0.037	$\mp$ 0.02346	$\mp$ 0.94274	$\pm$ 1.39254	$\mp$ 0.01873	$\pm$ 0.01124	$\pm$ 0.01270
	0.062	$\mp$ 0.01251	$\mp$ 1.03303	$\pm$ 1.32641	$\mp$ 0.01019	$\pm$ 0.00530	$\pm$ 0.00598
	0.087	$\pm$ 0.00016	$\mp$ 1.11640	$\pm$ 1.25641	$\mp$ 0.00028	$\pm$ 0.00013	$\pm$ 0.00012
0.087	0.012	$\mp$ 0.04776	$\mp$ 0.82848	$\pm$ 1.45550	$\mp$ 0.03784	$\pm$ 0.02705	$\pm$ 0.03053
	0.037	$\mp$ 0.04079	$\mp$ 0.92793	$\pm$ 1.39486	$\mp$ 0.03244	$\pm$ 0.01972	$\pm$ 0.02227
	0.062	$\mp$ 0.03165	$\mp$ 1.01707	$\pm$ 1.33074	$\mp$ 0.02533	$\pm$ 0.01336	$\pm$ 0.01509
	0.087	$\mp$ 0.02062	$\mp$ 1.09949	$\pm$ 1.26251	$\mp$ 0.01673	$\pm$ 0.00773	$\pm$ 0.00872

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Table 7. The real part of exact solutions of CTFMNLSE (1.1) obtained by 6 different methods, with several point sources through arbitrary.

$\mathfrak{X}$	$\tau$	$\overbrace{\Psi_{3, 4}^{\Re}(\mathfrak{X}, \tau)}^{\text{FIM}}$	$\overbrace{\Psi_{3, 4}^{\Re}(\mathfrak{X}, \tau)}^{\text{FVM}}$	$\overbrace{\Psi_{3, 4}^{\Re}(\mathfrak{X}, \tau)}^{\text{TEM}}$	$\overbrace{\Psi_{3, 4}^{\Re}(\mathfrak{X}, \tau)}^{\text{MTEM}}$	$\overbrace{\Psi_{3, 4}^{\Re}(\mathfrak{X}, \tau)}^{\text{DAM}}$	$\overbrace{\Psi_{3, 4}^{\Re}(\mathfrak{X}, \tau)}^{\text{SGEM}}$
0.012	0.012	$\mp$ 378.21073	$\mp$ 70.07373	$\pm$ 351.38054	$\mp$ 351.20376	$\pm$ 587.19354	$\pm$ 241.48348
	0.037	$\pm$ 38.81779	$\mp$ 30.84863	$\mp$ 39.15773	$\pm$ 39.17920	$\mp$ 55.91443	$\mp$ 22.75659
	0.062	$\pm$ 21.08640	$\mp$ 19.36377	$\mp$ 21.19424	$\pm$ 21.20629	$\mp$ 26.29074	$\mp$ 10.70930
	0.087	$\pm$ 15.32460	$\mp$ 14.48859	$\mp$ 15.38516	$\pm$ 15.39533	$\mp$ 16.74259	$\mp$ 6.82515
0.037	0.012	$\mp$ 21.73603	$\mp$ 28.35492	$\pm$ 21.46931	$\mp$ 21.47600	$\pm$ 36.46444	$\pm$ 14.88481
	0.037	$\mp$ 78.99292	$\mp$ 24.09697	$\pm$ 75.68598	$\mp$ 75.67674	$\pm$ 109.34813	$\pm$ 44.79797
	0.062	$\pm$ 80.05677	$\mp$ 18.81347	$\mp$ 83.68725	$\pm$ 83.77139	$\mp$ 105.25185	$\mp$ 42.62660
	0.087	$\pm$ 30.34363	$\mp$ 15.03660	$\mp$ 30.84459	$\pm$ 30.86541	$\mp$ 33.98710	$\mp$ 13.82036
0.062	0.012	$\mp$ 11.06619	$\mp$ 16.76490	$\pm$ 10.95095	$\mp$ 10.95634	$\pm$ 18.87873	$\pm$ 7.71124
	0.037	$\mp$ 19.30540	$\mp$ 16.27571	$\pm$ 18.97941	$\mp$ 18.98481	$\pm$ 27.8191	$\pm$ 11.36481
	0.062	$\mp$ 43.52313	$\mp$ 14.65664	$\pm$ 41.97445	$\mp$ 41.97341	$\pm$ 53.34247	$\pm$ 21.83730
	0.087	$\pm$ 3894.75067	$\mp$ 12.88518	$\pm$ 1783.25721	$\mp$ 1751.00331	$\pm$ 1903.01800	$\pm$ 950.20622
0.087	0.012	$\mp$ 7.36271	$\mp$ 11.81796	$\pm$ 7.29069	$\mp$ 7.29620	$\pm$ 12.76218	$\pm$ 5.21999
	0.037	$\mp$ 10.90080	$\mp$ 11.82677	$\pm$ 10.75601	$\mp$ 10.76155	$\pm$ 15.99035	$\pm$ 6.53613
	0.062	$\mp$ 16.93629	$\mp$ 11.28924	$\pm$ 16.60991	$\mp$ 16.61494	$\pm$ 21.40072	$\pm$ 8.74788
	0.087	$\mp$ 30.39447	$\mp$ 10.50579	$\pm$ 29.40686	$\mp$ 29.40818	$\pm$ 33.17835	$\pm$ 13.57924

