axioms Article

Exact Solutions to the Fractional Differential Equations with Mixed Partial Derivatives Jun Jiang 1 , Yuqiang Feng 2, * and Shougui Li 2 1 2

*

School of Science, Wuhan University of Science and Technology, Wuhan 430065, Hubei, China; [email protected] Hubei Province Key Laboratory of Systems Science in Metallurgical Process, Wuhan 430065, Hubei, China; [email protected] Correspondence: [email protected]

Received: 19 December 2017 ; Accepted: 8 February 2018; Published: 11 February 2018

Abstract: 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. Keywords: Caputo fractional derivative; the fractional partial differential equation; Laplace transform method; invariant subspace method MSC: 26D10

1. 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 [1–7]. It is of importance to find efficient methods for solving FDEs. More recently, much attention has been paid to the solutions of FDEs using various methods, such as the Adomian decomposition method (2005) [8], the first integral method (2014) [9], the Lie group theory method (2012, 2015) [10,11], the homotopy analysis method (2016) [12], the inverse differential operational method (2016) [13–15], the F-expansion method (2017) [16], M-Wright transforms (2017) [17], exponential differential operators (2017, 2018) [18,19], and so on. In reality, the finding of exact solutions of the FDEs is hard work and remains a problem. Recently, investigations have shown that a new method based on the invariant subspace provides an effective tool to find the exact solution of FDEs. This method was initially proposed by Galaktionov and Svirshchevskii (1995, 1996, 2007) [20–22]. The invariant subspace method was developed by Later Gazizov and Kasatkin (2013) [23], Harris and Garra (2013, 2014) [24,25], Sahadevan and Bakkyaraj (2015) [26], and Ouhadan and El Kinani (2015) [27]. 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.

Axioms 2018, 7, 10; doi:10.3390/axioms7010010

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∂α u ∂tα

= Fˆ [u], α > 0

∂α ˆ where ∂t α (·) is a fractional time derivative in the Caputo sense, and F [ 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.

( λ0

∂ α +1 ∂ β f ∂ β +1 f ∂ β+n f ∂α ∂α+m + λ ) f ( x, t ) = N ( x, f , , , · · · , ) + · · · + λ m 1 ∂tα ∂tα+m ∂tα+1 ∂x β ∂x β+1 ∂x β+n ∂α+ j f

∂ β +i f

where N [ f ] is the linear/nonlinear differential operator; ∂tα+ j , j = 0, 1, · · · , m and ∂x β+i , i = 0, 1, · · · , n are Caputo time derivatives and Caputo space derivatives, respectively; 0 < α, β ≤ 1 and λi ∈ R. 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).

( λ0

∂ β f ∂ β +1 f ∂α ∂ α +1 ∂ β + m2 f ∂α ∂ β f ∂ α + m1 + λ1 α+1 + · · · + λm α+m ) f ( x, t) = N ( x, f , β , β+1 , · · · , β+m ) + µ α ( β ) α 1 2 ∂t ∂t ∂t ∂x ∂t ∂x ∂x ∂x ∂α+ j f ∂tα+ j

, j = 0, 1, · · · , m1 , m1 ∈ N and ∂ β +i f , i = 0, 1, · · · , m2 , m2 ∈ N are Caputo time derivatives and Caputo space derivatives, respectively; ∂x β+i β ∂α ∂ f ∂tα ( ∂x β ) is the Caputo mixed partial derivative of space and time; k 1 < α ≤ k 1 + 1, k 2 < β ≤ k 2 + 1, k1 , k2 ∈ N and λi , µ ∈ R.

where f = f ( x, t), N [ f ] is a linear/nonlinear differential operator;

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. 2. Preliminaries and Notation In this section, we recall some standard definitions and notation. Definition 1. (See [7]) The Riemann–Liouville fractional integral of order α and function f is defined as I α f (t) =

1 Γ(α)

Z t 0

f (x) dx, t > 0 ( t − x )1− α

Definition 2. (See [7]) The Caputo fractional derivative of order α and function f is defined as dα f (t) = I n−α D n f (t ) = dtα

(

1 Γ(n−α)

f (n) ( x ) 0 (t− x )α−n+1 dx, f ( n ) ( t ),

Rt

n−1 < α < n α = n, n ∈ N

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The Riemann–Liouville fractional integral and the Caputo fractional derivative satisfy the following properties [3]: I α tβ = dα t β = dtα

(

Γ ( β + 1) β + α t , Γ ( β + α + 1)

dαe = n, β ∈ {0, 1, 2, · · · , n − 1}

0, Γ ( β +1) t β − α , Γ ( β − α +1)

Iα(

α > 0, β > −1, t > 0

/ N, andβ > n − 1 dαe = n, β ∈ N, andβ ≥ n; orβ ∈

n −1 k dα f (t) d f (0) t k ) = f (t) − ∑ , α k k! dt k =0 dt

n − 1 < α < n, t > 0

Definition 3. (See [7]) A two-parametric Mittag–Leffler function is defined as ∞

Eα,β (z) =

zk , Γ(kα + β) k =0

∑

α, β ∈ C, R(α), R( β) > 0

noting that Eα,1 (z) = Eα (z). Derivatives of the Mittag–Leffler function are given as (n)

Eα,β (z) =

∞ (k + n)!zk dn Eα,β (z) = ∑ , n dz k!Γ(αk + αn + β) k =0

d γ β −1 (t Eα,β ( atα )) = t β−γ−1 Eα,β−γ ( atα ), dtγ dα ( Eα ( atα )) = aEα ( atα ), dtα

n = 0, 1, 2, · · ·

α, γ > 0, a ∈ R

α > 0, a ∈ R

The Laplace transform of the αth order Caputo derivative is T{

n −1 dα f (t) α ˜ ; s } = s f ( s ) − ∑ s α − k −1 f ( k ) (0 ), dtα k =0

where f˜(s) = T { f (t); s} =

R∞ 0

n − 1 < α < n, n ∈ N, R(s) > 0

e−st f (t)dt,

s∈R

(n)

