Analytical Solutions for Nonlinear Dispersive Physical Model

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA School of Mathematics, South China University of Technology, Guangzhou 510640, China Department of Basic Science, Faculty of Engineering at Benha, Benha University, 13512, Egypt Department of Mathematics, Zagazig Faculty of Engineering, Zagazig University, Zagazig, Egypt

Getting exact solutions for these forms of equations became an important issue; then, most researchers try to achieve this target. e most effective method for obtaining exact solutions for NLFPDEs is the Lie symmetry reduction method. ere are many papers for using Lie's method to obtain explicit solutions for NLFPDEs [23][24][25][26].
In our paper, we drive the symmetry vectors for the time FBBM equation and present new closed-form solutions for it. e FBBM equation has many forms [27][28][29][30], and we choose to work on the following form: where ψ xxt is the dissipative term. e manuscript is prearranged as follows. In Section 2, Lie's group method for FPDEs is exposed. In Section 3, we apply the Lie group reduction method to obtain Lie point symmetry for the time FBBM equation (1). At the end of Section 4, we use these similarity variables to get the reduced equation. In Section 4, we use two methods for solving the resulting ordinary differential equation, the first method is the subequation method and the second method is the integrating factors method to get new solutions that have the properties and form the travelling wave form for the FBBME. In the end, conclusions are written in Section 5.

Fractional Riemann-Liouville Derivative.
In this section, we show some definitions for RL fractional derivative [31], which can be considered as follows: where D α t is the total differentiation of integer number of orders α, (α > 0), the Gamma function is Γ(n − α), and I n f(t) is the (RL) fractional integral of an order of n.
e partial derivative of order α for Riemann-Liouville definition is presented by

Notations for Lie Symmetry Reduction Method for the Time FPDEs.
In this section, we show in detail the main notations and definitions that will be used for obtaining the symmetries of NLFPDEs.
Here, we will consider timing NLFPDEs of the form [31][32][33] Assume, equation (2) has a Lie vector X in the form where ξ 1 , ξ 2 , and η can be called as the infinitesimals of the transformations the independent and the dependent variables (x, t, ψ) , respectively. Let a one-parameter Lie algebra of infinitesimal transformations be of the following form: where ε « 1 can be defined as a group parameter, in most cases we take it equal one. e explicit expressions of η x , η xx , and η xxt , which can be called the prolongation of the infinitesimals and are given by where D i is the total differentiation operator [34] with respect to the independent variables x i (i � 1, 2, then (2) with the Lie Vector X if and just if the accompanying infinitesimal conditions hold:

Theorem 1. Equation (1) concedes a one-parameter group of infinitesimal transformations in equation
where Δ � D α t u − F(t, x, u, u x , u xx , . . . . . .) and Pr is the 3rd prolongation of the infinitesimal generator X. Definition 2. Prolonged vector is given by [31] where q is the numbers of dependent variables, p is the numbers of independent variables, z/zu α j 1 � z/zu α x , and PDE involve derivatives up to order n. Also, the invariance condition [35] gives e αthextended infinitesimal, which deals with fractional derivatives, has the following form [36][37][38]: 2 Complexity where Remember that Due to linearization of the infinitesimal η in u and the presence of z k η/zψ k , μ will vanish, where k ≥ 2 in equation (14).

Lie Symmetry and Reduction of FBBM Equation
In this partition, the Lie symmetry reduction method was applied to find the similarity variables for a one-dimensional time (FBBM) equation. Suppose that (1) is an invariant under (2); we have that us, ψ(x, t) satisfies equation (1). Applying the third prolongation to (1), we have the accompanying deciding condition, which is given as Substituting (7) and (8) into (16) and equaling coefficients in derivatives for x and power of u to zero, the system of equations is obtained: α By solving the obtained equations in (18), we get the following infinitesimal: where c 1 and c 2 are constants. By the previous infinitesimal, equation (1) has two vector fields in the form Case 1. For the infinitesimal generator in (20a), we have a characteristic equation in the following form: By solving the previous equation, we get the variables t and ψ. Putting ψ � f(t) into (1), we obtain the following fractional ODE: By solving the above equation, we obtain where a 1 is constant of integration.
Case 2. For X 2 in equation (20b), the similarity variables for the infinitesimal generator X 2 can be obtained from the equation: e previous equation is called the characteristic equation; by solving it, we have the similarity variable as a result in the form: e group invariant solution where f(ξ) is a new arbitrary function of ξ and g(ξ) � t − ((α+1)/2) f(ξ). By using equation (26), equation (1) is transformed into FODE.

