Symmetry Groups, Similarity Reductions, and Conservation Laws of the Time-Fractional Fujimoto–Watanabe Equation Using Lie Symmetry Analysis Method

In this paper, the time-fractional Fujimoto–Watanabe equation is investigated using the Riemann–Liouville fractional derivative. Symmetry groups and similarity reductions are obtained by virtue of the Lie symmetry analysis approach. Meanwhile, the timefractional Fujimoto–Watanabe equation is transformed into three kinds of reduced equations and the third of which is based on Erdélyi–Kober fractional integro-differential operators. Furthermore, the conservation laws are also acquired by Ibragimov’s theory.

e Lie symmetry method also provides a way to seek solutions for NPDEs and NFDEs [29][30][31][32]. is is a way of using known (old) solutions to find new ones. If we obtain a solution for NPDEs or NFDEs, then by using group transformations, the new solutions can be derived. It means that if a NPDE or NFDE has a solution, then it will actually have infinitely many solutions. is method is so effective. Researchers obtain analytical solutions and conservation laws to equations with the help of this method, for example, seventh-order time-fractional Sawada-Kotera-Ito equation, time-fractional fifth-order modified Sawada-Kotera equation, Burridge-Knopoff equation, and so on [33][34][35][36][37]. e time-fractional Fujimoto-Watanabe equation [38] is where D c t u is the Riemann-Liouville fractional derivative of u � u(x, t) with respect to time variable t.
e Fujimoto-Watanabe equation is one important equation and applied in some fields [38]. Its analytical solutions are obtained, and these solutions can reveal many different natural phenomena [39,40]. For instance, its traveling wave solutions describe the propagation status of water waves in mathematical physics and oceanography. In geography, specialists can predict natural disasters with the help of its solutions. In fluid mechanics, researchers acquire its period solutions and study its dynamical behaviors [41].
is paper is organized as follows. In Section 2, we introduce basic concepts and properties about the Riemann-Liouville fractional derivative. In Section 3, symmetry groups are obtained with the help of the Lie symmetry analysis approach. In Section 4, similarity reductions are derived and the time-fractional Fujimoto-Watanabe equation is transformed into three kinds of reduced equations. In Section 5, based on Ibragimov's theory, the conservation laws of the time-fractional Fujimoto-Watanabe equation are constructed. In Section 6, some conclusions are given.

Basic Concept and Properties of the Riemann-Liouville Fractional Derivative
Definition 1 (Riemann-Liouville fractional derivative) (see [42]). Assuming x is the space variable and t is the time variable, then the Riemann-Liouville fractional derivative of f of order c(c > 0) is defined as follows: where V is sometimes called vector field. e third-order prolongation of V is where In order to satisfy the invariance condition of Lie symmetry, pr (3) V needs to meet the following identity: By direct calculation, we obtain From equation (7), we have Definition 2 (Generalized Leibniz rule of the Riemann-Liouville fractional derivative). Assuming u � u(t) and v � v(t) are real-valued functions, then the generalized Leibniz rule is where c Definition 3 (Generalized composite (chain) rule). Assuming u � u(t) and v � v(t) are real-valued functions, then the generalized composite rule is Because of equation (11), equation (6) can be rewritten as According to equations (11) and (13), we obtain where (14) can be rewritten as

According to equations (3)b and (3)c, equation
Substituting equations (14) and (21) into equation (13) and equating the coefficients of all powers of partial Complexity 3 derivatives of u to 0, we obtain a set of determining equations as follows: Solving equation (22), we acquire the solution as follows: where C 1 and C 2 are arbitrary constants. Based on the above results, the infinitesimal generator can be rewritten as If we let then V can also be rewritten as Introducing Lie bracket operation, i.e., for arbitrary vector fields A and B, [A From Table 1, we can find V 1 and V 2 are closed obviously. Consequently, the symmetry groups of the timefractional Fujimoto-Watanabe equation can be spanned by V 1 , V 2 .

Similarity Reductions for the Time-Fractional Fujimoto-Watanabe Equation
In this section, we investigate the similarity reductions for the time-fractional Fujimoto-Watanabe equation. us, we can obtain reduced equations. Because the symmetry groups are spanned by V 1 and V 2 , we need to discuss in two cases: Case 1: for V � (z/zx), we need to solve the following system of equations: From equation (27), we arrive at t and u are similarity variables. We can assume the solution of equation (1) has the form u � f(t).
Substituting u � f(t) into equation (1), we have the following reduced equation: Solving equation (28), we obtain the group invariant solutions: where l is an arbitrary constant. Case 2 (method 1): for V � cx(z/zx) − 3t(z/zt)+ 2cu(z/zu), similar to Case 1, we also need to solve the following system of equations: From equation (30), we arrive at similarity variables as follows: We can assume the solution of equation (1) has the form Substituting equation (32) into equation (1), we have the following reduced equation: where η � tx 3/c .
For Case 2, we have another method to obtain the reduced equation. We need to use Erdélyi-Kober fractional integro-differential operators. Case 2 (method 2): for V � cx(z/zx) − 3t(z/zt) + 2cu(z/zu), we also have the following similarity variables: en, the solution of equation (1) has the form Assuming n − 1 < c < n and substituting equation (35) into Definition 1, we have Introducing variable transformation, us, Substituting equations (37) and (38) into equation (36), we derive where the definition of the Erdélyi-Kober fractional integral operator is as follows: Repeating the same procedure n − 1 times, we derive

Complexity 5
where the definition of the Erdelyi-Kober fractional differential operator is as follows: Consequently, the time-fractional Fujimoto-Watanabe equation is transformed into the following fractional ordinary differential equation: where η � xt (c/3) .