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Table 8. The imaginary part of exact solutions of CTFMNLSE (1.1) obtained by 6 different methods, with several point sources through arbitrary.

$\mathfrak{X}$	$\tau$	$\overbrace{\Psi_{1, 2}^{\Im}(\mathfrak{X}, \tau)}^{\text{FIM}}$	$\overbrace{\Psi_{1, 2}^{\Im}(\mathfrak{X}, \tau)}^{\text{FVM}}$	$\overbrace{\Psi_{1, 2}^{\Im}(\mathfrak{X}, \tau)}^{\text{TEM}}$	$\overbrace{\Psi_{1, 2}^{\Im}(\mathfrak{X}, \tau)}^{\text{MTEM}}$	$\overbrace{\Psi_{1, 2}^{\Im}(\mathfrak{X}, \tau)}^{\text{DAM}}$	$\overbrace{\Psi_{1, 2}^{\Im}(\mathfrak{X}, \tau)}^{\text{SGEM}}$
0.012	0.012	$\pm$ 0.00165	$\pm$ 1.44929	$\pm$ 0.86568	$\pm$ 0.00140	$\pm$ 0.00034	$\pm$ 0.00038
	0.037	$\mp$ 0.01726	$\pm$ 1.38406	$\pm$ 0.96853	$\mp$ 0.01342	$\mp$ 0.00385	$\mp$ 0.00437
	0.062	$\mp$ 0.033041	$\pm$ 1.31780	$\pm$ 1.05867	$\mp$ 0.02579	$\mp$ 0.00851	$\mp$ 0.00965
	0.087	$\mp$ 0.04635	$\pm$ 1.24922	$\pm$ 1.13976	$\mp$ 0.03622	$\mp$ 0.01364	$\mp$ 0.01544
0.037	0.012	$\pm$ 0.02868	$\pm$ 1.45317	$\pm$ 0.85592	$\pm$ 0.02279	$\pm$ 0.00549	$\pm$ 0.00621
	0.037	$\pm$ 0.00844	$\pm$ 1.38971	$\pm$ 0.96009	$\pm$ 0.00691	$\pm$ 0.00196	$\pm$ 0.00220
	0.062	$\mp$ 0.00867	$\pm$ 1.32525	$\pm$ 1.05167	$\mp$ 0.00651	$\mp$ 0.00212	$\mp$ 0.00241
	0.087	$\mp$ 0.02338	$\pm$ 1.25853	$\pm$ 1.13434	$\mp$ 0.01804	$\mp$ 0.00671	$\mp$ 0.00762
0.062	0.012	$\pm$ 0.05593	$\pm$ 1.45494	$\pm$ 0.84497	$\pm$ 0.04436	$\pm$ 0.01054	$\pm$ 0.01191
	0.037	$\pm$ 0.03438	$\pm$ 1.39330	$\pm$ 0.95027	$\pm$ 0.02745	$\pm$ 0.00767	$\pm$ 0.00866
	0.062	$\pm$ 0.01591	$\pm$ 1.33069	$\pm$ 1.04316	$\pm$ 0.01295	$\pm$ 0.00417	$\pm$ 0.00470
	0.087	$\mp$ 0.00018	$\pm$ 1.26590	$\pm$ 1.12728	$\pm$ 0.00031	$\pm$ 0.00011	$\pm$ 0.00011
0.087	0.012	$\pm$ 0.08346	$\pm$ 1.45459	$\pm$ 0.83287	$\pm$ 0.06613	$\pm$ 0.01548	$\pm$ 0.01747
	0.037	$\pm$ 0.06058	$\pm$ 1.39481	$\pm$ 0.93913	$\pm$ 0.04818	$\pm$ 0.01327	$\pm$ 0.01499
	0.062	$\pm$ 0.04077	$\pm$ 1.33411	$\pm$ 1.03317	$\pm$ 0.03263	$\pm$ 0.01037	$\pm$ 0.01171
	0.087	$\pm$ 0.02328	$\pm$ 1.27129	$\pm$ 1.11860	$\pm$ 0.01889	$\pm$ 0.00685	$\pm$ 0.00773

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Table 9. The imaginary part of exact solutions of CTFMNLSE (1.1) obtained by 6 different methods, with several point sources through arbitrary.

