Fractional Complex Transform and exp-Function Methods for Fractional Differential Equations

Abstract

The exp-function method is presented for finding the exact solutions of nonlinear fractional equations. New solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie's modified Riemann-Liouville sense. We apply the exp-function method to both the nonlinear time and space fractional differential equations. As a result, some new exact solutions for them are successfully established.

1. Introduction

Fractional differential equations (FDEs) are viewed as alternative models to nonlinear differential equations. Varieties of them play important roles and serve as tools not only in mathematics but also in physics, biology, fluid flow, signal processing, control theory, systems identification, and fractional dynamics to create the mathematical modeling of many nonlinear phenomena. Besides, they are employed in social sciences such as food supplement, climate, finance, and economics. Oldham and Spanier first considered the fractional differential equations arising in diffusion problems [1]. The fractional differential equations have been investigated by many authors [2–4].

In recent decades, some effective methods for fractional calculus appeared in open literature, such as the exp-function method [5], the fractional subequation method [6–8], the (G′/G)-expansion method [9, 10], and the first integral method [11].

The fractional complex transform [12, 13] is the simplest approach; it is to convert the fractional differential equations into ordinary differential equations, making the solution procedure extremely simple. Recently, the fractional complex transform has been suggested to convert fractional-order differential equations with modified Riemann-Liouville derivatives into integer order differential equations, and the reduced equations can be solved by symbolic computation. The exp-function method [14–20] can be used to construct the exact solutions for fractional differential equations. The present paper investigates the applicability and efficiency of the exp-function method on fractional nonlinear differential equations. The aim of this paper is to extend the application of the exp-function method to obtain exact solutions to some fractional differential equations in mathematical physics and biology.

This paper is organized as follows. In Section 2, some basic properties of Jumarie’s modified Riemann-Liouville derivative are given. The main steps of the exp-function method are given in Section 3. In Sections 4–6, we construct the exact solutions of the fractional-order biological population model, fractional Burgers equation, and fractional Cahn-Hilliard equation via this method. Some conclusions are shown in Section 7.

2. Modified Riemann-Liouville Derivative

In the last few decades, in order to improve the local behavior of fractional types, a few local versions of fractional derivatives have been proposed, that is, Caputo’s fractional derivative [21], Grünwald-Letnikov’s fractional derivative [22], the Riemann-Liouville derivative [22], Jumarie’s modified Riemann-Liouville derivative [23, 24]. Jumarie’s derivative is defined as 𝐷^𝛼_𝑡𝑓(𝑡)=1Γ()𝑑1−𝛼𝑑𝑡^𝑡₀(𝑡−𝜉)^−𝛼𝑓(𝜉)−𝑓(0)𝑑𝜉,0<𝛼<1, (1) where 𝑓∶𝑅→𝑅, 𝑡→𝑓(𝑡) denotes a continuous (but not necessarily first-order-differentiable) function. We list some important properties for the modified Riemann-Liouville derivative as follows.(1)Assume that 𝑓(𝑡) denotes a continuous 𝑅→𝑅 function. We use the following equality for the integral with respect to (𝑑𝑡)^𝛼: 𝐼^𝛼_𝑡𝑓(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝜉)^𝛼−1𝑓(𝜉)𝐼𝑑𝜉,_𝛼𝑡𝑓(𝑡)=1Γ(𝛼+1)^𝑡₀𝑓(𝜉)(𝑑𝑡)^𝛼,0<𝛼≤1. (2)(2)Some useful formulas include 𝑓^(𝛼)𝑔(𝑡)=𝑑𝑓𝑔𝑑𝑡^(𝛼)(𝑡),𝐷^𝛼_𝑡𝑡^𝛾=Γ1+𝛾Γ𝑡1+𝛾−𝛼^𝛾−𝛼(),𝑑𝑡^𝛽=𝑡^𝛽. (3)(4)(5)(3)Let 𝑢(𝑡) and 𝑣(𝑡) satisfy the definition of the modified Riemann-Liouville derivative, and let 𝑓(𝑡) be an 𝛼-order-differentiable function: 𝐷^𝛼_𝑡(𝑢(𝑡)𝑣(𝑡(𝑡)𝐷))=𝑣^𝛼_𝑡𝑢(𝑡)+𝑢(𝑡)𝐷^𝛼_𝑡𝑣(𝑡),𝐷^𝛼_𝑡𝑓[𝑢(𝑡)]=𝑓^′_𝑢[𝑢(𝑡)]𝐷^𝛼_𝑡𝑢(𝑡)=𝐷^𝛼_𝑢𝑓[𝑢(𝑡)]𝑢^′(𝑡)^𝛼. (6)Function 𝑓(𝑡) should be differentiable with respect to 𝑔(𝑡), and 𝑔(𝑡) is fractional differentiable in (3). The previous results are employed in the following sections.

