Computational Study for Fiber Bragg Gratings with Dispersive Reﬂectivity Using Fractional Derivative

: In this paper, the new representations of optical wave solutions to ﬁber Bragg gratings with cubic–quartic dispersive reﬂectivity having the Kerr law of nonlinear refractive index structure are retrieved with high accuracy. The residual power series technique is used to derive power series solutions to this model. The fractional derivative is taken in Caputo’s sense. The residual power series technique (RPST) provides the approximate solutions in truncated series form for speciﬁed initial conditions. By using three test applications, the efﬁciency and validity of the employed technique are demonstrated. By considering the suitable values of parameters, the power series solutions are illustrated by sketching 2D, 3D, and contour proﬁles. The analysis of the obtained results reveals that the RPST is a signiﬁcant addition to exploring the dynamics of sustainable and smooth optical wave propagation across long distances through optical ﬁbers.


Introduction
Over the last few decades, the investigation of optical wave propagation has caught the attention of different scientists in applied mathematics. Many significant developments have been made in the field of nonlinear optics [1][2][3][4][5]. In the realm of nonlinear optics, it is widely known that the propagation of an optical pulse in nonlinear media, including Kerr law and non-Kerr law, may be represented by the nonlinear Schrödinger equation [6]. The Kerr law of nonlinearity arises when a light wave in an optical fiber experiences nonlinear responses due to nonharmonic motion of electrons caused by an external electric field [7][8][9]. Low chromatic dispersion is one of the fundamental issues with soliton transmission over intercontinental distances. To manage and lessen this effect, a number of strategies and approaches have been applied. The project was then successfully completed by adding gratings to optical fibers [10][11][12].
One of the most inventive technologies applied to nonlinear optics is fiber Bragg grating (FBG) [13]. FBG technology has been embraced by the field of fiber optics since its discovery. Presently, most fiber optic sensor systems use FBG technology. Due to the inherent benefits of FBGs, including their compact size, quick reaction, dispersed sensing, and immunity to the electromagnetic field, FBG-based sensing has attracted a lot of research interest [14]. Measurements of numerous physical parameters, including temperature, pressure, and strain are frequently performed using FBG technology [15]. For a wide range of applications, FBGs are also useful in signal shaping as well as filtering components [16].
Occasionally, another difficulty comes when the chromatic dispersion is gradually depleted, and hence, there is an unstable balance between the chromatic dispersion and the nonlinearity, being followed by a possible pulse collapse. To overcome this difficulty, the chromatic dispersion is replaced by a combination of third-order and fourth-order dispersions.
Optical waves in FBGs with different nonlinear refractive indices, where the chromatic dispersion is replaced by the third-and fourth-order dispersions together with the dispersive reflectivity have been retrieved analytically in the earlier studies. For example, Wang et al. [17] investigated this model by implementing the trial equation approach to retrieve cubic-quartic optical solitons in FBGs. Yildirim et al. [18] retrieved the soliton solutions of this model with the sine-Gordon equation method. Malik et al. [19] investigated the nonlinear Schrödinger equation, which considered the cubic-quartic form of nonlinear refractive index with parabolic law of nonlinearity by Lie symmetry. Arnous et al. [20] derived the optical solitons in FBGs with cubic-quartic dispersive reflectivity with the aid of enhanced Kudryashov's technique. Zayed et al. [21] used the auxiliary equation approach to explore solitons in FBGs with cubic-quartic dispersive reflectivity having the Kerr law of nonlinear refractive index.
In this paper, we consider a model in FBGs with cubic-quartic dispersive reflectivity having a Kerr nonlinear refractive index and Caputo's time-fractional derivative of order (0 < ≤ 1). The mathematical form of the considered model is expressed as where n( , t) and o( , t) denote wave profiles, and t and denote the time in dimensionless form and the non-dimensional distance, respectively. The coefficients a j , b j , α j , c j , d j and β j correspond to third-order dispersion, fourth-order dispersion, intermodal dispersion, self-phase modulation, cross-phase modulation, and detuning parameters, respectively. The exact solution of this model at = 1 is reported by Ming-Yue-Wang et al. [17]. The primary goal of this work is to present the optical wave solutions and their graphical observations for the considered model in FBGs with Caputo's time-fractional derivative using the residual power series technique (RPST). The obtained results are novel, and the proposed technique is utilized for the first time to study the considered fractional model in this work.
The discipline of fractional calculus has seen tremendous advancements in study at the intersection of probability, chaos, differential equations, and mathematical physics [22][23][24][25][26]. The nonlocal property of fractional differential equations (FDEs) is a significant advantage of employing them in various mathematical modeling. Retrieving the analytical solutions for FDEs is sometimes difficult to achieve due to the computational complexities of fractional operators. In this respect, numerous methods have been constructed and motivated to investigate the approximate solution for FDEs, among which are the Sinc-collocation method [27], the predictor-corrector compact difference scheme [28], the variational iteration method [29], the homotopy analysis technique [30], the homotopy asymptotic method [31], the homotopy perturbation scheme [32], the differential transform method [33], the linearly compact scheme [34], the Adomian decomposition method [35] etc.
One of the most efficient techniques for exploring approximate analytic solutions for linear and nonlinear FDEs is the residual power series technique (RPST) [36][37][38][39][40]. The RPST was introduced by the Jordan mathematician Abu Arqub [41]. The RPST has been successfully used to construct the approximate analytic solutions of FDEs without implementing linearization, perturbation, or discretization techniques, showing the reliability and simplicity of this technique. This proposed technique has been successfully applied to investigate the solutions of time-fractional Whitham-Broer-Kaup equations [42], Black-Scholes European option pricing equations [43], the KdV equation [44], the nonlinear Schrödinger equation [45], the Biswas-Milovic equation [46], the Caudrey-Dodd-Gibbon-Sawada-Kotera equation [47], etc. The suggested technique is a reliable, practical, and astonishingly effective tool for examining the approximate solutions of many types of real-life nonlinear models.
The RPST has several advantages. It is well-known for its accuracy and simplicity to obtain the desired results. It does not require variable discretization, and there is no requirement for huge computer memory or time to reach the solutions. Its nature is global regarding the approximate analytical solutions, which makes it useful to investigate various mathematical, engineering, and physical problems. Being analytic expressions, the results obtained through the RPST can be further explored with derivative calculations.
The arrangement of this paper is given in the following manner. Some preliminaries of fractional calculus and the RPST are presented in Section 2. A description of the fundamental steps of the proposed method and the construction of a solution to the suggested problem is illustrated in Section 3. Graphical findings for different applications are exhibited in Section 4. The conclusion is presented in the last section.

