Exact Solutions to the Fractional Differential Equations with Mixed Partial Derivatives

In this paper, the solvability of nonlinear fractional partial differential equations (FPDEs) with mixed partial derivatives is considered. The invariant subspace method is generalized and is then used to derive exact solutions to the nonlinear FPDEs. Some examples are solved to illustrate the effectiveness and applicability of the method.


Introduction
In recent years, fractional order calculus has been one of the most rapidly developing areas of mathematical analysis. In fact, a natural phenomenon may depend not only on the time instant but also on the previous time history, which can be successfully modeled by fractional calculus. Fractional-order differential equations are naturally related to systems with memory, as fractional derivatives are usually nonlocal operators. Thus fractional differential equations (FDEs) play an important role because of their application in various fields of science, such as mathematics, physics, chemistry, optimal control theory, finance, biology, engineering and so on [1][2][3][4][5][6][7].
In 2016, R. Sahadevan and P. Prakash [28] showed how the invariant subspace method could be extended to time fractional partial differential equations (FPDEs) and could construct their exact solutions. ∂ α u ∂t α =F [u], α > 0 where ∂ α ∂t α (·) is a fractional time derivative in the Caputo sense, andF[u] is a nonlinear differential operator of order k.
In 2016, S. Choudhary and V. Daftardar-Gejji [29] developed the invariant subspace method for deriving exact solutions of partial differential equations with fractional space and time derivatives.
In 2017, K.V. Zhukovsky [30] used the inverse differential operational method to obtain solutions for differential equations with mixed derivatives of physical problems.
Motivated by the above results, in this paper, we develop the invariant subspace method for finding exact solutions to some nonlinear partial differential equations with fractional-order mixed partial derivatives (including both fractional space derivatives and time derivatives).
∂x β+i , i = 0, 1, · · · , m 2 , m 2 ∈ N are Caputo time derivatives and Caputo space derivatives, respectively; Using the invariant subspace method, the FPDEs are reduced to the systems of FDEs that can be solved by familiar analytical methods.
The rest of this paper is organized as follows. In Section 2, the preliminaries and notations are given. In Section 3, we develop the invariant subspace method for solving fractional space and time derivative nonlinear partial differential equations with fractional-order mixed derivatives. In Section 4, illustrative examples are given to explain the applicability of the method. Initial value problems are considered. Finally in Section 5, we give conclusions.

Preliminaries and Notation
In this section, we recall some standard definitions and notation. [7]) The Riemann-Liouville fractional integral of order α and function f is defined as [7]) The Caputo fractional derivative of order α and function f is defined as

Definition 1. (See
The Riemann-Liouville fractional integral and the Caputo fractional derivative satisfy the following properties [3]: Definition 3. (See [7]) A two-parametric Mittag-Leffler function is defined as Derivatives of the Mittag-Leffler function are given as The Laplace transform of the αth order Caputo derivative is The Laplace transform of the function t αn+β−1 E (n) α,β (±at α ) is as follows [9]: We let I n be the n-dimensional linear space over R. It is spanned by n linearly independent functions ϕ 0 (x), ϕ 1 (x), · · · , ϕ n−1 (x): We let M be a differential operator; if M[ f ] ∈ I n , ∀ f ∈ I n , then a finite-dimensional linear space I n is invariant with respect to a differential operator M.
Remark: Theorems 1 and 2 in [27] are special cases of our results for µ = 0.

Illustrative Examples
In this section, we give several examples to illustrate Theorems 1 and 2.

Example 1. The fractional diffusion equation is as follows
: where C = constant.
Diffusion is a process in which molecules move around until they are evenly spread out in the area. For α > 1, the phenomenon is referred to as super-diffusion, and for α = 1, it is called normal diffusion, whereas α < 1 describes subdiffusion.
We consider two cases of Equation ( The subspace It follows from Theorem 1 applied to Equation (19) that Equation (19) has the exact solution that follows: where k 0 (t) and k 1 (t) satisfy the system of FDEs as follows: Solving the above FDE (22), we obtain Substituting Equation (23) into Equation (21), we obtain Then k 0 (t) = a + bCΓ(β + 1) Substituting Equations (23) and (25) into Equation (19), we obtain Equation (19) with the solution as follows: where a and b are arbitrary constants.
It is clearly verified that the subspace We let Equation (19) have the exact solution that follows: where k 0 (t), k 1 (t) and k 2 (t) satisfy the system of FDEs as follows: Equation (30) implies that k 2 (t) = a 2 . Thus Equation (29) takes the form which has the following solution: Similarly, Equation (28) yields Thus, Equation (19) has the following solution: where a 0 , a 1 and a 2 are arbitrary constants. It can be easily verified that I 2 = L{1, E β (x β )} is also an invariant subspace with respect to We consider the exact solution of the form where k 0 (t) and k 1 (t) satisfy the following system of FDEs: Clearly, k 0 (t) = a. Solving Equation (33) with the Laplace transform method, we obtain the following: If µC = 1, Thus Equation (19) has the exact solution that follows: where a and b are arbitrary constants. We find that Equations (26), (31) and (34) are distinct particular solutions of Equation (19) under distinct invariant subspaces. Subspace I n+1 = L{1, x β , x 2β , · · · , x nβ }, n ∈ N is invariant under Thus we obtain infinitely many invariant subspaces for Equation (19), which in turn yield infinitely many particular solutions.
Clearly, subspace We look for the exact solution that follows: where k 0 (t), k 1 (t) and k 2 (t) are unknown functions to be determined; k 0 (t), k 1 (t) and k 2 (t) satisfy the system of FDEs as follows: Solving Equations (35)-(37), we obtain Then, we obatin the exact solution of Equation (19) as where a 1 , a 2 , b 1 , b 2 , d 1 and d 2 are arbitrary constants.
When α and β are other numbers, we can similarly obtain the exact solution of Equation (19). Next, we find the closed-form solutions of FPDEs satisfying initial conditions using the invariant subspace method.