3. Fractional Complex Transform and exp-Function Method

We consider the following nonlinear FDE of the type 𝐹𝑢,𝐷^𝛼_𝑡𝑢,𝐷^𝛽_𝑥𝑢,𝐷^𝛼_𝑡𝐷^𝛼_𝑡𝑢,𝐷^𝛼_𝑡𝐷^𝛽_𝑥𝑢,𝐷^𝛽_𝑥𝐷^𝛽_𝑥𝑢,…=0,0<𝛼,𝛽<1, (7) where 𝑢 is an unknown function and 𝐹 is a polynomial of 𝑢 and its partial fractional derivatives, in which the highest order derivatives and the nonlinear terms are involved. In the following, we give the main steps of the exp-function method.

Step 1. Li and He [25, 26] suggested a fractional complex transform to convert fractional differential equations into ordinary differential equations, so all analytical methods devoted to the advanced calculus can be easily applied to the fractional calculus. The complex wave variable was as follows: 𝑢(𝑥,𝑡)=𝑈(𝜉),𝜉=𝜏𝑥^𝛽Γ+1+𝛽𝜆𝑡^𝛼Γ(),1+𝛼 (8) where 𝜏 and 𝜆 are nonzero arbitrary constants; we can rewrite (7) in the following nonlinear ordinary differential equation: 𝑄𝑈,𝑈^′,𝑈^′′,𝑈^′′′,…=0, (9) where the prime denotes the derivation with respect to 𝜉. If possible, we should integrate (9) term by term one or more times.

Step 2. According to exp-function method, which was developed by He and Wu [14], we assume that the wave solution can be expressed in the following form: 𝑈(𝜉)=∑^𝑑_𝑛=−𝑐𝑎_𝑛exp𝑛𝜉∑^𝑞_𝑚=−𝑝𝑏_𝑚,exp𝑚𝜉 (10) where 𝑝,𝑞,𝑐, and 𝑑 are positive integers which are known to be further determined and 𝑎_𝑛 and 𝑏_𝑚 are unknown constants. We can rewrite (10) in the following equivalent form: 𝑈(𝜉)=𝑎_−𝑐exp−𝑐𝜉+⋯+𝑎_𝑑exp𝑑𝜉𝑏_−𝑝exp−𝑝𝜉+⋯+𝑏_𝑞.exp𝑞𝜉 (11)

Step 3. This equivalent formulation plays a significant and fundamental part for finding the exact solution of mathematical problems. To determine the values of 𝑐 and 𝑝, we balance the linear term of highest order of (9) with the highest order nonlinear term. Similarly, to determine the value of 𝑑 and 𝑞, we balance the linear term of lowest order of (9) with lowest order nonlinear term [27–29].
In the following sections, we present three examples to illustrate the applicability of the exp-function method and fractional complex transform to solve nonlinear fractional differential equations.

4. Fractional-Order Biological Population Model

We consider a time fractional biological population model of the form [30, 31] 𝜕^𝛼𝑢𝜕𝑡^𝛼=𝜕²𝜕𝑥²𝑢²+𝜕²𝜕𝑦²𝑢²+ℎ𝑢²−𝑟,𝑡>0,0<𝛼≤1,𝑥,𝑦∈𝑅, (12) where 𝑢 denotes the population density ℎ(𝑢²−𝑟) represents the population supply due to births and deaths, and ℎ,  𝑟 are constants.

For our goal, we present the following transformation: 𝑢𝑥,𝑦,𝑡=𝑈(𝜉),𝜉=𝜈𝑥+𝑖𝜈𝑦−𝑐𝑡^𝛼Γ(),1+𝛼 (13) where 𝑐 and 𝜈 are constants and 𝑖²=−1.

Then by the use of (13), (12) can be turned into an ODE: 𝑐𝑈^′+ℎ𝑈²−ℎ𝑟=0, (14) where “𝑈^′” =𝑑𝑈/𝑑𝜉.

Balancing the order of 𝑈^′ and 𝑈² in (14), we get 𝑈^′=𝑐₁exp−𝑐+𝑝𝜉+⋯𝑐₂,𝑈exp−2𝑝𝜉+⋯₂=𝑐₃exp−2𝑐𝜉+⋯𝑐₄,exp−2𝑝𝜉+⋯ (15) where 𝑐_𝑖 are determined coefficients only for simplicity. Balancing highest order of exp-function in (15), we obtain −𝑝+𝑐=−2𝑐, (16) which leads to the result that 𝑝=𝑐. (17) In the same way to determine the values of 𝑑 and 𝑞, we balance the linear term of the lowest order in (14): 𝑈^′=⋯+𝑑₁exp𝑞+𝑑𝜉⋯+𝑑₂,𝑈exp2𝑞𝜉₂=⋯+𝑑₃exp2𝑑𝜉⋯+𝑑₄,exp2𝑞𝜉 (18) where 𝑑_𝑖 are determined coefficients only for simplicity. From (18), we have 𝑞+𝑑=2𝑑, (19) and this gives 𝑞=𝑑. (20)