The Laplace transform of the function tαn+ β−1 Eα,β (± atα ) is as follows [9]: (n)

T {tαn+ β−1 Eα,β (± atα ); s} =

n!sα− β , ( s α ∓ a ) n +1

1

R(s) > | a| α , n = 0, 1, 2, · · ·

We let In be the n-dimensional linear space over R. It is spanned by n linearly independent functions ϕ0 ( x ), ϕ1 ( x ), · · · , ϕn−1 ( x ): n −1

In = L{ ϕ0 ( x ), ϕ1 ( x ), · · · , ϕn−1 ( x )} = { ∑ k i ϕi ( x )|k i ∈ R, i = 0, 1, · · · , n − 1} i =0

We let M be a differential operator; if M[ f ] ∈ In , ∀ f ∈ In , then a finite-dimensional linear space In is invariant with respect to a differential operator M. 3. Invariant Subspace Method; Fractional Partial Differential Equations with Fractional-Order Mixed Partial Derivative The FPDE with fractional-order mixed partial derivative is as follows:

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( λ0

∂ α +1 ∂ α + m1 ∂α ∂ β ∂α + λ 1 α +1 + · · · + λ m1 α + m ) f = N [ f ] + µ α ( β f ) α 1 ∂t ∂t ∂t ∂x ∂t

where f = f ( x, t), N [ f ] = N ( x, f ,

and and

∂ β +1 ∂ β + m2 ∂β f , f , · · · , f) ∂x β ∂x β+1 ∂x β+m2

∂ β +i f , i = 0, 1, · · · , m2 ; m2 ∈ N are Caputo time derivatives ∂x β+i ∂α ∂ β Caputo space derivatives respectively; ∂tα ( ∂x β f ) is the Caputo mixed partial derivative of space time; k1 < α ≤ k1 + 1, k2 < β ≤ k2 + 1 , k1 , k2 ∈ N and λi , µ ∈ R.

Here,

∂α+ j f ∂tα+ j

(1)

, j = 0, 1, · · · , m1 ; m1 ∈ N and

Theorem 1. Suppose In+1 = L{ ϕ0 ( x ), ϕ1 ( x ), · · · , ϕn ( x )} is a finite-dimensional linear space, and it is ∂β invariant with respect to the operators N [ f ] and ∂x β f ; then FPDE (1) has an exact solution as follows: n

f ( x, t) =

∑ k i ( t ) ϕi ( x )

(2)

i =0

where {k i (t)} satisfies the following system of FDEs: m1

∑ λj

j =0

dα ψn+1+i (k0 (t), k1 (t), · · · , k n (t)) dα+ j k i (t ) − µ = ψi (k0 (t), k1 (t), · · · , k n (t)) dtα dtα+ j

(3)

Here i = 0, 1, · · · , n, {ψ0 , ψ1 , · · · , ψn } are the expansion coefficients of N [ f ] with respect to ∂β { ϕ0 ( x ), ϕ1 ( x ), · · · , ϕn ( x )}; {ψn+1 , ψn+2 , · · · , ψ2n+1 } are the expansion coefficients of ∂x β f with respect to { ϕ0 ( x ), ϕ1 ( x ), · · · , ϕn ( x )} . Proof. Using Equation (2) and the linearity of Caputo fractional derivatives, we obtain m1

∑ λj

j =0

∂α+ j f ( x, t) = ∂tα+ j

m1

∑ λj

j =0

m1 n ∂α+ j n dα+ j k i (t ) ( k ( t ) ϕ ( x )) = ( λj ) ϕi ( x ) i i ∑ ∑ ∑ α + j ∂t dtα+ j i =0 i =0 j =0

Further, as In+1 is an invariant space under the operator N [ f ] and ψ0 , ψ1 , · · · , ψn ; ψn+1 , ψn+2 , · · · , ψ2n+1 such that n

N ( ∑ k i (t) ϕi ( x )) = i =0

∂β ∂x β

(4)

f , there exist 2n + 2 functions

n

∑ ψi (k0 (t), k1 (t), · · · , kn (t)) ϕi (x)

(5)

i =0

n ∂β f ( x, t) = ∑ ψn+1+i (k0 (t), k1 (t), · · · , k n (t)) ϕi ( x ) β ∂x i =0

(6)

where {ψ0 , ψ1 , · · · , ψn } are the expansion coefficients of N [ f ] with respect to { ϕ0 (x), ϕ1 (x), · · · , ϕn (x)}; ∂β {ψn+1 , ψn+2 , · · · , ψ2n+1 } are the expansion coefficients of ∂x β f with respect to { ϕ0 ( x ), ϕ1 ( x ), · · · , ϕn ( x )}. In view of Equations (2), (5) and (6), α

β

∂ ∂ N [ f ( x, t)] + µ ∂t f ( x, t)) α( ∂x β n

α

n

= ∑ ψi (k0 (t), k1 (t), · · · , k n (t)) ϕi ( x ) + µ ∂t∂ α ( ∑ ψn+1+i (k0 (t), k1 (t), · · · , k n (t)) ϕi ( x )) i =0 n

i =0 n dα ψ n+1+i ( k 0 ( t ),k 1 ( t ),··· ,k n ( t )) ϕi ( x )) dtα i =0 dα ψ (k (t),k (t),··· ,k n (t)) , k n (t)) ϕi ( x ) + µ n+1+i 0 dtα 1 ϕi ( x )))

= ∑ ψi (k0 (t), k1 (t), · · · , k n (t)) ϕi ( x ) + µ( ∑ i =0 n

= ∑ (ψi (k0 (t), k1 (t), · · · i =0

(7)

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Equations (4) and (7) are substituted in Equation (1) to obtain m1

n

∑ ( ∑ λj

i =0 j =0

dα+ j k i (t) dtα+ j

− ψi (k0 (t), k1 (t), · · · , k n (t)) − µ

dα ψn+1+i (k0 (t),k1 (t),··· ,k n (t)) ) ϕi ( x ) dtα

(8)

