A Spectral Collocation Technique for Riesz Fractional Chen-Lee- Liu Equation

This paper discusses the study of optical solitons that are modeled by Riesz fractional Chen-Lee-Liu model, one of the versions of the famous nonlinear Schrödinger equation. This model is solved by the assistance of consecutive spectral collocation technique with two independent approaches. The first is the approach of the spatial variable, while the other is the approach of the temporal variable. It is concluded that the method of the current paper is far more efficient and credible for the proposed problem. Numerical results illustrate the performance efficiency of the algorithm. The results also point out that the scheme can lead to spectral accuracy of the studied model.


Introduction
Several numerical methods, including local and global methods, have been listed as approximation techniques for treating the differential equations. The local methods listed the approximate solution at specific points, while the global methods give the approximate solution in whole the mentioned interval. The numerical approximations for differential equations [1][2][3][4] are listed at specific points using finite difference methods. While the finite element methods subdivide the whole interval into subintervals and give the approximate solution in them. The finite element methods are used for various types of differential equation; see for example [5][6][7].
Recently, there are more interests of appointing the spectral methods to treat with various kinds of differential and integral equations [8,9], due to their applicability to bounded and unbounded domains [10,11]. The convergence speed is one of the major advantages of spectral method. Spectral methods are promising candidates for solving fractional differential equations since their global nature fits well with the nonlocal definition of fractional operators. They have gained new popularity in automatic computations for a wide class of different problems which included linear and nonlinear differential equation of integer or fractional (fixed, variable, Riesz, tempered, and distributed orders); see [12,13]. Also, they are more reliable to treat the integral and integro-differential equations. Spectral methods have exponential convergence rates as well as a high accuracy level. The spectral method has been classified into four classes, collocation [14], tau [15], Galerkin [16], and Petrov-Galerkin [17] methods.
The theory of optical solitons [18][19][20][21] is mainly governed by the well-known nonlinear Schrödinger equation (NLSE) [22][23][24][25]. However, there exists a wide variety of its manifestations and modifications that also govern pulse transfer across the globe through optical fibers, PCF, metamaterials, and couplers. A few such models are Schrödinger-Hirota equation [26], Manakov equation, complex Ginzburg-Landau equation, Fokas-Lenells equation, Gabitov-Turitsyn equation, and many others. These models are considered under different circumstances such as dispersive solitons, differential group delay, and dispersion-managed solitons. Besides these familiar models, there is another class of versions of NLSE that is referred to as derivative NLSE (DNLSE) [27][28][29] that appears in three forms. One such form is the Chen-Lee-Liu equation [30][31][32] that incorporates higher order perturbations from optics and is going to be the focus of today's paper. While a plethora of preexisting work has been already reported in regard to this model, today's focus is going to be handling the model by the aid of fully shifted Legendre collocation method.
Shifted Legendre collocation schemes are used to numerically solve the Riesz fractional Chen-Lee-Liu model. The solution Θðξ, τÞ is firstly placed in its real Uðξ, τÞ and imaginary V ðξ, τÞ parts. Accordingly, the real Uðξ, τÞ and imaginary V ðξ, τÞ parts of such equation are approximated as U N ,M ðξ, τÞ and U N ,M ðξ, τÞ, respectively, which can be expressed as a finite expansion of shifted Legendre polynomials for spatial variable. Subsequently, the Chen-Lee-Liu equation with boundary conditions is reduced to temporal differential system with initial conditions. Then, the shifted Legendre-Gauss-Radau collocation is assigned for temporal discretization, which is more reliable for treating with such problems. Substituting these discretizations in the mentioned equation gets a nonlinear system of algebraic equations which solved numerically using the Newton-Raphson approach.
This paper is arranged as follows. In Section 1, some properties of Riemann-Liouville fractional derivatives, shifted Legendre polynomials, and shifted Chebyshev polynomials are listed. The mentioned scheme is implemented for the Chen-Lee-Liu equation with initial-boundary conditions in Section 2. In Section 3, two test examples are discussed. The competence of our numerical approach is exhibited by diverse examples in Section 4. Few remarks are mentioned in the last section (Section 5).

Riemann-Liouville Fractional Derivative
The fractional integration of order μ > 0 exists in different formulas [33]. Riemann-Liouville formula, the most common and widely used, is defined as follows: Here, we introduce some properties of the fractional operators. The left-sided and the right-sided fractional derivatives of Riemann-Liouville type of order β ðn − 1 < β < nÞ are defined as follows: The Riesz fractional derivative is defined as follows: where c β = −1/2 cos ðπβ/2Þ. The fractional Laplacian operator in Equation (3) can be represented in the following equivalent Fourier form on the spatial variable ξ: If ψ is defined on ½A, B and satisfies ψðA, τÞ = ψðB, τ Þ = 0, then the function can be extended by taking ψðξ, τÞ ≡ 0 for x ≪ a and x ≫ b. Moreover, as shown in [34], if ψ ξ ðA , τÞ = ψ ξ ðB, τÞ = 0, then the Riesz fractional derivative can be written as follows: The left and right RL-FDs of the Legendre polynomial are given by the following:

Chen-Lee-Liu Equation
In this section, we treat the next nonlinear Riesz space Chen-Lee-Liu equation with the following conditions: We now split the complex function Θðξ, τÞ into two real functions Uðξ, τÞ and V ðξ, τÞ as follows: where Uðξ, τÞ, V ðξ, τÞ, Δ 1 ðξ, τÞ, Delta 2 ðξ, τÞ, η 1 ðτÞ, η 3 ðτÞ, η 2 Journal of Function Spaces 2ðτÞ, η 4 ðτÞ, φ 1 ðxÞ, and φ 2 ðxÞ are the real functions. Thereafter, with the next conditions: 3.1. Spatial Discretization. The distribution of shifted Legendre-Gauss-Lobatto nodes in ½0, ξ end is the major feature of considering them in our discretization. Here, we list the basic main of implementing our Legendre-Gauss-Lobatto collocation scheme for converting the nonlinear system (Equations (10) and (11)) into temporal ordinary differential system. The spectral approximation of P ðξ, τÞ and Qðξ, τÞ is given as follows: where the orthogonal property and discrete inner product permit the following: In that case, Equation (14) takes the form: Over and above that, the partial derivative of first order in space evaluated at shifted Legendre-Gauss-Lobatto colloca-tion is as follows: where Comparable procedure can be performed to the Riesz fractional derivative ∂ μ ϕ N ,M ðξ, η, τÞ/∂jξj μ for space variable to get where Combining the boundary conditions with the abovementioned equations and equalizing the residual of Equation (7) with initial values The numerical approach of such system will be listed in Subsection 3.2.

Temporal Discretization.
Here, we numerically treat the temporal differential system with initial conditions: where G r ðt, W 1 ðτÞ,⋯,W R ðτÞÞ, r = 1, ⋯, R are given functions. Shifted Legendre-Gauss-Radau collocation is assigned for temporal discretization, which is more reliable for    Journal of Function Spaces treating with such problems. We approximate W r ðτÞ as follows: The temporal derivative _ W r ðτÞ is evaluated as follows: Thus, we get the following: Combining the initial conditions with the abovementioned equations and equalizing the residual of Equation        Journal of Function Spaces Finally, we have ðRðK + 1ÞÞ algebraic equations as follows: The numerical approach of the previous system will be acquired by using Newton's iterative method.

Applications and Numerical Results
Here, the adequacy of the spectral collocation algorithms is verified by the obtained results. Problems including initialboundary conditions are examined. Mathematica version 10 is utilized to carry out the code.
Example 1. We test the next problem: where the function Δðξ, τÞ, initial condition, and the boundary conditions are given such as the continuous problem has the next exact solution: In Table 1, the numerical results based on the maximum absolute errors of Equation (32) obtained using the previous algorithms are listed, where Space graphs of real and imaginary parts of the numerical solution of Equation (32) are shown in Figures 1(a) and 1(b), respectively, where N = M = 12. While in Figures 2(a) and 2(b), we recognize the outright matching of numerical and exact solutions in its real and imaginary parts of Equation (32), where N = M = 12. Also, ξ-direction curves for real and imaginary parts of the absolute errors of Equation (32) are plotted in Figures 3(a) and 3(b), respectively, where τ = 0:5, N = M = 12. Moreover, we sketched in Figure 4 the logarithmic graphs of M ε (i.e., log 10 M ε ) of Equation (32) obtained by the present method with different values of ðN = M = 2, 4, 6,⋯,12Þ.
Example 2. Now, consider the following: where the function Δðξ, τÞ, initial condition, and the boundary conditions are given such as the continuous problem has the next exact solution: In Table 2, the numerical results based on the maximum absolute errors of Equation (35)  Even though few values of N and M, the accurate results have been spotted in these tables. This is consistent with which was predicted in case of using a spectral collocation method. Likewise, these results bring to light the responsibility convergence of the shifted Legendre collocation method for such problems.

Conclusions
This paper adopted fully collocation method to study Riesz fractional Chen-Lee-Liu equation that discusses soliton propagation down the optical fibers with perturbation terms incorporated into the waveguides. The powerful numerical scheme gave way to a number of impressive numerical results that prove high efficiency of the algorithm. The study was carried out with initial-boundary conditions.
The results of the algorithm pave way to conduct further additional research in this field to display additional results in future. One avenue is to consider Riesz fractional Chen-Lee-Liu equation with differential group delay and then further along study the model with additional optoelectronic devices such as in magneto-optic waveguides. Subsequently, this model will be treated with the same algorithm for dense wavelength division multiplexing (DWDM) topology.
Thus, a lot lies in the bucket list!