For simplicity, we set 𝑝=𝑐=1 and 𝑞=𝑑=1, so (11) reduces to 𝑈(𝜉)=𝑎₁exp(𝜉)+𝑎₀+𝑎₋₁exp(−𝜉)𝑏₁exp(𝜉)+𝑏₀+𝑏₋₁().exp−𝜉 (21)

Substituting (21) into (14) and by the help of symbolic computation, we have 1𝐴𝑅₂exp(2𝜉)+𝑅₁exp(𝜉)+𝑅₀+𝑅₋₁exp(−𝜉)+𝑅₋₂exp(−2𝜉)=0, (22) where 𝐴=𝑏₋₁exp(−𝜉)+𝑏₀+𝑏₁exp(𝜉)²,𝑅₂=ℎ𝑎²₁−ℎ𝑟𝑏²₁,𝑅₁=𝑐𝑎₁𝑏₀−𝑐𝑎₀𝑏₁+2ℎ𝑎₁𝑎₀−2ℎ𝑟𝑏₁𝑏₀,𝑅₀=−2𝑐𝑎₋₁𝑏₁+2𝑐𝑎₁𝑏₋₁−2ℎ𝑟𝑏₁𝑏₋₁−ℎ𝑟𝑏²₀+2ℎ𝑎₁𝑎₋₁+ℎ𝑎²₀,𝑅₋₁=−𝑐𝑎₋₁𝑏₀+𝑐𝑎₀𝑏₋₁+2ℎ𝑎₀𝑎₋₁−2ℎ𝑟𝑏₀𝑏₋₁,𝑅₋₂=ℎ𝑎²₋₁−ℎ𝑟𝑏²₋₁. (23)

Solving this system of algebraic equations by using symbolic computation, we get the following results.

Case 1. Consider 𝑎₁=𝑟𝑏²₀−𝑎²₀4√𝑟𝑏₋₁,𝑎₀=𝑎₀,𝑎₋₁=−√𝑟𝑏₋₁,𝑏₁=𝑟𝑏²₀−𝑎²₀4𝑟𝑏₋₁,𝑏₀=𝑏₀,𝑏₋₁=𝑏₋₁,𝑐=2ℎ√𝑟, (24) where 𝑎₀,𝑏₀, and 𝑏₋₁ are free parameters which exist provided that 𝑏₋₁≠0. Substituting these results into (21), we get the following exact solution: 𝑢=𝑥,𝑦,𝑡𝑟𝑏²₀−𝑎²₀4√𝑟𝑏₋₁√exp𝜈𝑥+𝑖𝜈𝑦−2ℎ𝑟𝑡^𝛼Γ(1+𝛼)+𝑎₀−√𝑟𝑏₋₁−√exp𝜈𝑥+𝑖𝜈𝑦−2ℎ𝑟𝑡^𝛼Γ()×1+𝛼𝑟𝑏²₀−𝑎²₀4𝑟𝑏₋₁√exp𝜈𝑥+𝑖𝜈𝑦−2ℎ𝑟𝑡^𝛼Γ(1+𝛼)+𝑏₀+𝑏₋₁−√exp𝜈𝑥+𝑖𝜈𝑦−2ℎ𝑟𝑡^𝛼Γ(1+𝛼)⁻¹. (25)

Case 2. Consider 𝑎₁=𝑎²₀4√𝑟𝑏₋₁,𝑎₀=𝑎₀,𝑎₋₁=−√𝑟𝑏₋₁,𝑏₁𝑎=−²⁰4𝑟𝑏₋₁,𝑏₀=0,𝑏₋₁=𝑏₋₁,𝑐=2ℎ√𝑟, (26) where 𝑎₀ and 𝑏₋₁ are free parameters, which exist provided that 𝑏₋₁≠0. Substituting these results into (21), we obtain the following exact solution: 𝑢=𝑎𝑥,𝑦,𝑡₂₀4√𝑟𝑏₋₁√exp𝜈𝑥+𝑖𝜈𝑦−2ℎ𝑟𝑡^𝛼Γ(1+𝛼)+𝑎₀−√𝑟𝑏₋₁−√exp𝜈𝑥+𝑖𝜈𝑦−2ℎ𝑟𝑡^𝛼Γ()×−𝑎1+𝛼₂₀4𝑟𝑏₋₁√exp𝜈𝑥+𝑖𝜈𝑦−2ℎ𝑟𝑡^𝛼Γ(1+𝛼)+𝑏₋₁−√exp𝜈𝑥+𝑖𝜈𝑦−2ℎ𝑟𝑡^𝛼Γ(1+𝛼)⁻¹. (27)