Clarifications for the Subequation Method.
e subequation method [39] is presented in this section. Consider the NLFPDE in the form 4 Complexity where ψ is a dependent variable, P is a series of ψ and its fractional derivatives, and D α t ψ and D α x ψ are the Riemann-Liouville (RL) derivatives of ψ w.r.t t and x. Here, we present the principles for the subequation technique. By using the d'Alembert transformation, where c is constant that will be determined later, and we can rewrite (41) as NLFODE: According to the subequation procedure, assume that the wave solution will be written in the following form: where a i , ( i � 1, . . . , n) are constants, which will be determined later, n belongs to integers numbers, which are determined by equaling the highest order derivatives and nonlinear terms in (44) together, and the function ϕ(ζ) achieves the Riccati equation of fractional order where σ is a constant. Some trigonometric solutions of the fractional Riccati equation (46) are By substituting forms (45) into (44) and setting the coefficients of ϕ(ζ) to be zero, we obtain an algebraic system in a i , ( i � 1, . . . , n)and c. By solving the determinate system, we obtain the constants a i , ( i � 1, . . . , n) and c. Substituting these constants and the solutions of (47) into (45), we obtain the closed form solutions of (42).

Applying the Subequation Method to the Time FBBM Equation.
We now implement a subequation method to (1). We will use the transformation where c is a constant, and this will transform (1) into an NLFODE: We now assume that (49) has the solution in the form where a i (i � 1, ..., n) are constants, which will be determined, and ϕ(ζ) achieves equation (46). Balancing the highest order derivative terms with nonlinear terms in equation (49), we obtain n � 2. Hence, We then substitute (51) along with (46) into (49), then collect the coefficients of ϕ(ζ), and set them to equal zero. A set of algebraic equations are obtained in knowns c, a 0 , a 1 , and a 2 . Solving these algebraic equations with the help of the software program (Maple), we get the following values.
us, from (47), we obtain five forms of explicit travelling wave solutions of (1), namely, us, from (47), we obtain five forms of explicit travelling wave solutions of (1), namely, where a 0 is arbitrary constant. We plot the result in equation (57) in the three dimensions, contour plot, and density plot, as shown in Figures 1-3, respectively.

Applying Simple Transformation.
We solve the conformable FBBM equation using simple transformation to change the fraction order in partial derivative to nonsolvable ODE. For the reduction of (1) to ODE, we use the following transformation: where v and k are arbitrary constants; we can rewrite (1) as NLODE: is equation has no implicit solution but possesses two integrating factors. We apply the integrating factor technique to obtain an analytical solution for (59).
Equation (59) has two integrating factors (IF) as follows: Using these integrating factors by the same steps in [39] and neglecting the constants of integration, equation (59) will be reduced to By solving this equation, we obtain travelling wave solution for (1):

Complexity
Replacing ζ � vx − k(t α /α), In other manner, equation (59) have two Lie vectors. e first one of them reduces it to Equation (64) has closed form solution, but, in the back substitution step, we are unable to get ψ(x, t) even if we neglect the values of constants. So, from here, we can say the integrating factor method for reducing and solve ODEs, occasionally, more effectiveness than the Lie reduction method. Result obtained in (63) is plotted in Figure 4 at different values of α. We observe that, by decreasing the value of α, the top of the wave has a parabolic shape.
Comparing our result in (63) with results in [5], specially equation (17), we find that the two solutions are travelling wave solutions, but the amplitude and direction of flow are different.

Conclusions
In this paper, we show the importance and the effective of the Lie symmetry reduction method on the FBBM equations. We obtain time FBBM equation's Lie symmetry generators and then reduce the equation to FODE using these symmetry vectors. e projected analysis is extremely effective and dependable for getting similarity solutions for fractional differential equations. New travelling solutions were derived for the FBBM equation using the subequation method.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.