=0 Using Equation (8) and the fact that ϕ0 ( x ), ϕ1 ( x ), · · · , ϕn ( x ) are linearly independent, we have the system of FDEs that follows: m1

∑ λj

j =0

dα+ j k i (t ) dα ψn+1+i (k0 (t), k1 (t), · · · , k n (t)) − µ = ψi (k0 (t), k1 (t), · · · , k n (t)) dtα dtα+ j

(9)

where i = 0, 1, · · · , n. If FPDE (1) satisfies the conditions of Theorem 1, then FPDE (1) has a particular solution given by Equation (2). We consider the following FPDE: m α

2α

α

α

β

∂ (λ1 ∂t∂ α + λ2 ∂t∂ 2α + · · · + λm1 ∂t∂ m11 α ) f = N [ f ] + µ ∂t∂ α ( ∂x β f) β

2β

∂ ∂ = N [ x, f , ∂x β f , ∂x2β f , · · · ,

∂ m2 β ∂x m2 β

α

β

∂ ∂ f ] + µ ∂t f) α( ∂x β

where f = f ( x, t), N [ f ] is a linear/nonlinear differential operator; ∂iβ f

∂ jα f ∂t jα

(10)

, j = 1, 2, · · · , m1 ; m1 ∈ N and

, i = 1, 2, · · · , m2 ; m2 ∈ N are Caputo time derivatives and Caputo space derivatives, respectively; ∂xiβ k1 < α ≤ k1 + 1, k2 < β ≤ k2 + 1 , k1 , k2 ∈ N and λi , µ ∈ R. Theorem 2. Suppose In = L{ ϕ1 ( x ), ϕ2 ( x ), · · · , ϕn ( x )} is a finite-dimensional linear space, and it is invariant ∂β with respect to the operator N [ f ] and ∂x β f ; then FPDE (10) has an exact solution as follows: n

f ( x, t) =

∑ k i ( t ) ϕi ( x )

(11)

i =1

where {k i (t)} satisfy the following system of FDEs: m1

∑ λj

j =1

d jα k i (t) dα ψn+i (k1 (t), k2 (t), · · · , k n (t)) −µ = ψi (k1 (t), k2 (t), · · · , k n (t)) jα dtα dt

(12)

Here, i = 1, 2, · · · , n, {ψ1 , ψ2 , · · · , ψn } are the expansion coefficients of N [ f ] with respect to ∂β { ϕ1 ( x ), ϕ2 ( x ), · · · , ϕn ( x )}; {ψn+1 , ψn+2 , · · · , ψ2n } are the expansion coefficients of ∂x β f with respect to { ϕ1 ( x ), ϕ2 ( x ), · · · , ϕn ( x )}. Proof. Using Equation (11) and the linearity of Caputo fractional derivatives, we obtain m1

∑ λj

j =1

∂ jα f ( x, t) = ∂t jα

m1

∂ jα

n

n

m1

∑ λ j ∂t jα ( ∑ ki (t) ϕi (x)) = ∑ ( ∑ λ j

j =1

i =1

i =1 j =1

Further, as In is an invariant space under the operator N [ f ] and ψ1 , ψ2 , · · · , ψn ; ψn+1 , ψn+2 , · · · , ψ2n such that n

N ( ∑ k i (t) ϕi ( x )) = i =1

d jα k i (t) ) ϕi ( x ) dt jα ∂β ∂x β

(13)

f , there exist 2n functions

n

∑ ψi (k1 (t), k2 (t), · · · , kn (t)) ϕi (x)

i =1

(14)

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n ∂β f ( x, t) = ∑ ψn+i (k1 (t), k2 (t), · · · , k n (t)) ϕi ( x ) β ∂x i =1

(15)

where {ψ1 , ψ2 , · · · , ψn } are the expansion coefficients of N [ f ] with respect to { ϕ1 ( x), ϕ2 ( x), · · · , ϕn ( x)}; ∂β {ψn+1 , ψn+2 , · · · , ψ2n } are the expansion coefficients of ∂x β f with respect to { ϕ1 ( x ), ϕ2 ( x ), · · · , ϕn ( x )}. In view of Equations (11), (14) and (15), α

β

∂ ∂ f ( x, t)) N [ f ( x, t)] + µ ∂t α( ∂x β n

= ∑ (ψi (k1 (t), k2 (t), · · · , k n (t)) ϕi ( x ) + µ i =1

dα ψn+i (k1 (t),k2 (t),··· ,k n (t)) ϕi ( x )) dtα

(16)

Equations (13) and (16) are substituted into Equation (10), to obtain n

m1

∑ ( ∑ λj

i =1 j =1

d jα k i (t) dt jα

− ψi (k1 (t), k2 (t), · · · , k n (t)) − µ

dα ψn+i (k1 (t),k2 (t),··· ,k n (t)) ) ϕi ( x ) dtα

(17)

=0 Using Equation (17) and the fact that ϕ1 ( x ), ϕ2 ( x ), · · · , ϕn ( x ) are linearly independent, we have the system of FDEs as follows: m1

∑ λj

j =1

d jα k i (t) dα ψn+i (k1 (t), k2 (t), · · · , k n (t)) = ψi (k1 (t), k2 (t), · · · , k n (t)) − µ dtα dt jα

(18)

here i = 0, 1, · · · , n. Remark: Theorems 1 and 2 in [27] are special cases of our results for µ = 0. 4. Illustrative Examples In this section, we give several examples to illustrate Theorems 1 and 2. Example 1. The fractional diffusion equation is as follows: ∂β f ∂α f ∂α ∂ β = C β + µ α( β f) α ∂t ∂t ∂x ∂x

(19)

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 (19): case 1: α ∈ (0, 1], β ∈ (1, 2]; case 2: α ∈ (1, 2], β ∈ (1, 2]. Case 1: α ∈ (0, 1], β ∈ (1, 2]. ∂β f