5. Time Fractional Burgers Equation

We consider the one-dimensional time fractional Burgers equation with the value problem [32] 𝜕^𝛼𝑢𝜕𝑡^𝛼+𝜀𝑢𝜕𝑢𝜕𝜕𝑥−𝜈²𝑢𝜕𝑥²𝑢()(𝑥),=0,𝑡>0,0<𝛼≤1,𝑥,0=𝑔 (28)(29) where 𝛼 is a parameter describing the order of the fractional time derivative. The function 𝑢(𝑥,𝑡) is assumed to be a causal function of time.

For our purpose, we introduce the following transformations: 𝑢(𝑥,𝑡)=𝑈(𝜉),𝜉=𝜆𝑥−𝑐𝑡^𝛼Γ(),1+𝛼 (30) where 𝜆 and 𝑐 are nonzero constants.

Substituting (30) into (28), we can show that (28) reduced into the following ODE: −𝑐𝑈^′+𝜆𝜀𝑈𝑈^′−𝜆²𝜈𝑈^′′=0, (31) where “𝑈^′” =𝑑𝑈/𝑑𝜉.

Integrating (31) with respect to 𝜉 yields −𝑐𝑈+𝜆𝜀2𝑈²−𝜆²𝜈𝑈^′+𝜉₀=0, (32) where 𝜉₀ is a constant of integration.

By the same procedure as illustrated in Section 3, we can determine values of 𝑐 and 𝑝 by balancing terms 𝑈² and 𝑈^′ in (32). By symbolic computation, we have 𝑈^′=𝑐₁exp−𝑐+𝑝𝜉+⋯𝑐₂,𝑈exp−2𝑝𝜉+⋯₂=𝑐₃exp−2𝑐𝜉+⋯𝑐₄,exp−2𝑝𝜉+⋯ (33) where 𝑐_𝑖 are determined coefficients only for simplicity. Balancing the highest order of exp-function in (33), we have −𝑝+𝑐=−2𝑐, (34) which leads to the result that 𝑝=𝑐. (35) Similarly, to determine the values of 𝑑 and 𝑞, we balance the linear term of the lowest order in (32): 𝑈^′=⋯+𝑑₁exp𝑞+𝑑𝜉⋯+𝑑₂,𝑈exp2𝑞𝜉₂=⋯+𝑑₃exp2𝑑𝜉⋯+𝑑₄,exp2𝑞𝜉 (36) where 𝑑_𝑖 are determined coefficients only for simplicity. From (36), we obtain 𝑞+𝑑=2𝑑, (37) and this gives 𝑞=𝑑. (38)

For simplicity, we set 𝑝=𝑐=1 and 𝑞=𝑑=1, so (11) reduces to 𝑈(𝜉)=𝑎₁exp(𝜉)+𝑎₀+𝑎₋₁exp(−𝜉)𝑏₁exp(𝜉)+𝑏₀+𝑏₋₁().exp−𝜉 (39)

Substituting (39) into (32) and by the help of symbolic computation, we have 1𝐴𝑅₂exp(2𝜉)+𝑅₁exp(𝜉)+𝑅₀+𝑅₋₁exp(−𝜉)+𝑅₋₂exp(−2𝜉)=0, (40) where 𝐴=𝑏₋₁exp(−𝜉)+𝑏₀+𝑏₁exp(𝜉)²,𝑅₂=𝜉₀𝑏²₁−𝑐𝑎₁𝑏₁+12𝜆𝜀𝑎²₁,𝑅₁=−𝜆²𝜈𝑎₁𝑏₀+𝜆𝜀𝑎₁𝑎₀+𝜆²𝜈𝑎₀𝑏₁−𝑐𝑎₀𝑏₁−𝑐𝑎₁𝑏₀+2𝜉₀𝑏₁𝑏₀,𝑅₀=−2𝜆²𝜈𝑎₁𝑏₋₁+2𝜆²𝜈𝑎₋₁𝑏₁−𝑐𝑎₀𝑏₀+𝜆𝜀𝑎₁𝑎₋₁+𝜉₀𝑏²₀−𝑐𝑎₁𝑏₋₁−𝑐𝑎₋₁𝑏₁+12𝜆𝜀𝑎²₀+2𝜉₀𝑏₁𝑏₋₁,𝑅₋₁=𝜆𝜀𝑎₀𝑎₋₁+𝜆²𝜈𝑎₋₁𝑏₀−𝜆²𝜈𝑎₀𝑏₋₁+2𝜉₀𝑏₀𝑏₋₁−𝑐𝑎₀𝑏₋₁−𝑐𝑎₋₁𝑏₀,𝑅₋₂=𝜉₀𝑏²₋₁−𝑐𝑎₋₁𝑏₋₁+12𝜆𝜀𝑎²₋₁ (41)

Solving this system of algebraic equations by using symbolic computation, we obtain the following results.

