4.1 Fractional sub-equation method
Now, we outline the main steps of the fractional sub-equation method for solving fractional differential equations.
For a given NFDE, consider two variables x and t,
(41)
where and are the modified Riemann-Liouville derivatives of u with respect to t and x, respectively.
Step 1: By making use of the traveling wave transformation
where c is a nonzero constant to be determined later, (41) can be reduced to a nonlinear fractional ordinary differential equation (NFODE)
(43)
Step 2: Suppose that Eq. (43) has the following solution:
(44)
where () are constants to be determined later, positive integer n can be determined by balancing the highest order derivatives and nonlinear terms in Eq. (41) or Eq. (43). The function satisfies the following Bäklund transformation of the fractional Riccati equation [38]:
(45)
where B, D are arbitrary parameters, and . Meanwhile, are decided by
(46)
where σ is a constant. Eq. (46) has the following solutions:
(47)
with the generalized hyperbolic and trigonometric functions
(48)
here () is the Mittag-Leffler function in one parameter.
Step 3: Substituting (44), (45) and (46) into (43) and setting the coefficients of the powers of to be zero, one can obtain an over-determined nonlinear algebraic system in () and c.
Step 4: With the aid of Maple, solving the nonlinear algebraic system yields the explicit expressions of the parameters () and c. Then substituting these constants and the solutions of Eq. (47) into Eq. (44), we can get the exact and explicit solutions of the nonlinear fractional partial differential equation (NFPDE) (41).
4.2 Applications to the time fractional KdV equation
According to the above steps, firstly, we introduce the following transformations:
where c is a constant. Substituting (49) into (1), then (1) can be reduced to the following nonlinear fractional ordinary differential equation (NFODE):
(50)
We suppose that Eq. (50) has the following solution:
(51)
where () are constants to be determined later. Balancing the highest order derivative terms with nonlinear terms in Eq. (50), we get
(52)
Substituting (52) along with (45) into (50) and then letting the coefficients of to zero, one can get some algebraic equations about c, , and . Solving the algebraic equations by Maple, one can get the following.
Case 1:
(53)
Case 2:
(54)
Case 3:
(55)
Case 4:
(56)
In view of (54), one can get new types of explicit solutions of Eq. (1) as follows:
(57)
where , ,
(58)
where , ,
(59)
where , ,
(60)
where , .
If , one can get .
Using (55), one can get new types of explicit solutions of Eq. (1) as follows:
(61)
where , ,
(62)
where , ,
(63)
where , ,
(64)
where , ,
(65)
where , .
Remark 2 Using (53) and (56), we can also get other exact solutions of (1). Here we do not list all of them.
Remark 3 To the best of our knowledge, the solutions obtained in this paper have not been reported in previous literature. Therefore, these solutions are new.
Remark 4 It is interesting to note that if , FDEs (1) can be reduced to the conventional integer order KdV equation, and the obtained exact solutions can be reduced to the conventional hyperbolic and trigonometric functions.