Lie Symmetry Analysis, Conservation Laws, Power Series Solutions, and Convergence Analysis of Time Fractional Generalized Drinfeld-Sokolov Systems

: In this work, we investigate invariance analysis, conservation laws, and exact power series solutions of time fractional generalized Drinfeld–Sokolov systems (GDSS) using Lie group analysis. Using Lie point symmetries and the Erdelyi–Kober (EK) fractional differential operator, the time fractional GDSS equation is reduced to a nonlinear ordinary differential equation (ODE) of fractional order. Moreover, we have constructed conservation laws for time fractional GDSS and obtained explicit power series solutions of the reduced nonlinear ODEs that converge. Lastly, some ﬁgures are presented for explicit solutions.


Introduction
Because of the great importance of nonlinear fractional partial differential equations (NFPDEs) in physics, mechanics, hydrology, viscoelasticity, image processing, electromagnetics, and other fields, researchers have long been aware of the solutions and applications of fractional partial differential equations [1][2][3][4][5][6][7][8][9][10][11][12]. In recent years, parallel to the increase in mathematical techniques and the use of computer programs, many authors have an increased desire to work on fractional analysis. Therefore, many methods are used to solve the NFPDEs-for example, the finite-difference method [13], the multiple exp-function method [14], the homotopy perturbation method [15,16], the variational-iteration method, the exp method [17], the fractional sub-equation method [18], and the Lie invariance method [19][20][21]. When we look through the literature, we realize Sophus Lie firstly put forward a methodology about symmetry analysis at the end of the nineteenth century [22]. After that, some impressive Lie group methods were considered in order to obtain symmetries, symmetry groups, and symmetry reduction. These are the classical and nonclasical Lie group approaches [23,24] and the Clarkson and Kruksal direct methods [25,26]. The main role of Lie symmetry methods is to construct invariance properties having partial equations as invariant forms. With the aid of these properties, we can reduce an NFPDE into a nonlinear ODE of fractional order with the help of the Riemann-Liouville (RL) derivative.
The link between Lie symmetry analysis and conservation laws of differential equations was revealed by Noether [27]. A generalized Noether theorem was used in [28] to construct conservation laws of NFPDEs with fractional Lagrangians. However, some differential equations do not arise from Lagrangians. To overcome this problem,İbragimov [29] put forward a new method without Lagrangians. Lukashchuk [30] showed important results towards obtaining conservation laws of NFPDEs.
In this study, we deal with exact solutions of the time fractional GDSS by using Lie symmetry analysis and conservation laws. The time fractional GDSS that models 1D nonlinear wave processes in two-component media has the form where a 1 , a 2 , b 1 , b 2 , c, and q are arbitrary constants, and D α t is the RL fractional derivative defined in (2). We could not take b 1 = b 2 = 0, because the time fractional GDSS may develop finite time singularities [31][32][33]. According to our research, Lie symmetry methods have not been applied to the time fractional GDSS until now. With some special choices of a 1 , a 2 , b 1 , b 2 , c, and q, the equation can be reduced to the time fractional Drinfeld-Sokolov-Satsuma-Hirota equations discussed in [34].
The RL fractional partial derivative [35] is defined by where Γ(z) is a Gamma function defined by which converges in the complex plane when Re(z) > 0.

Preliminaries for Symmetry Analysis
In this chapter, we give the basic idea of the Lie symmetry method. Consider a system of time fractional partial differential equations with independent variables x and t as follows: where the subscripts denote partial derivatives. Equations (3) and (4) are invariant under a one-parameter Lie group of point transformations given as follows: where wherein D x and D t are total differential operators given as follows: The infinitesimal generator of Equations (3) and (4) consists of a set of vector fields given by One can prove that the infinitesimal generators defined above must satisfy the following invariance criterion for Equations (3) and (4): Pr n X(∆ 1 )| ∆ 1 =0 = 0 and Pr n X(∆ 2 )| ∆ 2 =0 = 0, n = 1, 2, .., where .. ). As the lower limit of integral in (1) is fixed, it is going to be invariant under the transformations given in (5), so the corresponding invariance condition becomes Under the condition (9), the α-th infinitesimal related to RL fractional time derivative [36] is given by (3) and (4) if and only if and v = ν 2 (x, t) satisfy the following expressions: (3) and (4), respectively.