Case 1. Consider 𝑎₁=𝑏₁𝜀𝑎₋₁−4𝜆𝜈𝑏₋₁𝜀𝑏₋₁,𝑎₀=0,𝑎₋₁=𝑎₋₁,𝑏₁=𝑏₁,𝑏₀=0,𝑏₋₁=𝑏₋₁,𝜉𝜆=𝜆,𝜀=𝜀,𝜈=𝜈,₀=𝜆𝑎₋₁𝜀𝑎₋₁−4𝜆𝜈𝑏₋₁2𝑏²₋₁𝜆,𝑐=𝜀𝑎₋₁−2𝜆𝜈𝑏₋₁𝑏₋₁, (42) where 𝑎₋₁ and 𝑏₋₁ are free parameters which exist provided that 𝑏₋₁≠0and 𝜀𝑎₋₁−2𝜆𝜈𝑏₋₁≠0. Substituting these results into (39), we obtain the following exact solution: 𝑢()=𝑏𝑥,𝑡₁𝜀𝑎₋₁−4𝜆𝜈𝑏₋₁𝜀𝑏₋₁exp𝜆𝑥−𝑐𝑡^𝛼Γ(1+𝛼)+𝑎₋₁exp−𝜆𝑥−𝑐𝑡^𝛼Γ()×𝑏1+𝛼₁exp𝜆𝑥−𝑐𝑡^𝛼Γ(1+𝛼)+𝑏₋₁exp−𝜆𝑥−𝑐𝑡^𝛼Γ(1+𝛼)⁻¹. (43)

Case 2. Consider 𝑎₁=−𝜀𝑎²₋₁𝑏²₀−2𝑎₋₁𝑏²₀𝜆𝜈𝑏₋₁−2𝜀𝑎₀𝑏₀𝑏₋₁𝑎₋₁+2𝜆𝜈𝑎₀𝑏₀𝑏²₋₁+𝜀𝑎²₀𝑏²₋₁×𝜀𝑎₋₁−𝜆𝜈𝑏₋₁×4𝑏²₋₁𝜆²𝜈²⁻¹,𝑏₁=−𝜀𝜀𝑎²₋₁𝑏²₀−2𝑎₋₁𝑏²₀𝜆𝜈𝑏₋₁−2𝜀𝑎₀𝑏₀𝑏₋₁𝑎₋₁+2𝜆𝜈𝑎₀𝑏₀𝑏²₋₁+𝜀𝑎²₀𝑏²₋₁×4𝑏³₋₁𝜆²𝜈²⁻¹,𝑎₀=𝑎₀,𝑎₋₁=𝑎₋₁,𝑏₀=𝑏₀,𝑏₋₁=𝑏₋₁𝜉,𝜆=𝜆,𝜀=𝜀,𝜈=𝜈,₀=𝜆𝑎₋₁𝜀𝑎₋₁−2𝜆𝜈𝑏₋₁2𝑏²₋₁𝜆,𝑐=𝜀𝑎₋₁−𝜆𝜈𝑏₋₁𝑏₋₁, (44) where 𝑎₋₁,  𝑎₀,  𝑏₋₁, and 𝑏₀ are free parameters which exist provided that 𝑏₋₁≠0and 𝜀𝑎₋₁−2𝜆𝜈𝑏₋₁≠0. Substituting these results into (39), we get the following exact solution: 𝑢(𝑥,𝑡)=−𝜀𝑎²₋₁𝑏²₀−2𝑎₋₁𝑏²₀𝜆𝜈𝑏₋₁−2𝜀𝑎₀𝑏₀𝑏₋₁𝑎₋₁+2𝜆𝜈𝑎₀𝑏₀𝑏²₋₁+𝜀𝑎²₀𝑏²₋₁𝜀𝑎₋₁−𝜆𝜈𝑏₋₁×4𝑏²₋₁𝜆²𝜈²⁻¹×exp𝜆𝑥−𝑐𝑡^𝛼Γ(1+𝛼)+𝑎₀+𝑎₋₁exp−𝜆𝑥−𝑐𝑡^𝛼Γ()×1+𝛼−𝜀𝜀𝑎²₋₁𝑏²₀−2𝑎₋₁𝑏²₀𝜆𝜈𝑏₋₁−2𝜀𝑎₀𝑏₀𝑏₋₁𝑎₋₁+2𝜆𝜈𝑎₀𝑏₀𝑏²₋₁+𝜀𝑎²₀𝑏²₋₁×4𝑏³₋₁𝜆²𝜈²⁻¹×exp𝜆𝑥−𝑐𝑡^𝛼Γ(1+𝛼)+𝑏₀+𝑏₋₁exp−𝜆𝑥−𝑐𝑡^𝛼Γ(1+𝛼)⁻¹. (45)