The subspace I2 = L{1, x β } is invariant under N [ f ] = C ∂x β and

∂β ∂x β

f as

N [k0 + k1 x β ] = Ck1 Γ( β + 1) ∈ I2 ∂ β (k0 + k1 x β ) = k1 Γ( β + 1) ∈ I2 ∂x β It follows from Theorem 1 applied to Equation (19) that Equation (19) has the exact solution that follows: f ( x, t) = k0 (t) + k1 (t) x β (20)

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where k0 (t) and k1 (t) satisfy the system of FDEs as follows: dα k0 (t) dα k1 (t) − µCΓ( β + 1) = k1 (t)CΓ( β + 1) dtα dtα

(21)

dα k1 (t) =0 dtα

(22)

k1 (t) = b

(23)

Solving the above FDE (22), we obtain

Substituting Equation (23) into Equation (21), we obtain dα k0 (t) = bCΓ( β + 1) dtα Then k0 (t) = a +

(24)

bCΓ( β + 1) α t Γ ( α + 1)

(25)

Substituting Equations (23) and (25) into Equation (19), we obtain Equation (19) with the solution as follows: bCΓ( β + 1) α f ( x, t) = a + t + bx β (26) Γ ( α + 1) where a and b are arbitrary constants. ∂β f

It is clearly verified that the subspace I3 = L{1, x β , x2β } is invariant under N [ f ] = C ∂x β and ∂β ∂x β

f as N [k0 + k1 x β + k2 x2β ] = Ck1 Γ( β + 1) + Ck2

Γ(2β + 1) β x ∈ I3 Γ ( β + 1)

Γ(2β + 1) β ∂ β (k0 + k1 x β + k2 x2β ) = k 1 Γ ( β + 1) + k 2 x ∈ I3 β Γ ( β + 1) ∂x We let Equation (19) have the exact solution that follows: f ( x, t) = k0 (t) + k1 (t) x β + k2 (t) x2β

(27)

where k0 (t), k1 (t) and k2 (t) satisfy the system of FDEs as follows: dα k0 (t) dα k1 (t) − µ Γ( β + 1) = CΓ( β + 1)k1 (t) dtα dtα

(28)

dα k1 (t) dα k2 (t) Γ(2β + 1) CΓ(2β + 1) − µ = k2 (t) α α dt dt Γ ( β + 1) Γ ( β + 1)

(29)

dα k2 (t) =0 dtα

(30)

Equation (30) implies that k2 (t) = a2 . Thus Equation (29) takes the form a CΓ(2β + 1) dα k1 (t) = 2 dtα Γ ( β + 1)

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which has the following solution: k 1 ( t ) = a1 +

a2 CΓ(2β + 1) α t Γ ( β + 1)

Similarly, Equation (28) yields k 0 ( t ) = a0 +

a1 CΓ( β + 1)2 + a2 µCΓ(2β + 1) α a2 C2 Γ(2β + 1) 2α t + t Γ ( α + 1) Γ ( β + 1) Γ(2α + 1)

Thus, Equation (19) has the following solution: f ( x, t) =

( a0 +

a1 CΓ( β+1)2 + a2 µCΓ(2β+1) α t Γ ( α +1) Γ ( β +1)

+( a1 +

a2 CΓ(2β+1) α β t )x Γ ( β +1)

+

a2 C2 Γ(2β+1) 2α t ) Γ(2α+1)

(31)

+ a2 x2β

where a0 , a1 and a2 are arbitrary constants. It can be easily verified that I2 = L{1, Eβ ( x β )} is also an invariant subspace with respect to ∂β f

N [ f ] = C ∂x β and

∂β ∂x β

f , as

N [k0 + k1 Eβ ( x β )] = C

∂β (k0 + k1 Eβ ( x β )) = Ck1 Eβ ( x β ) ∈ I2 ∂x β

∂ β (k0 + k1 Eβ ( x β )) ∂x β

= k1 Eβ ( x β ) ∈ I2

We consider the exact solution of the form f ( x, t) = k0 (t) + k1 (t) Eβ ( x β ) where k0 (t) and k1 (t) satisfy the following system of FDEs: dα k0 (t) =0 dtα

(32)

(1 − µC )dα k1 (t) = Ck1 (t) dtα

(33)

Clearly, k0 (t) = a. Solving Equation (33) with the Laplace transform method, we obtain the following: If µC 6= 1, C k1 (t) = bEα ( tα ) 1 − µC Thus Equation (19) has the exact solution that follows: f ( x, t) = a + bEα (

C t α ) Eβ ( x β ) 1 − µC

(34)

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 In+1 = L{1, x β , x2β , · · · , x nβ }, n ∈ N is invariant under ∂β f

N [ f ] = C ∂x β and

∂β ∂x β

f , as

N [k0 + k1 x β + · · · + k n x nβ ]

= Ck1 Γ( β + 1) + ∈ In+1

Ck2 Γ(2β+1) β x Γ ( β +1)

+···+

Ck n Γ(nβ+1) (n−1) β x Γ((n−1) β+1)

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Thus we obtain infinitely many invariant subspaces for Equation (19), which in turn yield infinitely many particular solutions. Case 2: α ∈ (1, 2], β ∈ (1, 2]. ∂β f

Clearly, subspace I3 = L{1, x β , x2β } is an invariant subspace under N [ f ] = C ∂x β and N [k0 + k1 x β + k2 x2β ] = Ck1 Γ( β + 1) + Ck2

∂β ∂x β

f , as

Γ(2β + 1) β x ∈ I3 Γ ( β + 1)

∂ β (k0 + k1 x β + k2 x2β ) Γ(2β + 1) β x ∈ I3 = k 1 Γ ( β + 1) + k 2 β Γ ( β + 1) ∂x We look for the exact solution that follows: f ( x, t) = k0 (t) + k1 (t) x β + k2 (t) x2β where k0 (t), k1 (t) and k2 (t) are unknown functions to be determined; k0 (t), k1 (t) and k2 (t) satisfy the system of FDEs as follows: dα k1 (t) dα k0 (t) −µ Γ( β + 1) = CΓ( β + 1)k1 (t) α dt dtα

