Abstract
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 time-fractional 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.
1. Introduction
Nowadays, nonlinear partial differential equations (NPDEs) have become more and more significant in fluid mechanics, mathematical physics, oceanography, and so on [1–4]. As we know, NPDEs are usually of integer order and researchers have proposed abundant methods to obtain solutions of NPDEs, including inverse scattering transformation [5], Riemann–Hilbert method [6–10], Hirota direct method [11, 12], Darboux transformation [13, 14], Bäcklund transformation [15], Frobenius integrable decompositions [16, 17], and so on [18–24]. As a generalization, the notions of fractional derivatives are put forward and the classical are Riemann–Liouville and Caputo fractional derivatives. Nonlinear fractional differential equations (NFDEs) are also introduced and have a large number of applications in mathematical physics and automation. The exploration of solutions for NFDEs is a crucial aspect. A variety of methods are presented, for instance, the first integral method [25], functional variable method [26], auxiliary equation method [27], and exponential function method [28].
The Lie symmetry method also provides a way to seek solutions for NPDEs and NFDEs [29–32]. This 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. This 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 [33–37].
The time-fractional Fujimoto–Watanabe equation [38] iswhere is the Riemann–Liouville fractional derivative of with respect to time variable .
The 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].
This 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.
2. Basic Concept and Properties of the Riemann–Liouville Fractional Derivative
Definition 1. (Riemann–Liouville fractional derivative) (see [42]). Assuming is a real-valued function, where is the space variable and is the time variable, then the Riemann–Liouville fractional derivative of of order is defined as follows:where is the gamma function.
The Riemann–Liouville fractional derivative has many properties, for instance,where are real-valued functions, , and
From Definition 1, we find that the Riemann–Liouville fractional derivative is a generalized form of the ordinary integer-order derivative. Property (b) is the composite rule of the Riemann–Liouville fractional derivative. Property (c) is the Leibniz rule of the Riemann–Liouville fractional derivative.
3. Symmetry Group of the Time-Fractional Fujimoto–Watanabe Equation
In this section, we seek symmetry groups for the time fractional Fujimoto–Watanabe equation with the help of the Lie symmetry analysis method.
The general time-fractional differential equation is as follows:where , the subscripts represent partial derivatives, i.e., , and the corresponding one-parameter transformations arewhere is an infinitesimal parameter and , , and are real-valued infinitesimal functions with respect to variables , and . , , , and are extended infinitesimal functions. These extended infinitesimal functions can be determined using the following expressions:where represents the total Riemann–Liouville fractional derivative with respect to , represents the total derivative with respect to , and represents the total derivative with respect to , i.e.,
In order to obtain the symmetry groups, let the infinitesimal generator be as follows:where is sometimes called vector field.
The third-order prolongation of iswhere .
Assume
In order to satisfy the invariance condition of Lie symmetry, 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 and are real-valued functions, then the generalized Leibniz rule iswhere
Definition 3. (Generalized composite (chain) rule). Assuming and are real-valued functions, then the generalized composite rule isBecause of equation (11), equation (6) can be rewritten asAccording to equations (11) and (13), we obtainwhereAccording to equations (3)b and (3)c, equation (14) can be rewritten asSubstituting equations (14) and (21) into equation (13) and equating the coefficients of all powers of partial derivatives of to 0, we obtain a set of determining equations as follows:Solving equation (22), we acquire the solution as follows:where are arbitrary constants.
Based on the above results, the infinitesimal generator can be rewritten asIf we letthen can also be rewritten asIntroducing Lie bracket operation, i.e., for arbitrary vector fields and , .
From Table 1, we can find are closed obviously. Consequently, the symmetry groups of the time-fractional Fujimoto–Watanabe equation can be spanned by .
4. Similarity Reductions for the Time-Fractional Fujimoto–Watanabe Equation
In this section, we investigate the similarity reductions for the time-fractional Fujimoto–Watanabe equation. Thus, we can obtain reduced equations.
Because the symmetry groups are spanned by and , we need to discuss in two cases: Case 1: for , we need to solve the following system of equations: From equation (27), we arrive at and are similarity variables. We can assume the solution of equation (1) has the form . Substituting into equation (1), we have the following reduced equation: Solving equation (28), we obtain the group invariant solutions: where is an arbitrary constant. Case 2 (method 1): for , 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 . 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 , we also have the following similarity variables:
Then, the solution of equation (1) has the form
Assuming and substituting equation (35) into Definition 1, we have
Introducing variable transformation,
Thus,
Substituting equations (37) and (38) into equation (36), we derivewhere the definition of the Erdélyi–Kober fractional integral operator is as follows:
Repeating the same procedure times, we derivewhere 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 .
5. Conservation Laws of the Time-Fractional Fujimoto–Watanabe Equation
Conservation laws have always been one significant aspect of the investigation on NPDEs and NFDEs. In this section, we construct conservation laws for the time-fractional Fujimoto–Watanabe equation. In this process, Ibragimov’s theory plays a key role [43].
Assume
Introduce a new dependent variable and a formal Lagrangian:
We can present action integral of equation (45) in the following form:
The corresponding adjoint equation of equation (44) is obtained as follows:where
is the adjoint operator of , and it is defined as follows:where
In order to construct conservation laws, we need to find conservation vector which satisfies the following conservation law equation:where and are the total derivatives with respect to and .
According to Ibragimov’s theory, conservation vectors can be acquired as follows:where and s.t. infinitesimal generator , and operator is defined by Case 1: infinitesimal generator , , , , and . Then, we have Case 2: infinitesimal generator , , , , and . Then, we have
6. Conclusions
In this investigation, we explore the time-fractional Fujimoto–Watanabe equation in the perspective of the Riemann–Liouville derivative. We acquire symmetry groups and similarity reductions by means of the Lie symmetry approach. Meanwhile, three kinds of reduced equations have been obtained. For the second infinitesimal generator, we use two distinct methods to derive two different reduced equations, one of which is based on fractional integral operators and fractional differential operators. At last, with the help of Ibragimov’s theory, the conservation laws are constructed. These results reveal this approach is very effective to obtain reduced equations for the fractional differential equations.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant nos. 11975143 and 61602188), Shandong Provincial Natural Science Foundation (Grant no. ZR2019QD018), and Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents (Grant nos. 2017RCJJ068 and 2017RCJJ069).