Preliminaries
This section introduces the fundamental properties of fractional calculus theory, allowing us to follow the solutions of a coupled nonlinear Schrödinger equation for the Kerr law of nonlinear refractive index in FBGs.
The Caputo technique is chosen in this study because it is appropriate for real-world physical problems, and it specifies integer-order initial conditions for FDEs.

Definition 3.
A multiple fractional power series about t = t 0 , for 0 ≤ ζ − 1 < ≤ ζ is defined as follows: where the coefficients of the series are δ k and are functions of .

Theorem 2.
Assume that δ has a fractional power series representation of the form at t = t 0 , If D k δ(t) are continuous on (t 0 , t 0 + R), k = 0, 1, 2, . . . , then the coefficients "e k " appearing in Equation (5) can be determined as where R is considered as the series's radius of convergence.
Other results related to the RPST can be found in [37,48,49].

RPST to FBGs for Cubic-Quartic Dispersive Reflectivity with Time-Fractional Derivative
Consider a coupled nonlinear Schrödinger equation with the cubic-quartic dispersive reflectivity having the Kerr law of nonlinear refractive index as subject to the initial conditions where D t represents Caputo's time-fractional derivative of order . The fundamental goal of this work is to develop the solution for Equations (6) and (7) by its power series expansion among its truncated residual function. For the construction of power series solutions of Equations (6) and (7), let where n( , 0) = p( , 0) + ιs( , 0), and o( , 0) = u( , 0) + ιw( , 0). The following system can be obtained using the above equations in (6) and (7) as with initial conditions where q( ) = σ 1 ( ) + ισ 2 ( ), and r( ) = ξ 1 ( ) + ιξ 2 ( ).