(35)

dα k1 (t) dα k2 (t) Γ(2β + 1) CΓ(2β + 1) − µ = k2 (t) dtα dtα Γ( β + 1) Γ ( β + 1)

(36)

dα k2 (t) =0 dtα

(37)

Solving Equations (35)–(37), we obtain k 2 ( t ) = d1 + d2 t k1 (t) = b1 + b2 t +

Cd1 Γ(2β + 1) α Cd2 Γ(2β + 1) α+1 t + t Γ ( β + 1) Γ ( α + 1) Γ ( β + 1) Γ ( α + 2)

Cb1 Γ( β+1)+µCd2 Γ(2β+1) α t Γ ( α +1) C2 d2 Γ(2β+1) 2α+1 t Γ(2α+2)

k 0 ( t ) = a1 + a2 t +

+

+

Cb2 Γ( β+1)+µCd2 Γ(2β+1) α+1 t Γ ( α +2)

+

C2 d1 Γ(2β+1) 2α t Γ(2α+1)

Then, we obatin the exact solution of Equation (19) as f ( x, t) = ( a1 + a2 t +

+

Cb1 Γ( β+1)+µCd2 Γ(2β+1) α t Γ ( α +1)

C2 d2 Γ(2β+1) 2α+1 t ) + (b1 Γ(2α+2)

+ b2 t +

+

Cb2 Γ( β+1)+µCd2 Γ(2β+1) α+1 t Γ ( α +2)

Cd1 Γ(2β+1) α t Γ ( β +1) Γ ( α +1)

+

Cd2 Γ(2β+1) α+1 β t )x Γ ( β +1) Γ ( α +2)

+

C2 d1 Γ(2β+1) 2α t Γ(2α+1)

+ (d1 + d2 t) x2β

where a1 , a2 , b1 , b2 , d1 and d2 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: ∂α f ∂β f ∂β f ∂α ∂ β = ( β )2 − f ( β ) + µ α ( β f ), α, β ∈ (0, 1] α ∂t ∂t ∂x ∂x ∂x

(38)

5 f ( x, 0) = 3 + Eβ ( x β ) 2

(39)

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∂β f

∂β f

The subspace I2 = L{1, Eβ ( x β )} is invariant under N [ f ] = ( ∂x β )2 − f ( ∂x β ) and

∂β ∂x β

f , as

N [k0 + k1 Eβ ( x β )] = (k1 Eβ ( x β ))2 − (k0 + k1 Eβ ( x β ))k1 Eβ ( x β ) = −k0 k1 Eβ ( x β ) ∈ I2 ∂ β (k0 + k1 Eβ ( x β )) ∂x β

= k1 Eβ ( x β ) ∈ I2

We consider the exact solution that follows: f ( x, t) = k0 (t) + k1 (t) Eβ ( x β )

(40)

where k0 (t) and k1 (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: dα k0 (t) =0 dtα

(1 − µ )

(41)

dα k1 (t) = −k0 (t)k1 (t) dtα

(42)

We obtain k0 (t) = a, and Equation (42) takes the following form: If µ 6= 1, dα k1 (t) a = k (t) α dt µ−1 1 Then using the Laplace transform technique, we obtain sα k˜ 1 (s) − sα−1 k1 (0) = k˜ 1 (s) =

sα

a ˜ k (s) µ−1 1

s α −1 k (0) − µ−a 1 1

Using the inverse Laplace transform, we obtain k1 (t) = k1 (0) Eα (

a α t ) µ−1

which leads to the exact solution of Equation (38) that follows: f ( x, t) = a + bEα (

a α t ) Eβ ( x β ) µ−1

where a and b are arbitrary constants. Thus the exact solution of Equation (38) along with the initial condition of Equation (39) is 5 3 α f ( x, t) = 3 + Eα ( t ) Eβ ( x β ) 2 µ−1 Example 3. 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: ∂β ∂2α f ∂β f ∂α ∂ β = ( f ) − 1 + µ ( f ), α, β ∈ (0, 1] ∂tα ∂x β ∂t2α ∂x β ∂x β f ( x, 0) = e +

xβ , f t ( x, 0) = 1 − x β Γ ( β + 1)

(43) (44)

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Clearly, the subspace I2 = L{1, x β } is invariant under N [ f ] = N [k0 + k1 x β ] =

β ∂β ( f ∂∂x βf ) − 1 ∂x β

and

∂β ∂x β

f , as

∂β ((k0 + k1 x β )k1 Γ( β + 1)) − 1 = k21 Γ2 ( β + 1) − 1 ∈ I2 ∂x β ∂ β (k0 + k1 x β ) = k1 Γ( β + 1) ∈ I2 ∂x β

By an application of Theorem 2, we know that Equation (43) has the exact solution as follows: f ( x, t) = k0 (t) + k1 (t) x β where k0 (t) and k1 (t) satisfy the system of FDEs as follows: dα k1 (t) d2α k0 (t) − µ Γ( β + 1) = k21 (t)Γ2 ( β + 1) − 1 dtα dt2α

(45)

d2α k1 (t) =0 dt2α

(46)

Solving Equations (45) and (46) we obtain the following: Case 1: when 0 < α ≤ 12 : k1 (t) = b1 k 0 ( t ) = a1 +

b12 Γ2 ( β + 1) − 1 2α t Γ(2α + 1)

Thus Equation (43) has the exact solution that follows: f ( x, t) = ( a1 +

b12 Γ2 ( β + 1) − 1 2α t ) + b1 x β Γ(2α + 1)

where a1 and b1 are arbitrary constants. By the initial conditions of Equation (44), we obtain a1 = e and b1 = Hence the exact solution of Equations (40) and (41) is f ( x, t) = e + Case 2: when

1 2

1 . Γ ( β +1)

xβ Γ ( β + 1)

< α ≤ 1: k1 (t) = b1 + b2 t

k0 (t) = a1 + a2 t + µb2

Γ( β + 1) α+1 b12 Γ2 ( β + 1) − 1 2α 2b1 b2 Γ2 ( β + 1) 2α+1 2b22 Γ2 ( β + 1) 2α+2 t + t + t + t Γ ( α + 2) Γ(2α + 1) Γ(2α + 2) Γ(2α + 3)