Lie Symmetry Analysis and Reduction of Time Fractional GDSS
In this part of the work, we acquire an infinitesimal generator of the Generalized Drinfeld-Sokolov systems by applying Lie point symmetries. Assume that Equation (1) are invariant under one parameter transformations (5). We then have that Applying the second prolongation Pr 2 transformation to Equation (1), then using transformation (6), we obtain invariant equations as follows: Putting the values of ρ 0 α , µ 0 α ,ρ,µ µ x , µ xxx , ρ x , and ρ xxx from Equation (6) into Equation (12) and then isolating the coefficients in partial derivatives with respect to u and v, we obtain a determined system of linear equations stated as Solving all these determination equations, we obtain an explicit form of infinitesimal symmetry for Equation (1) as where c 1 and c 2 are arbitrary constants. Thus, we can construct corresponding vector fields: It is stated that there are two vector fields spanning Equation (1): Case 1. For the symmetry X 1 , we can write characteristics equations as follows: Solving these characteristic equations, we can easily obtain a trivial solution.
Case 2. Lastly we focus on symmetry X 2 , so we can write characteristic equations as Solving these characteristic equations, we get both the similarity variable and the similarity transformation as follows: where f and g are arbitrary functions of ξ.
where the Erdelyi-Kober (EK) fractional differential operators [37] are given as and with the EK Fractional integral operators [38,39] defined as and Proof. Let n − 1 < α < n, n = 1, 2, 3, .... Using the transformations defined in (15) and the definition of the RL fractional derivative, we have Let v = t s , so ds = −t v 2 dv. Equation (22) becomes Applying the above procedure n − 1 times, we obtain Now, by using Equation (17), we can immediately write Consequently, we prove that the first equation defined in (1) reduces to an ODE of a fractional order: Similarly, using transformations (15) and the definition of the RL fractional derivative, we easily reduced the second equation of (1) into an ODE: The proof is completed.

Conservation Laws for the Time Fractional GDSS
Before obtaining the conserved vector for the time fractional GDSS, some important definitions should be given. We can start with the RL right-sided time fractional derivative, defined as follows: oD n t (oI n−α f ), where D t is a total differential operator with respect to t, and the oI n−α is the right-sided time-fractional integral of n − α [40] given by where n = [α] + 1. All solutions of u(x, t) and v(x, t) provide the following conservation equation: where N t = N t (x, t, u, ..) and N x = N x (x, t, u, ..). We use theİbragimov method [29] to construct conservation laws of Equation (1). The formal Langrangian of Equation (1) is defined as the following: where σ and φ define the new dependent variable of x and t. Now, we express Euler-Langrangian operators [41] as follows: and δ δv where (D α t ) * is the adjoint operator of D α t . Using the Euler Lagrange Equations (35) and (36), we can write the following expression: δL δu = 0, and δL δv = 0.
Thus, we havē where I denotes the identity operator, δ δu and δ δv represents the Euler Lagrange operators, C t and C x are the Noether operators, andX is defined as and we can give Lie characteristics functions W for vector field X 2 as and Using the RL time fractional derivative in Equation (1), we now write components of conserved vectors [40,41], as follows: where J is defined as follows: and the explicit form of N x for Equation (1) is given as We can now derive the corresponding conserved vectors respectively as follows:

Series Solutions of Equations (29) and (30)
In this section, we examine the power series solution of the system. This method is more accurate and efficient for obtaining an exact analytical solution. The procedures of the method are given in [42]. We can construct and Substituting Equations (44)-(46) into Equations (29) and (30), we have and When n = 0 in Equations (47) and (48), we obtain the following coefficients: and When n 1, we have the following coefficients ...

Convergence Analysis of the Power Series Solution
In this part of the work, we will prove that the power series solutions (53) and (54) ... and Using the properties of Γ, we can easily show that Therefore, we can write ... and where and let y i =| r i | and z i =| s i |, i = 0, 1, ... . We can then easily obtain y n+3 ≤ M y n + n ∑ k=0 y n−k+1 y k ... and Thus, it is obvious that | y n |≤ r n and | z n |≤ s n for n = 0, 1, 2, .... This also confirms that the series Equation (59) are the majorant series of Equation (44). We now have to prove that the series K(ξ) and L(ξ) have positive radius of convergence. By elementary calculations, we have the following: and L(ξ) = z 0 + z 1 ξ + z 2 ξ 2 + z 3 ξ 3 + N ∞ ∑ n=0 z n + ∞ ∑ n=0 n ∑ k=0 z n−k+1 y k ξ n+3 .

Conclusions
In this paper, we used a Lie point symmetry method in order to reduce a time fractional generalized coupled Drinfeld-Sokolov system to a time fractional coupled ODE system with the aid of the Riemann Liouville derivative and the fractional EK differential operator. By using theİbragimov conservation theorem, we obtained conservation vectors of the system. We then acquired explicit exact solutions of the reduced time fractional coupled ODE system by using a power series expansion method and proved that the series solutions are convergent.