$\mathfrak{X}$	$\tau$	$\overbrace{\Psi_{3, 4}^{\Im}(\mathfrak{X}, \tau)}^{\text{FIM}}$	$\overbrace{\Psi_{3, 4}^{\Im}(\mathfrak{X}, \tau)}^{\text{FVM}}$	$\overbrace{\Psi_{3, 4}^{\Im}(\mathfrak{X}, \tau)}^{\text{TEM}}$	$\overbrace{\Psi_{3, 4}^{\Im}(\mathfrak{X}, \tau)}^{\text{MTEM}}$	$\overbrace{\Psi_{3, 4}^{\Im}(\mathfrak{X}, \tau)}^{\text{DAM}}$	$\overbrace{\Psi_{3, 4}^{\Im}(\mathfrak{X}, \tau)}^{\text{SGEM}}$
0.012	0.012	$\pm$ 633.10822	$\pm$ 12.28177	$\mp$ 588.19565	$\pm$ 587.89974	$\pm$ 350.78189	$\pm$ 144.25913
	0.037	$\mp$ 55.38382	$\pm$ 19.40644	$\pm$ 55.86884	$\mp$ 55.89947	$\mp$ 39.18969	$\mp$ 15.94980
	0.062	$\mp$ 26.13300	$\pm$ 14.23268	$\pm$ 26.26665	$\mp$ 26.28158	$\mp$ 21.21368	$\mp$ 8.64120
	0.087	$\mp$ 16.65633	$\pm$ 10.67754	$\pm$ 16.72216	$\mp$ 16.73321	$\mp$ 15.40396	$\mp$ 6.27945
0.037	0.012	$\pm$ 36.90715	$\mp$ 7.51554	$\mp$ 36.45427	$\pm$ 36.46563	$\pm$ 21.47530	$\pm$ 8.76623
	0.037	$\pm$ 114.21647	$\pm$ 1.26633	$\mp$ 109.43494	$\pm$ 109.42158	$\pm$ 75.62594	$\pm$ 30.98259
	0.062	$\mp$ 100.49552	$\pm$ 4.11852	$\pm$ 105.05287	$\mp$ 105.15849	$\mp$ 83.84576	$\mp$ 33.95722
	0.087	$\mp$ 33.39707	$\pm$ 4.55124	$\pm$ 33.94843	$\mp$ 33.97135	$\mp$ 30.87972	$\mp$ 12.55679
0.062	0.012	$\pm$ 19.06153	$\mp$ 6.31111	$\mp$ 18.86303	$\pm$ 18.87231	$\pm$ 10.96007	$\pm$ 4.47677
	0.037	$\pm$ 28.29015	$\mp$ 2.73162	$\mp$ 27.81244	$\pm$ 27.82036	$\pm$ 18.98398	$\pm$ 7.75543
	0.062	$\pm$ 55.34093	$\mp$ 0.47397	$\mp$ 53.37174	$\pm$ 53.37043	$\pm$ 41.95143	$\pm$ 17.17404
	0.087	$\mp$ 4340.87665	$\pm$ 0.69068	$\mp$ 1987.52122	$\pm$ 1951.57277	$\pm$ 1707.43866	$\pm$ 852.55044
0.087	0.012	$\pm$ 12.86682	$\mp$ 5.00514	$\mp$ 12.74097	$\pm$ 12.75059	$\pm$ 7.30283	$\pm$ 2.98701
	0.037	$\pm$ 16.19046	$\mp$ 3.32263	$\mp$ 15.97541	$\pm$ 15.98364	$\pm$ 10.76607	$\pm$ 4.40068
	0.062	$\pm$ 21.81420	$\mp$ 2.01126	$\mp$ 21.39382	$\pm$ 21.40030	$\pm$ 16.61527	$\pm$ 6.79175
	0.087	$\pm$ 34.30495	$\mp$ 1.10820	$\mp$ 33.19028	$\pm$ 33.19177	$\pm$ 29.39629	$\pm$ 12.03132

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5. Conclusions

Using the DAM and SGEM, firstly we found the exact solutions of CTFMNLSE (1.1) and finally, we presented numerical results in tables and charts. Also, we compared the real and imaginary parts of the exact solutions of CTFMNLSE (1.1) obtained by DAM and SGEM with four other different methods: FIM, FVM, TEM, MTEM. For some values the results obtained through DAM and SGEM, were near to the results obtained in four other methods. Overall, the performance of the proposed methods (DAM and SGEM) is reliable and effective and gives more solutions. These methods are direct and concise. Therefore, we conclude these methods can be extended to solve many nonlinear conformable fractional PDEs which are arising in the theory of solitons and other areas.

Acknowledgments

The authors T. Abdeljawad and N. Mlaiki would like to thank Prince Sultan University for paying the APC and for the support through the TAS research lab.

Conflict of interest

The authors declare no conflicts of interest.

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