On deferred statistical convergence of sequences of sets

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:

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

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,

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

Remark 2.5. Let

The solutions of ODE (2.3) are:

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

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

where generalized hyperbolic and triangular functions are given by

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,

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

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

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

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

and

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

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

From above, it can be followed that

Rewrite (3.5) into following form

or

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

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

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

where

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

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

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 $\mathfrak{P}_{2}\neq 0,$

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

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]

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:

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

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

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

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

and

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

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

and

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

and

Also we have

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

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

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

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

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

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

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

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.

Figure 1 (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, Figure 1 (b), (f) and (c), (d) 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, Figure 2 (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 $\tau$ and also Figure 2 (b) and (d) 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.

Moreover, Figure 3 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 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 Figure 4 (b) 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.

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.

In Table 5, based on 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. 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.

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.

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.