The invariant subspace method for solving nonlinear fractional partial differential equations with generalized fractional derivatives

In this paper, we show that the invariant subspace method can be successfully utilized to get exact solutions for nonlinear fractional partial differential equations with generalized fractional derivatives. Using the invariant subspace method, some exact solutions have been obtained for the time fractional Hunter–Saxton equation, a time fractional nonlinear diffusion equation, a time fractional thin-film equation, the fractional Whitman–Broer–Kaup-type equation, and a system of time fractional diffusion equations.


Introduction
Fractional calculus has several applications in science and engineering [1,2]. It is extensively used in modeling physical and engineering phenomena in the form of fractional partial differential equations [3][4][5][6]. Many definitions of the fractional derivative have been introduced in the literature, such as the Riemann-Liouville definition [2], the Caputo definition [2], the Riesz definition [2], the the Caputo-Fabrizio definition [7], and Atangana-Baleanu definition [8]. In recent years, a novel fractional derivative has appeared in the literature called the generalized fractional derivative [9,10] which generalizes the Riemann-Liouville fractional derivative. This generalized fractional derivative has attracted the interest of many researchers. Many properties and applications of this generalized fractional derivative can be found in [9][10][11][12][13][14][15][16]. Some basic properties of the generalized fractional derivative are given in the Appendix.
The invariant subspace method (ISM) is a very effective method that can be used for obtaining exact solutions of fractional partial differential equations. It is widely used in getting exact solutions of fractional differential equations with Riemann-Liouville and Caputo fractional derivatives [17][18][19][20]. It is also successfully utilized for getting exact solutions of fractional partial differential equations with conformable derivatives [21]. In this paper, we adapt the ISM to be utilized for obtaining exact solutions for some fractional partial differential equations with the generalized fractional Riemann-Liouville derivative. In the next section, we will introduce the ISM.

The invariant subspace method
The ISM can be used for solving the following fractional system of PDEs: The ISM can be summarized in the following steps: Step 1. Assume the solution of Eq. (1) in the form where j and k depend upon the dimension of the invariant subspace.
Step 2. Determine the functions B i (x), D i (x) as follows: • Solve the system of determining equations to obtain the coefficients c 0 (x), . . . , c j-1 (x) and r 0 (x), . . . , r k-1 (x); • Solve the system of ordinary differential equations to obtain the solution where h i , i = 1, . . . , j, s i , i = 1, . . . , k are arbitrary constants.
Step 3. Substitute Eq. (2) into Eq. (1) to obtain a system of fractional ordinary differential equations in A i (t) and C i (t).
In the following section, we solve some fractional differential equations using ISM.
The exact solution of the system (44) can't be obtained, in general. A special solution of the system (44) when μ = -2ρ 1 b 2 13 is given by In this case, the solution of Eq. (39) is given by where, b 13 , b 14 , b 15 , b 16 are constants.

Conclusions
In this paper, we have utilized the ISM for getting exact solutions for some nonlinear fractional partial differential equations with generalized fractional derivatives. The obtained solutions in this paper are given in generalized forms which depend upon the parameter ρ. We can retrieve the obtained solutions in [19,20,22] by putting ρ = 1 in our obtained solutions. The ISM is a very powerful method that can be used to solve various fractional Using the relation Γ (γ + 1) = γ Γ (γ ), we obtain Remark In [10], the generalized fractional derivative of f (t) = t v is obtained as (see Eq. (5.7)) as So we can see that there is a misprint in this relation. The correct relation is given by Eq. (A1).