6. Space-Time Fractional Cahn-Hilliard Equation

We consider the space-time fractional Cahn-Hilliard equation [33] 𝐷^𝛼_𝑡𝑢−𝛾𝐷^𝛼_𝑥𝑢−6𝑢𝐷^𝛼_𝑥𝑢²−3𝑢²−1𝐷_𝑥^2𝛼𝑢+𝐷_𝑥^4𝛼𝑢=0, (46) where 0<𝛼≤1 and 𝑢 is the function of (𝑥,𝑡). For the case corresponding to 𝛼=1, this equation is related to a number of interesting physical phenomena like the spinodal decomposition, phase separation, and phase ordering dynamics. Moreover, it becomes important in material sciences [34]. Nevertheless we notice that this equation is very difficult to be solved and several articles investigated it [35].

Firstly, we consider the following transformations: 𝑢()(𝜉)𝑥𝑥,𝑡=𝑈,𝜉=^𝛼Γ()−1+𝛼𝑐𝑡^𝛼Γ(),1+𝛼 (47) where 𝑐 is a nonzero constant.

Substituting (47) into (46), we can know this equation reduced into an ODE: −𝑐𝑈^′−𝛾𝑈^′−6𝑈𝑈^′2−3𝑈²𝑈^′′+𝑈^′′+𝑈^′′′′=0, (48) where “𝑈^′” =𝑑𝑈/𝑑𝜉.

Integrating (48) with respect to 𝜉 yields −𝑐𝑈−𝛾𝑈−3𝑈²𝑈^′+𝑈^′+𝑈^′′′+𝜉₀=0, (49) where 𝜉₀ is a constant of integration.

Here take notice of nonlinear term in (49), and we can balance 𝑈²𝑈^′ and 𝑈^′′′ by the idea of the exp-function method [14] to determine the values of 𝑝,𝑞,𝑐, and 𝑑. By simple calculation, we have 𝑈^′′′=𝑐₁exp−7𝑝−𝑐𝜉+⋯𝑐₂=𝑐exp−8𝑝𝜉+⋯₁exp−7𝑝+𝑐𝜉+⋯𝑐₂,𝑈exp−8𝑝𝜉+⋯₂𝑈^′=𝑐₃exp−𝑝−3𝑐𝜉+⋯𝑐₄=𝑐exp−4𝑝𝜉+⋯₃exp−5𝑝+3𝑐𝜉+⋯𝑐₄,exp−8𝑝𝜉+⋯ (50) where 𝑐_𝑖 are determined coefficients only for simplicity. Balancing the highest order of exp-function in (50), we have −7𝑝+𝑐=−5𝑐+3𝑝, (51) which leads to the result that 𝑝=𝑐. (52) Similarly, to determine the values of 𝑑 and 𝑞, we balance the linear term of lowest order in (49) 𝑈^′′′=⋯+𝑑₁exp7𝑞+𝑑𝜉⋯+𝑑₂,𝑈exp8𝑞𝜉₂𝑈^′=⋯+𝑑₃exp2𝑑+6𝑞𝜉⋯+𝑑₄,exp8𝑞𝜉 (53) where 𝑑_𝑖 are determined coefficients only for simplicity. From (53), we obtain 6𝑞+2𝑑=𝑑+7𝑞, (54) and this gives 𝑞=𝑑. (55) For simplicity, we set 𝑝=𝑐=1 and 𝑞=𝑑=1, so (11) reduces to 𝑢(𝜉)=𝑎₁exp(𝜉)+𝑎₀+𝑎₋₁exp(−𝜉)𝑏₁exp(𝜉)+𝑏₀+𝑏₋₁().exp−𝜉 (56)