Thus Equation (43) has the exact solution that follows: Γ ( β +1)

f ( x, t) = ( a1 + a2 t + µb2 Γ(α+2) tα+1 +

b12 Γ2 ( β+1)−1 2α t Γ(2α+1)

+

2b1 b2 Γ2 ( β+1) 2α+1 t Γ(2α+2)

+

2b22 Γ2 ( β+1) 2α+2 t ) Γ(2α+3)

+ (b1 + b2 t) x β where a1 , a2 , b1 and b2 are arbitrary constants. Substituting the initial conditions of Equation (44), we obtain a1 = e, a2 = 1, b1 = b2 = −1. Thus the exact solution of Equations (43) and (44) is f ( x, t) = (e + t − µ

1 Γ ( β +1)

Γ( β + 1) α+1 2Γ( β + 1) 2α+1 2Γ2 ( β + 1) 2α+2 1 t − t + t )+( − t) x β Γ ( α + 2) Γ(2α + 2) Γ(2α + 3) Γ ( β + 1)

and

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We consider the following fractional generalization of the wave equation with a constant absorption term: ∂ α +1 f ∂β ∂β f ∂α ∂ β = β ( f β ) − 1 + µ α ( β f ), α, β ∈ (0, 1] (47) α + 1 ∂t ∂x ∂t ∂x ∂x We know that the subspace I2 = L{1, x β } is invariant from the above. In view of Theorem 1, Equation (47) has the exact solution that follows: f ( x, t) = k0 (t) + k1 (t) x β where k0 (t) and k1 (t) satisfy the system of FDEs as follows: dα k1 (t) d α +1 k 0 ( t ) −µ Γ( β + 1) = k21 (t)Γ2 ( β + 1) − 1 α + 1 dtα dt

(48)

d α +1 k 1 ( t ) =0 dtα+1

(49)

Solving the system of FDEs (48) and (49), we obtain k1 (t) = b1 + b2 t k 0 ( t ) = a1 + a2 t +

b12 Γ2 ( β + 1) − 1 α+1 µb2 Γ( β + 1) 2 2b1 b2 Γ2 ( β + 1) α+2 2b22 Γ2 ( β + 1) α+3 t + t + t + t Γ ( α + 2) 2 Γ ( α + 3) Γ ( α + 4)

Therefore Equation (47) has the exact solution that follows: b12 Γ2 ( β+1)−1 α+1 t Γ ( α +2) t) x β

f ( x, t) = ( a1 + a2 t +

+ (b1 + b2

+

µb2 Γ( β+1) 2 t 2

+

2b1 b2 Γ2 ( β+1) α+2 t Γ ( α +3)

+

2b22 Γ2 ( β+1) α+3 t ) Γ ( α +4)

where a1 , a2 , b1 and b2 are arbitrary constants. Example 4. The Korteweg–de Vries (KdV) equation describes the evolution in time of long, unidirectional, nonlinear shallow water waves. We considered the fractional KdV equation that follows: ∂α f ∂β f ∂β ∂β f 2 ∂α ∂ β = ( ( ( ))) + µ ( f ), α, β ∈ (0, 1] ∂tα ∂tα ∂x β ∂x β ∂x β ∂x β 2 I3 = L{1, x β , x2β } is an invariant subspace under N [ f ] = N [k0 + k1 x β + k2 x2β ] = k1 k2 Γ(3β + 1) +

2 β ∂β f ∂β ( ( ∂ ( f ))) ∂x β ∂x β ∂x β 2

(50) and

∂β ∂x β

f , as

k22 Γ(4β + 1) β x ∈ I3 2Γ( β + 1)

∂ β (k0 + k1 x β + k2 x2β ) k Γ(2β + 1) β = k 1 Γ ( β + 1) + 2 x ∈ I3 Γ ( β + 1) ∂x β We consider an exact solution that follows: f ( x, t) = k0 (t) + k1 (t) x β + k2 (t) x2β where k0 (t), k1 (t) and k2 (t) are unknown functions to be determined. It follows from Theorem 1 applied to Equation (47) that k0 (t), k1 (t) and k2 (t) satisfy the FDEs as follows: dα k0 (t) dα k1 (t) − µ Γ( β + 1) = Γ(3β + 1)k1 (t)k2 (t) dtα dtα

(51)

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dα k2 (t) Γ(2β + 1) Γ(4β + 1) 2 dα k1 (t) −µ = k (t) α α dt dt Γ ( β + 1) 2Γ( β + 1) 2

(52)

dα k2 (t) =0 dtα

(53)

Solving Equation (53), we obtain k2 (t) = c. Hence Equation (52) has the form c2 Γ(4β + 1) dα k1 (t) = dtα 2Γ( β + 1) We obtain k1 (t) = b +

c2 Γ(4β + 1) tα 2Γ( β + 1)Γ(α + 1)

Similarly, we obtain k0 (t) = a +

2bcΓ(3β + 1) + µc2 Γ(4β + 1) α c3 Γ(4β + 1)Γ(3β + 1) 2α t + t 2Γ(α + 1) 2Γ( β + 1)Γ(2α + 1)

Thus the exact solution of Equation (50) is 2bcΓ(3β+1)+µc2 Γ(4β+1) α t 2Γ(α+1) 2β + cx

f ( x, t) = ( a +

+

c3 Γ(4β+1)Γ(3β+1) 2α c2 Γ(4β+1) t ) + (b + 2Γ( β+1)Γ(α+1) tα ) x β 2Γ( β+1)Γ(2α+1)

where a, b and c are arbitrary constants. Example 5. The fractional version of the nonlinear heat equation is as follows: ∂β f ∂α f ∂β ∂α ∂ β = ( ( f ) + µ f ), α, β ∈ (0, 1] ∂tα ∂tα ∂x β ∂x β ∂x β Clearly, the subspace I2 = L{1, x β } is invariant under N [ f ] = N [k0 + k1 x β ] =