Example 2.
We have the following FPDE with the initial condition as follows: The subspace We consider the exact solution that follows: where k 0 (t) and k 1 (t) are unknown functions to be determined. By substituting Equation (40) into Equation (38) and equating coefficients of different powers of x, we obtain the following system of FDEs: We obtain k 0 (t) = a, and Equation (42) takes the following form: Then using the Laplace transform technique, we obtain Using the inverse Laplace transform, we obtain which leads to the exact solution of Equation (38) that follows: where a and b are arbitrary constants. Thus the exact solution of Equation (38) along with the initial condition of Equation (39) is The fractional wave equation is used as an example to model the propagation of diffusive waves in viscoelastic solids. We considered the fractional wave equation with a constant absorption term as follows: Clearly, the subspace By an application of Theorem 2, we know that Equation (43) has the exact solution as follows: where k 0 (t) and k 1 (t) satisfy the system of FDEs as follows: Solving Equations (45) and (46) we obtain the following: Case 1: when 0 < α ≤ 1 2 : Thus Equation (43) has the exact solution that follows: where a 1 and b 1 are arbitrary constants. By the initial conditions of Equation (44), we obtain a 1 = e and b 1 = 1 Γ(β+1) . Hence the exact solution of Equations (40) and (41) is Thus Equation (43) has the exact solution that follows: where a 1 , a 2 , b 1 and b 2 are arbitrary constants.
Thus the exact solution of Equations (43) and (44) is Solving Equation (53), we obtain k 2 (t) = c. Hence Equation (52) has the form We obtain Similarly, we obtain Thus the exact solution of Equation (50) is where a, b and c are arbitrary constants.

Example 5. The fractional version of the nonlinear heat equation is as follows
: Clearly, the subspace It follows from Theorem 1 that we consider the exact solution of Equation (54) as follows: Solving Equations (55) and (56), we obtain is not linearly dependent on x, Equations (57) and (58) do not have the form of the solution given by Equation (61).
Clearly, the subspace We suppose the exact solution that follows: where k 0 (t) and k 1 (t) are unknown functions to be determined; k 0 (t) and k 1 (t) satisfy the system of differential equations as follows: Case 1: when β > 0: Thus Equation (62) has the exact solution that follows: where a 1 , a 2 , b 1 and b 2 are arbitrary constants.
Case 2: when β = 0: Thus Equation (62) has the exact solution that follows: where a 1 , a 2 , b 1 and b 2 are arbitrary constants. Case 3: when β < 0: Thus Equation (62) has the exact solution that follows: where a 1 , a 2 , b 1 and b 2 are arbitrary constants.

Conclusions
The present article develops the invariant subspace method for solving certain fractional space and time derivative nonlinear partial differential equations with fractional-order mixed partial derivatives. Using the invariant subspace method, FPDEs are reduced to systems of FDEs; then they are solved by known analytic methods. In general, FPDEs admit more than one invariant subspace, each of which that has the exact solution. In fact, FPDEs admit infinitely many invariant subspaces. The invariant subspace method is used to derive closed-form solutions of fractional space and time derivative nonlinear partial differential equations with fractional-order mixed partial derivatives along with certain kinds of initial conditions. Thus, the invariant subspace method represents an effective and powerful tool for exact solutions of a wide class of linear/nonlinear FPDEs.
The bases of invariant subspaces usually are orthogonal polynomials, Mittag-Leffler functions, trigonometric functions, and so on. What kinds of spaces are the invariant subspaces of one FPDE? At present we can only try one by one. Although we have found some invariant subspaces of the equations examples above, are there any more invariant subspaces of the equations? We hope to find a simple discriminant method for finding the correct invariant subspaces for FPDEs.