Abstract
Using the trial equation method (TEM) and modified trial equation method (MTEM), firstly, we find the analytical solutions of the conformable time-fractional modified nonlinear Schrödinger equation (CTFMNLSE), and finally, we present numerical results in tables and charts.
1. Introduction
In the past ten years, NLSEs have received a great deal of interest from scholars and researchers. Actually, simplified versions of Zakharov’s system give rise to certain NLSHs. For more details, see [1]. NLSEs have applications in different subjects, e.g., quantum mechanics, 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, and nonlinear optics.
Lately, the investigation of the CTFNLSE in the formincluding numerics, analysis, and applications becomes a significant issue in applied mathematics, and diverse computational methods have been designed to discuss the exact solutions of it. Every method has its own proportion formulas and merits for the application to the administering equation for exploring the exact solutions.
In [2–4], the exact solution of CTFNLSE (1) is obtained through different methods such as the first integral method, functional variable method, sine-Gordon method, and direct algebraic method. In [5], Younas et al. presented the CTFMNLSE and studied the exact solutions of it by the generalized exponential rational function method.
Motivated by Younas et al. [5], we consider the following CTFMNLSE:where , and are the frequency and the wave number of the carrier wave, respectively, and the operator of order , where , represents the conformable fractional derivative.
Here, using the TEM and MTEM, we investigate the exact solutions for the above CTFMNLSE. For more details, you can see [6–11].
2. Preliminaries
Here, it would be helpful to present some properties and definitions of the conformal derivative and other preliminaries.
Definition 1 (see [12]). Suppose is a function. Then, the conformal fractional derivative of of order is presented asfor all .
Definition 2 (fractional integral; see [12]). Suppose and . Also, suppose is a function defined on and . Then, the fractional integral of is defined byif the Riemann improper integral exists.
Theorem 1 (see [12]). Suppose and and are differentiable at a point ; then,(i)(ii)(iii)(iv)Furthermore, if is differentiable, then .
Theorem 2 (see [12]). Suppose is a function such that is differentiable and also differentiable. Suppose is a function defined in the range of and also differentiable; then, one has the following rule:
3. Methods and Applications
In this section, we present the first step of the TEM and the MTEM for finding analytical solutions of the CTFMNLSE defined as (2). For more details, see [5].
Suppose a CTFNLPDEwhere and are an unknown function and a polynomial in its arguments, respectively.
Using a fractional travelling wave transformationwhere is the velocity and substituting (7) into (6), we have a NLODE given bywhere denotes the derivative with respect to .
Here, since in (2) is a complex function, for proceeding, we begin with the following travelling wave assumption:where and , and , and are parameters, representing the phase constant, frequency, and wave number, respectively. Substituting (9) into (2), we get real and imaginary parts as follows:
Now, integrating the imaginary part of the equation and taking constant equal to zero, one may have
From (10) and (12), it can be followed that
From the above, it can be followed that
Rewrite (10) into the following form:where and .
In the next two sections, we investigate the primary steps for detecting the exact solution of (10) by using the TEM and the MTEM.
The exact solution of (12) can be found in a similar way.
3.1. Trial Equation Method (TEM)
Firstly, CTFNLPDE (6) is reduced to NLODE (8) under transformation (7). Secondly, consider the trial equationwhere and are constants, which are derived from the solution of the system and the balancing principle, respectively. Finally, the solution of (17) can be given by the integral form:
Note 1. Ifthen integrating (19) with respect to once, we getwhere , and integration constant are to be determined.
Now, considering (16) and balancing and , we obtain . So, the trial equation isin whichso we have
Now, (18) is rewritten with (23):
If we set in (24) and integrate this equation, the exact solution of (2) is obtained:
If , then we have bright and singular solutions, respectively:
If , the singular periodic solutions appear aswhere is an arbitrary constant. For more details, see [13–15].
3.2. Modified Trial Equation Method (MTEM)
In this section, instead of using (17), the trial equation can be chosen aswhich is seen as the modified trial equation method when the same procedure is applied with the integral form solution of (29) as
Similarly, one can solve integral (30) and find the exact travelling wave solution of equation (6).
Note 3.2. If equation (29) holds, then we have
If we now use trial (29) in (16) for balancing the procedure, we get . Choosing and , the trial equation isand we havewhere and .
The corresponding system is
Solving the corresponding system, we obtain
Using these coefficients in (30), we get
When we integrate this equation and use the wave transformation, the exact travelling wave solutions of (2) are obtained as follows.
If , then we haveand if , then we havewhere is an arbitrary constant. For more details, see [16–18].
4. Numerical Results in Tables and Charts
Here, we let . Therefore, we have and Now, in the following sections, we calculate the exact solutions of CTFMNLSE (2) obtained through the two different methods presented in Section 3.
Figures 1(a), 1(c), 2(a), and 2(c) and also Figures 1(b), 1(d), 2(b), and 2(d) display 3D and 2D with the real part of solutions , and obtained by the TEM and MTEM under the values presented in the above paragraph. Tables 1–4 present the numerical results of analytical solutions of CTFMNLSE (2) obtained by the TEM and MTEM with several point sources through arbitrary. In addition, Figures 2–5 display 2D with the real part of solutions and for fixed and different values of . Figure 3 displays 3D with the imaginary and the real part of solutions and Also, Figure 6 displays 3D with the imaginary and real part of solution and also 2D with the imaginary part of the solution for fixed and different values of .
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4.1. The Exact Solutions of CTFMNLSE (2) Obtained through the TEM
Considering the given values in Section 4, we get Tables 1 and 2.
4.2. The Exact Solutions of CTFMNLSE (2) Obtained through the MTEM
Considering the given values in Section 4, we get Tables 3 and 4.
5. Concluding Remarks
Using the TEM and MTEM, firstly, we found the exact solutions of CTFMNLSE (2), and finally, we presented numerical results in tables and charts.
According to Table 5, we can observe that the differences between solutions and also for fixed and different values of are considerable. It is clear that these differences obtained through the MTEM are less than those of the TEM. In other words, for fixed , by changing the value of , the MTEM results in more minor changes than the TEM. Also, based on Figure 7, we can observe that the results gained by the MTEM have higher accuracy than the TEM.
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Figure 8 displays the differences between the real part of solutions obtained by the TEM and MTEM for fixed and . As you can see, the difference obtained among and is more minor than the difference obtained among and
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.