∂β f ∂β ( f ∂x β) ∂x β

(54) and

∂β ∂x β

f , as

∂β ((k0 + k1 x β )k1 Γ( β + 1)) = k21 Γ2 ( β + 1) ∈ I2 ∂x β ∂ β (k0 + k1 x β ) = k1 Γ( β + 1) ∈ I2 ∂x β

It follows from Theorem 1 that we consider the exact solution of Equation (54) as follows: f ( x, t) = k0 (t) + k1 (t) x β such that

dα k1 (t) dα k0 (t) −µ Γ( β + 1) = k21 (t)Γ2 ( β + 1) α dt dtα

(55)

dα k1 (t) =0 dtα

(56)

Solving Equations (55) and (56), we obtain k1 (t) = b k0 (t) = a +

b2 Γ2 ( β + 1) α t Γ ( α + 1)

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We obtain an exact solution as follows: f ( x, t) = ( a +

b2 Γ2 ( β + 1) α t ) + bx β Γ ( α + 1)

where a and b are arbitrary constants. Next, we consider the integer-order differential equations in [30]. We can obtain some new different solutions using the invariant subspace method. Example 6. The modified hyperbolic heat conduction equation with the mixed derivative term is as follows ([30]): ∂2 f ∂2 ∂2 f 2 = + κ f − ε f ∂t∂x ∂t2 ∂x2

(57)

f ( x, 0) = g( x ), f ( x, ∞) < ∞

(58)

where ε, κ = const. Clearly, the subspace I2 = L{1, x } is an invariant subspace under N [ f ] =

∂2 f ∂x2

+ κ 2 f and

∂ ∂x

f , as

N [k0 + k1 x ] = κ 2 k0 + κ 2 k1 x ∈ I2 ∂(k0 + k1 x ) = k1 ∈ I2 ∂x We let the exact solution be as follows: f ( x, t) = k0 (t) + k1 (t) x where k0 (t), k1 (t) are unknown functions to be determined, and k0 (t) and k1 (t) satisfy the system of differential equations as follows: d2 k 0 ( t ) dk (t) +ε 1 = κ 2 k0 (t) dt dt2

(59)

d2 k 1 ( t ) = κ 2 k1 (t) dt2

(60)

Solving Equation (60), we obtain k1 (t) = b1 e−κt + b2 eκt Hence Equation (59) has the form d2 k 0 ( t ) − κ 2 k0 (t) = b1 εκe−κt + b2 εκeκt dt2 We obtain k0 (t) = a1 e−κt + a2 eκt −

a1 ε −κt a2 ε κt te − te 2 2

Then, we obtain the exact solution of Equation (57) as f ( x, t) = a1 e−κt + a2 eκt −

a1 ε −κt a2 ε κt te − te + (b1 e−κt + b2 eκt ) x 2 2

(61)

where a1 , a2 , b1 and b2 are arbitrary constants. Substituting the conditions of Equation (58) into Equation (61), we obtain a2 = b2 = 0 and a1 + b1 x = g( x ). When g( x ) has linear dependence on x, Equations (57) and (58) have the partial solution

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f ( x, t) = a1 e−κt −

a1 ε −κt te + b1 xe−κt 2

where a1 + b1 x = g( x ). When g( x ) is not linearly dependent on x, Equations (57) and (58) do not have the form of the solution given by Equation (61). The subspace In+1 = L{1, x, x2 , · · · , x n }, n ∈ N is invariant under N [ f ] =

∂2 f ∂x2

+ κ 2 f and

∂ ∂x

f , as

N [k0 + k1 x + · · · + k n x n ] = (κ 2 k0 + 2k2 ) + · · · + (κ 2 k n−2 + n(n − 1)k n ) x n−2 + κ 2 k n−1 x n−1 + κ 2 k n x n ∈ In+1 ∂(k0 + k1 x + · · · + k n x n ) = k1 + k2 x + · · · + k n−1 x n−1 ∈ In+1 ∂x Thus we obtain infinitely many invariant subspaces for Equation (57), which in turn yield infinitely many solutions. If g( x ) is a polynomial, we can obtain an exact solution of Equations (57) and (58). Example 7. The Fokker–Planck equation is the following ([30]): ∂2 f ∂f ∂2 ∂2 f = α + βx − ε f ∂x ∂t∂x ∂t2 ∂x2

(62)

where α, β, κ = const. ∂2 f

∂f

Clearly, the subspace I2 = L{1, x } is an invariant subspace under N [ f ] = α ∂x2 + βx ∂x and

∂ ∂x

f , as

N [k0 + k1 x ] = βk1 x ∈ I2 ∂(k0 + k1 x ) = k1 ∈ I2 ∂x We suppose the exact solution that follows: f ( x, t) = k0 (t) + k1 (t) x where k0 (t) and k1 (t) are unknown functions to be determined; k0 (t) and k1 (t) satisfy the system of differential equations as follows:

Case 1: when β > 0:

d2 k 1 ( t ) = βk1 (t) dt2

(63)

d2 k 0 ( t ) dk (t) = −ε 1 2 dt dt

(64)

√ + b2 e βt √ εb εb2 √ k0 (t) = a1 + a2 t + p1 e− βt − p e βt β β k1 (t) = b1 e−

√

βt

Thus Equation (62) has the exact solution that follows:

√ √ √ εb εb2 √ f ( x, t) = ( a1 + a2 t + p1 e− βt − p e βt ) + (b1 e− βt + b2 e βt ) x β β where a1 , a2 , b1 and b2 are arbitrary constants.