Substituting (56) into (49) and by the help of symbolic computation, we obtain 1𝐴𝑅₄exp(4𝜉)+𝑅₃exp(3𝜉)+𝑅₂exp(2𝜉)+𝑅₁exp(𝜉)+𝑅₀+𝑅₋₁exp(−𝜉)+𝑅₋₂exp(−2𝜉)+𝑅₋₃exp(−3𝜉)+𝑅₋₄exp(−4𝜉)=0, (57) where 𝐴=𝑏₋₁exp(−𝜉)+𝑏₀+𝑏₁exp(𝜉)⁴,𝑅₄=𝜉₀𝑏⁴₁−𝑐𝑎₁𝑏³₁−𝛾𝑎₁𝑏³₁,𝑅₃=−3𝑎³₁𝑏₀−2𝑎₀𝑏³₁−𝑐𝑎₀𝑏³₁−𝛾𝑎₀𝑏³₁+2𝑎₁𝑏²₁𝑏₀+3𝑎²₁𝑎₀𝑏₁+4𝜉₀𝑏³₁𝑏₀−3𝛾𝑎₁𝑏²₁𝑏₀−3𝑐𝑎₁𝑏²₁𝑏₀,𝑅₂=−10𝑎₋₁𝑏³₁−6𝑎³₁𝑏₋₁−𝑐𝑎₋₁𝑏³₁−𝛾𝑎₋₁𝑏³₁+10𝑎₁𝑏²₁𝑏₋₁+6𝑎²₁𝑎₋₁𝑏₁−2𝑎₁𝑏₁𝑏²₀+4𝜉₀𝑏³₁𝑏₋₁+2𝑎₀𝑏²₁𝑏₀−6𝑎²₁𝑎₀𝑏₀+6𝑎₁𝑎²₀𝑏₁+6𝜉₀𝑏²₁𝑏²₀−3𝑐𝑎₁𝑏₁𝑏²₀−3𝑐𝑎₀𝑏²₁𝑏₀−3𝛾𝑎₁𝑏₁𝑏²₀−3𝛾𝑎₀𝑏²₁𝑏₀−3𝑐𝑎₁𝑏²₁𝑏₋₁−3𝛾𝑎₁𝑏²₁𝑏₋₁,𝑅₁=−3𝑐𝑎₀𝑏₁𝑏²₀−3𝛾𝑎₀𝑏₁𝑏²₀−2𝑎₀𝑏₁𝑏²₀−𝑐𝑎₁𝑏³₀−𝛾𝑎₁𝑏³₀−3𝑎²₀𝑎₁𝑏₀+4𝜉₀𝑏₁𝑏³₀+2𝑎₁𝑏³₀+3𝑎³₀𝑏₁−6𝑐𝑎₁𝑏₁𝑏₀𝑏₋₁−6𝛾𝑎₁𝑏₁𝑏₀𝑏₋₁+22𝑎₀𝑏²₁𝑏₋₁−10𝑎₋₁𝑏²₁𝑏₀−3𝑎²₁𝑎₋₁𝑏₀−15𝑎²₁𝑎₀𝑏₋₁−12𝑎₁𝑏₁𝑏₀𝑏₋₁−3𝑐𝑎₀𝑏²₁𝑏₋₁−3𝑐𝑎₋₁𝑏²₁𝑏₀−3𝛾𝑎₀𝑏²₁𝑏₋₁−3𝛾𝑎₋₁𝑏²₁𝑏₀+18𝑎₁𝑏₁𝑎₀𝑎₋₁+12𝜉₀𝑏²₁𝑏₀𝑏₋₁,𝑅₀=−3𝑐𝑎₋₁𝑏₁𝑏²₀−6𝛾𝑎₀𝑏₁𝑏₀𝑏₋₁+𝜉₀𝑏⁴₀−𝑐𝑎₀𝑏³₀+28𝑎₋₁𝑏²₁𝑏₋₁−28𝑎₁𝑏²₋₁𝑏₁+8𝑎₁𝑏₋₁𝑏²₀−6𝑐𝑎₀𝑏₁𝑏₀𝑏₋₁−8𝑎₋₁𝑏₁𝑏²₀−3𝑐𝑎₁𝑏²₋₁𝑏₁−3𝑐𝑎₁𝑏²₀𝑏₋₁−3𝑐𝑎₋₁𝑏²₁𝑏₋₁−𝛾𝑎₀𝑏³₀−3𝛾𝑎₁𝑏²₋₁𝑏₁−3𝛾𝑎₁𝑏²₀𝑏₋₁−3𝛾𝑎₋₁𝑏²₁𝑏₋₁−3𝛾𝑎₋₁𝑏²₀𝑏₁+12𝜉₀𝑏₁𝑏²₀𝑏₋₁−12𝑎₁𝑎²₀𝑏₋₁−12𝑎²₁𝑏₋₁𝑎₋₁+12𝑎₁𝑎²₋₁𝑏₁+12𝑏₁𝑎²₀𝑎₋₁,𝑅₋₁=−3𝑐𝑎₀𝑏₋₁𝑏²₀−3𝛾𝑎₀𝑏₋₁𝑏²₀+2𝑎₀𝑏₋₁𝑏²₀−𝑐𝑎₋₁𝑏³₀−𝛾𝑎₋₁𝑏³₀+3𝑎²₀𝑎₋₁𝑏₀+4𝜉₀𝑏³₀𝑏₋₁−2𝑎₋₁𝑏³₀−3𝑎³₀𝑏₋₁−6𝑐𝑎₋₁𝑏₁𝑏₀𝑏₋₁−6𝛾𝑎₋₁𝑏₁𝑏₀𝑏₋₁+10𝑎₁𝑏²₋₁𝑏₀−22𝑎₀𝑏²₋₁𝑏₁+3𝑎₁𝑎²₋₁𝑏₀+15𝑎₀𝑎²₋₁𝑏₁−12𝑎₋₁𝑏₁𝑏₀𝑏₋₁−3𝑐𝑎₁𝑏₀𝑏²₋₁−3𝑐𝑎₀𝑏₁𝑏²₋₁−3𝛾𝑎₁𝑏²₋₁𝑏₀−3𝛾𝑎₀𝑏₁𝑏²₋₁−18𝑎₁𝑎₋₁𝑎₀𝑏₋₁+12𝜉₀𝑏²₋₁𝑏₀𝑏₁,𝑅₋₂=10𝑎₁𝑏³₋₁+6𝑎³₋₁𝑏₁−𝑐𝑎₁𝑏³₋₁−𝛾𝑎₁𝑏³₋₁−10𝑎₋₁𝑏₁𝑏²₋₁−6𝑎₁𝑎²₋₁𝑏₋₁+4𝜉₀𝑏³₋₁𝑏₁−2𝑎₀𝑏₀𝑏²₋₁+2𝑎₋₁𝑏²₀𝑏₋₁+6𝑎₀𝑎²₋₁𝑏₀−6𝑎₋₁𝑎²₀𝑏₋₁+6𝜉₀𝑏²₋₁𝑏²₀−3𝑐𝑎₀𝑏₀𝑏²₋₁−3𝑐𝑎₋₁𝑏₋₁𝑏²₀−3𝛾𝑎₀𝑏₀𝑏²₋₁−3𝛾𝑎₋₁𝑏²₀𝑏₋₁−3𝑐𝑎₋₁𝑏²₋₁𝑏₁−3𝛾𝑎₋₁𝑏²₋₁𝑏₁,𝑅₋₃=2𝑎₀𝑏³₋₁+3𝑎³₋₁𝑏₀−𝑐𝑎₀𝑏³₋₁−𝛾𝑎₀𝑏³₋₁−2𝑎₋₁𝑏₀𝑏²₋₁−3𝑎²₋₁𝑎₀𝑏₋₁+4𝜉₀𝑏₀𝑏³₋₁−3𝑐𝑎₋₁𝑏₀𝑏²₋₁−3𝛾𝑎₋₁𝑏₀𝑏²₋₁,𝑅₋₄=𝜉₀𝑏⁴₋₁−𝑐𝑎₋₁𝑏³₋₁−𝛾𝑎₋₁𝑏³₋₁. (58) Solving