General Procedure of the RPST
The following are the primary steps for the recommended process.
Step A. Assume the fractional power series solutions of above system regarding the initial point (t = 0) as For the numerical values of fractional power series solutions for the system of equations with initial conditions, take the z − th truncated series of p( , t), s( , t), u( , t), and w( , t) as p z ( , t), s z ( , t), u z ( , t), and w z ( , t), respectively, i.e., Using the initial conditions, the 0 − th fractional power series solutions can be obtained as The system of Equations (12) can be rewritten as The required expressions for z − th fractional power series solutions will be obtained by varying k = 1, 2, 3, . . . , z in p( , t), s( , t), u( , t), and w( , t).
Step B. For Equations (9) and (10), the residual functions will be constructed. This construction will be made in the following manner as Taking , and w z = w z ( , t), then the z − th residual functions z = 1, 2, 3, . . . can be defined as Some important results of Res p ( , t), Res s ( , t), Res u ( , t), and Res w ( , t) that are useful for the residual power series solutions for i = 1, 2, . . . , z are stated below: 1.
Step C. Substituting p z ( , t), s z ( , t), u z ( , t), and w z ( , t) into (16) and calculating the fractional derivative D , and Res w,z ( , t), z = 1, 2, 3, . . . at the initial point (t = 0), together with results mentioned in Step B, the resulting algebraic systems are as follows: Step D. The required values of g k ( ), f k ( ), l k ( ), and m k ( ), k = 1, 2, . . . , z can be derived by solving Systems (17). Finally, the z − th residual power series solutions can be obtained.
In the next discussion, the first, second, and third residual power series solutions of the suggested problem are developed in detail by following the above steps.

Graphical Representation
To depict the reliability of the RPST to obtain the optical solutions to fiber Bragg gratings with cubic-quartic dispersive reflectivity having the Kerr law of nonlinear refractive index structures, three applications are examined. The fractional derivative is defined by Caputo's sense.

Example 1. Examine the system,
where 0 < < 1, subject to the initial conditions, This problem has an analytical solution for = 1 that is provided in [17] as For graphical observations, 2D plotting, contour plotting, and 3D plotting are constructed for Example 1 and shown in the Figures 1-6. It can be seen that the power series solutions very accurately agreed with the analytical solutions. Finally, by employing a large number of terms from the residual power series approximations, a lower error can be attained. The 2D graphs for Example 1 illustrate that the solution had nearly identical behavior for the standard case = 1 and various values of in terms of accuracy.
where 0 < < 1, subject to the initial conditions The advantage of using the RPST is that it establishes the continuous approximated solutions. In the Figures 7-9, the power series generated solutions of (27) are depicted geometrically by 3D plotting, contour plotting and 2D plotting. Error can be reduced by taking many terms, and the efficiency of the method can be increased.

Example 3. Examine the system,
where 0 < < 1, subject to the initial conditions, Figure 10 represents the mathematical behavior of Equation (30) subject to initial conditions Equations (31) and (32). It can be clearly observed that the obtained solution also depends on the initial conditions. By varying the initial conditions, the mathematical behavior of a problem also changes. Consequently, the proposed technique is an effective technique for the approximated solutions of different fractional-order models.

Conclusions
In this paper, the RPST with Caputo's time-fractional derivative is applied to develop the approximate solutions to FBGs with cubic-quartic dispersive reflectivity having the Kerr law of nonlinear refractive index, with high accuracy for the first time. The efficiency and reliability of the RPST are established by three test applications with different initial conditions. It is worth mentioning that the RPST structure has a rapidly converging series with components that are easily calculated with symbolic calculation software. The obtained results are also demonstrated through 2D and 3D representations by taking different values of . Since the exact solution of the problem for = 1 is available in the literature, a graphical comparison is made to confirm that the RPS approximation solutions are in agreement with the exact solutions for = 1. The variation in the solutions for change in the value of is also observed. The consequence found by using the active RPST is that it can be successfully used to investigate the dynamics of nonlinear models in the field of optical fibers.