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Case 2: when β = 0: k1 (t) = b1 + b2 t k0 (t) = −

εb2 2 t + a1 + a2 t 2

Thus Equation (62) has the exact solution that follows: f ( x, t) = (−

εb2 2 t + a1 + a2 t) + (b1 + b2 t) x 2

where a1 , a2 , b1 and b2 are arbitrary constants. Case 3: when β < 0: p p k1 (t) = b1 cos − βt + b2 sin − βt p p εb2 εb k0 (t) = a1 + a2 t − p 1 sin − βt + p cos − βt −β −β Thus Equation (62) has the exact solution that follows: p p p p εb εb2 f ( x, t) = ( a1 + a2 t − p 1 sin − βt + p cos − βt) + (b1 cos − βt + b2 sin − βt) x −β −β where a1 , a2 , b1 and b2 are arbitrary constants. as

∂2 f

∂f

The subspace In+1 = L{1, x, x2 , · · · , x n }, n ∈ N is invariant under N [ f ] = α ∂x2 + βx ∂x and

∂ ∂x

f,

N [k0 + k1 x + · · · + k n x n ] = αk2 + (αk3 + βk1 ) x + · · · + ( βk n−1 (n − 1)) x n−1 + βk n nx n ∈ In+1 ∂(k0 + k1 x + · · · + k n x n ) = k1 + k2 x + · · · + k n−1 x n−1 ∈ In+1 ∂x

Thus we obtain infinitely many invariant subspaces for Equation (62), which in turn yield infinitely many solutions. 5. 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. Acknowledgments: This research is partially supported by the Doctoral Fund of Education Ministry of China (20134219120003) and the National Natural Science Foundation of China (61473338). Author Contributions: All authors made equal contribution to this paper. Conflicts of Interest: The authors declare no conflict of interest.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

Oldham, K.B., Spanier, J. The Fractional Calculus; Academic Press: New York, NY, USA, 1974. Miller, K.S.; Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations; Wiley: New York, NY, USA, 1993. Diethelm, K. The Analysis of Fractional Differential Equations; Springer: New York, NY, USA, 2010. Kilbas, A.A.; Trujillo, J.J.; Srivastava, H.M. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006. Hilfer, R. Applications of Fractional Calculus in Physics; World Scientific: Singapore, 2000. Ansari, A.; Askari, H. On fractional calculus of A2n+1(x) function. Appl. Math. Comput. 2014, 232, 487–497. Podlubny, I. Fractional Differential Equations; Academic Press: New York, NY, USA, 1998. Daftardar-Gejji, V.; Jafari, H. Adomian decomposition: A tool for solving a system of fractional differential equations. J. Math. Anal. Appl. 2005, 301, 508–518. Eslami, M.; Vajargah, B.F.; Mirzazadeh, M.; Biswas, A. Applications of first integral method to fractional partial differential equations. Indian J. Phys. 2014, 88, 177–184. Sahadevan, R.; Bakkyaraj, T. Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations. J. Math. Anal. Appl. 2012, 2, 341–347. Ouhadan, A.; El Kinani, E.H. Lie symmetry analysis of some time fractional partial differential equations. Int. J. Mod. Phys. Conf. Ser. 2015, 38, 1560075. Bakkyaraj, T.; Sahadevan, R. Approximate analytical solution of two coupled time fractional nonlinear Schrodinger equations. Int. J. Appl. Comput. Math. 2016, 2, 113–135. Zhukovsky, K.V.; Srivastava, H.M. Operational solution of non-integer ordinary and evolution-type partial differential equations. Axioms 2016, 5, 29, doi:10.3390/axioms5040029. Zhukovsky, K.V. The Operational Solution of Fractional-Order Differential Equations as well as Black-Scholes and Heat-Conduction Equations. Mosc. Univ. Phys. Bull. 2016, 71, 237–244. Zhukovsky, K.V. Operational method of solution of linear non-integer ordinary and partial differential equations. SpringerPlus 2016, 5, 119, doi:10.1186/s40064-016-1734-3. Pandir, Y.; Huseyin Duzgun, H. New Exact Solutions of Time Fractional Gardner Equation by Using New Version of F-Expansion Method. Commun. Theor. Phys. 2017, 67, 9–14. Moslehi, L.; Ansari, A. On M-Wright transforms and time-fractional diffusion equations. Integral Transforms Spec. Funct. 2017, 28, 113–129. Aghili, A.; Aghili, J. Exponential differential operators for singular integral equations and space fractional Fokker-Planck equation. Bol. Soc. Parana. Mat. 2018, 36, 223–233. Aghili, A. Fractional Black-Scholes equation. Int. J. Financ. Eng. 2017, 4, 1750004. Galaktionov, V.A. Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities. Proc. R. Soc. Edinb. Sect. A Math. 1995, 125, 225–246. Svirshchevskii, S.R. Invariant linear spaces and exact solutions of nonlinear evolution equations. J. Nonlinear Math. Phys. 1996, 3, 164–169. Galaktionov, V.A.; Svirshchevskii, S.R. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics; Chapman and Hall/CRC: London, UK, 2007. Gazizov, R.K.; Kasatkin, A.A. Construction of exact solutions for fractional order differential equations by invariant subspace method. Comput. Math. Appl. 2013, 66, 576–584. Harris, P.A.; Garra, R. Analytic solution of nonlinear fractional Burgers-type equation by invariant subspace method. Nonlinear Stud. 2013, 20, 471–481. Harris, P.A.; Garra,R. Nonlinear time-fractional dispersive equations. arXiv 2014, arXiv:1410.8085v1. Sahadevan, R.; Bakkyaraj, T. Invariant subspace method and exact solutions of certain nonlinear time fractional partial differential equations. Fract. Calc. Appl. Anal. 2015, 18, 146–162. Ouhadan, A.; El Kinani, E.H. Invariant subspace method and fractional modified Kuramoto-Sivashinsky equation. arXiv 2015, arXiv:1503.08789v1. Sahadevan, R.; Prakash, P. Exact solution of certain time fractional nonlinear partial differential equations. Nonlinear Dyn. 2016, 85, 659–673.

Axioms 2018, 7, 10

29. 30.

18 of 18

Choudhary, S. Daftardar-Gejji, V. Invariant Subspace Method: A tool for solving fractional partial differential equations. arXiv 2016, arXiv:1609.04209v1. Zhukovsky, K.V. Operational solution for some types of second order differential equations and for relevant physical problems. J. Math. Anal. Appl. 2016, 446, 628–647. c 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access

article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).