Some Exact Solutions and Conservation Laws of the Coupled Time-Fractional Boussinesq-Burgers System

In this paper, we investigate the invariant properties of the coupled time-fractional Boussinesq-Burgers system. The coupled time-fractional Boussinesq-Burgers system is established to study the fluid flow in the power system and describe the propagation of shallow water waves. Firstly, the Lie symmetry analysis method is used to consider the Lie point symmetry, similarity transformation. Using the obtained symmetries, then the coupled time-fractional Boussinesq-Burgers system is reduced to nonlinear fractional ordinary differential equations (FODEs), with Erdélyi-Kober fractional differential operator. Secondly, we solve the reduced system of FODEs by using a power series expansion method. Meanwhile, the convergence of the power series solution is analyzed. Thirdly, by using the new conservation theorem, the conservation laws of the coupled time-fractional Boussinesq-Burgers system is constructed. In particular, the presentation of the numerical simulations of q-homotopy analysis method of coupled time fractional Boussinesq-Burgers system is dedicated.


Introduction
Fractional differential equations (FDEs) come from the generalization of classical differential equations of integer order.It is well known that fractional calculus was widely applied to describe many complex nonlinear phenomena arising in the areas of heat transfer, diffusion, solid mechanics, wave propagation and other topics.Therefore, the fractional partial differential equations play an important role in describing physics, engineering and other scientific fields [1][2][3][4].
In 2009, Gazizov and Kasatkin [5] extended Lie symmetry approach to investigate several FDEs.Based on the symmetry, many useful properties of FDEs, such as symmetry generators, similarity transformation, explicit solutions and conservation laws which can be analyzed successively [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].Komal and Gupta [21,22] extended the symmetry approach from single time fractional PDEs to nonlinear systems of time fractional PDEs.The famous Noether theorem [23] established a connection between Lie symmetries and conservation laws of differential equations.Recently, constructing conservation laws via a new conservation Noether theorem to the FPDEs without Lagrangian has been introduced [24,25].In spite of the symmetry approach and conservation laws have made some progress in FDEs, the research for coupled time fractional FDEs are not very well explored.There are still many unknown results having not been reported before.The main aim of this paper is to investigate the Lie point symmetry, similarity reduction and conservation laws under the definition of Riemann-Liouville fractional differential.In addition, numerical results and verification of the correctness of the presented method are presented.In this paper, the fractional Lie symmetry scheme and the new conservation Noether theorem are developed to research the coupled time-fractional Boussinesq-Burgers system, that is [26] D where 0 < α ≤ 1, t > 0. ∂ α f (x,t) is the Riemann-Liouville partial fractional derivative, u(x, t) is the horizontal velocity field and v(x, t) is the height of the water surface above a horizontal level at the bottom.The coupled time-fractional Boussinesq-Burgers system is an interesting mathematical model that arises in the study of fluids flow in a dynamic system and describes the propagation of shallow water waves.
The organization of the paper is as follows.In Section 2, we recall some definitions of fractional derivatives given in [27][28][29][30][31][32][33][34][35][36][37][38]; we highlight some steps for the Lie symmetry analysis of PDEs.In Section 3, we obtain the Lie point symmetries and symmetry reductions of Equation (1).In Section 4, explicit solutions of reduction equation for Equation (1) are obtained by using the power series expansion method.In addition, the convergence of power series solution is analyzed.Section 5 deals with the application of the proposed approach for investigating conservation laws for Equation (1).In Section 6, the well-known q-homotopy analysis method is used to investigate numerical approximations for the coupled time fractional Boussinesq-Burgers system.The concluding remarks are presented in the last section.

Preliminaries
In this section, we discuss the main points of fractional Lie symmetry analysis of the coupled time fractional PDEs.Consider a system of time fractional PDEs as follows: where α > 0, subscripts represent partial derivatives.
Definition 1 ([29-38]).The Riemann-Liouville partial fractional derivative is defined as follows: where Γ(α) is the Euler's gamma function.According to the Riemann-Liouville partial fractional derivative operators, we have where (D α t ) * is the adjoint operator for the D α t , and (D α c ) C t is the right-sided Caputo operator [39,40].Then we consider the single parameter Lie group with infinitesimal transformation given by here ξ, τ, η and φ are the infinitesimals operators, η α,t , φ α,t are the extended infinitesimal of order α and η x , φ x , η xx , φ xx ,η xxx , φ xxx are extended infinitesimals of integer-order.Consider the following vector fields: The αth order developed infinitesimal η α,t has the following form Using the generalized Leibnitz rule and generalized chain rule [41][42][43][44], the final expression for αth order developed infinitesimal η α,t for system of fractional PDEs of the form Equation (2) can be calculated as follows: where D t represents the total derivative operator, µ 1 , µ 2 , µ 3 , µ 4 are given by Equation ( 5) represents a point symmetry of Equation ( 2) as long as where i is the order of system Equation ( 5).The invariance condition should be held:

Lie Symmetry Analysis
Let us consider the invariance of the group transformations (5).The invariance criterion takes the following forms: Then, substituting the values of prolongations and equating the coefficients of various linearly independent variables to zero, we have: where c 1 , c 2 are arbitrary constants.Therefore, one infers the following corresponding infinitesimal generators of Lie algebra: It is easy to check that the vector fields ( 14) are closed under the Lie bracket, respectively Considering the vector field V 1 , we can write the characteristic equations Solving the above equations, the similarity solutions are given by
Definition 2 ([45-47]). where is the Erd élyi-Kober fractional integral operator.To calculated , the definition of Riemann-Liouville fractional derivative (3) can be written as following: Let p = t s , then the above expression is converted to the following: Based on the definition of Erd élyi-Kober fractional integral operator, (21) can be written as Considering Hence, Equation ( 22) can be transformed as follows: According to the definition of Erd élyi-Kober fractional differential operator, it can be written as follows: At the same time, the ∂ α v ∂t α can be presented as , we have the following: Similarly, for g)(ω).Hence, expressions ( 25) and ( 26) Therefore, the coupled time-fractional Boussinesq-Burgers system (1) is reduced to

Power Series Solution
In what follows, we shall derive explicit solutions for system (1) by means of the power series expansion method.To find the exact power series solutions of system (1), we let where a n and b n are constants to be known later.Substituting Equation (29) into Equation ( 28), we can obtain In view of Equation ( 30), comparing coefficients for n = 0, we get When n ≥ 1, we have Then, we can write Hence, the explicit solution of Equation ( 1) is (see Figure 1).

Conservation Laws
The method of constructing conservation laws for fractional partial equations has been given in many papers [12,15,16,[19][20][21][22][23][24].In this section, we will study the conservation laws of the coupled time-fractional Boussinesq-Burgers system (1) by using the adjoint equation and symmetries of Equation (1).A formal Lagrangian for Equation (1) can be written in the following form: where p(x, t) and q(x, t) are the new dependent variables.For Equation ( 1), the adjoint equation has the form Contenting Ξ i = 0 with at least one i(i = 1, 2), that is where and their derivatives, system (43) have the following form: Equating the coefficients of various derivatives and powers of u, v in Equation ( 45) and thereafter solving simultaneously, we obtain where A and B are arbitrary constants.Hence, Equation ( 1) is nonlinearly self-adjoint.Subsequently, the character functions have the form: Using (46) and setting A = B = 1, the conserved vectors are given as follows.
The x-components C x i corresponding to V i (i = 1, 2) are given as follows: ), ). ( The t-component C t i are given as follows: Case I: when α ∈ (0, 1), the conserved vectors are Case II: when α ∈ (1, 2), the conserved vectors are (50)
Remark 1.Using the first two terms of the q-HAM series in Equation (64), when n = (65) Thus, we get exact solution to the Equation (1) given by just two terms of the series.
Remark 2. Figure 2 displays the solution plot of the coupled time-fractional Boussinesq-Burgers system obtained by the q-HAM, while Figure 3 displays the exact solutions for the same equation when α = 0.4, α = 0.6, α = 0.8, respectively.It should be noted that only three terms of the q-HAM series solution are used for the plot.
The results match comparatively with results of other analytical methods.It is easy to observe that the amplitude of u and v increase with the increase of α.

Conclusions
In this paper, the fractional Lie symmetry analysis to the coupled time-fractional Boussinesq-Burgers system has been performed.Based on the fractional Lie symmetry analysis approach, we have determined vector fields and reduced it to the system of FODEs.We have solved the reduced system of FODEs by using the power series expansion method.Meanwhile, the convergence of power series solution is analyzed.Especially, by using the new conservation theorem, the conservation laws of Equation ( 1) have also been constructed on the basis of the obtained symmetries.Finally, the approximate analytical solution was studied by employing the q-homotopy analysis method under the background of Caputo fractional differential.This method has achieved good results in practical application and could be easily applied to fractional fluid problem such as the Boussinesq-Burgers system and other fractional order nonlinear evaluation problems.