this system of algebraic equations by using symbolic computation, we obtain the following results: 𝑎₁=𝑎²₀√68𝑏₋₁,𝑎₀=𝑎₀,𝑎₋₁=𝑏₋₁23,𝑏₁=3𝑎²₀8𝑏₋₁,𝑏₀=−𝑎₀√6,𝑏₋₁=𝑏₋₁,𝑐=𝑐,𝛾=−𝑐,𝜉₀=0, (59) where 𝑎₀ and 𝑏₋₁≠0 are free parameters.

From (59), substituting these results into (56), we obtain the following exact solution: 𝑢()=𝑎𝑥,𝑡₂₀√68𝑏₋₁𝑥exp^𝛼Γ()−1+𝛼𝑐𝑡^𝛼Γ(1+𝛼)+𝑎₀+𝑏₋₁23−𝑥exp^𝛼Γ()−1+𝛼𝑐𝑡^𝛼Γ()×1+𝛼3𝑎²₀8𝑏₋₁𝑥exp^𝛼Γ()−1+𝛼𝑐𝑡^𝛼Γ(1+𝛼)−𝑎₀√6+𝑏₋₁−𝑥exp^𝛼Γ()−1+𝛼𝑐𝑡^𝛼Γ(1+𝛼)⁻¹. (60)

7. Conclusion

In this paper, we have successfully developed fractional complex transform with the help of exp-function method to obtain exact solution of some fractional differential equations. The fractional complex transform and exp-function methods are extremely simple but effective and powerful for solving fractional differential equations. These methods are accessible to solve other similar nonlinear equations in fractional calculus. To our knowledge, these new solutions have not been reported in former literature; they may be of significant importance for the explanation of some special physical phenomena.

Conflict of Interests

The authors declare that there is no conflict of interests in this paper.