AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 426462 10.1155/2013/426462 426462 Research Article Fractional Complex Transform and exp-Function Methods for Fractional Differential Equations 0000-0001-9394-4681 Bekir Ahmet 1 Güner Özkan 2 Cevikel Adem C. 3 Bayram Mustafa 1 Eskisehir Osmangazi University Art-Science Faculty Department of Mathematics and Computer Science 26480 Eskişehir Turkey ogu.edu.tr 2 Dumlupınar University School of Applied Sciences Department of Management Information Systems 43100 Kütahya Turkey dpu.edu.tr 3 Yildiz Technical University Education Faculty Department of Mathematics Education 34220 Istanbul Turkey yildiz.edu.tr 2013 9 6 2013 2013 01 03 2013 21 03 2013 2013 Copyright © 2013 Ahmet Bekir et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 . The fractional differential equations have been investigated by many authors .

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

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  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 46, 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 , Grünwald-Letnikov’s fractional derivative , the Riemann-Liouville derivative , Jumarie’s modified Riemann-Liouville derivative [23, 24]. Jumarie’s derivative is defined as (1)Dtαf(t)=1Γ(1-α)ddt0t(t-ξ)-α[f(ξ)-f(0)]dξ,0<α<1, where f:RR, tf(t) denotes a continuous (but not necessarily first-order-differentiable) function. We list some important properties for the modified Riemann-Liouville derivative as follows.

Assume that f(t) denotes a continuous RR function. We use the following equality for the integral with respect to (dt)α: (2)Itαf(t)=1Γ(α)0t(t-ξ)α-1f(ξ)dξ,Itαf(t)=1Γ(α+1)0tf(ξ)(dt)α,0<α1.

Some useful formulas include (3)f(α)[g(t)]=dfdtg(α)(t),(4)Dtαtγ=Γ(1+γ)Γ(1+γ-α)tγ-α,(5)(dt)β=tβ.

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: (6)Dtα(u(t)v(t))=v(t)Dtαu(t)+u(t)Dtαv(t),Dtαf[u(t)]=fu[u(t)]Dtαu(t)=Duα  f[u(t)](u(t))α.

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 (7)F(u,Dtαu,Dxβu,DtαDtαu,DtαDxβu,DxβDxβu,)=0,0<α,β<1, 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: (8)u(x,t)=U(ξ),ξ=τxβΓ(1+β)+λtαΓ(1+α), where τ and λ are nonzero arbitrary constants; we can rewrite (7) in the following nonlinear ordinary differential equation: (9)Q(U,U,U′′,U′′′,)=0, 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 , we assume that the wave solution can be expressed in the following form: (10)U(ξ)=n=-cdanexp[nξ]m=-pqbmexp[mξ], where p,q,c, and d are positive integers which are known to be further determined and an and bm are unknown constants. We can rewrite (10) in the following equivalent form: (11)U(ξ)=a-cexp[-cξ]++adexp[dξ]b-pexp[-pξ]++bqexp[qξ].

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 .

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] (12)  αutα=2x2(u2)+2y2(u2)+h(u2-r),t>0,0<α1,x,yR, 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: (13)u(x,y,t)=U(ξ),ξ=νx+iνy-ctαΓ(1+α), where c and ν are constants and i2=-1.

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

Balancing the order of U and U2 in (14), we get (15)U=c1exp[-(c+p)ξ]+c2exp[-2pξ]+,U2=c3exp[-2cξ]+c4exp[-2pξ]+, where ci are determined coefficients only for simplicity. Balancing highest order of exp-function in (15), we obtain (16)-(p+c)=-2c, which leads to the result that (17)p=c. In the same way to determine the values of d and q, we balance the linear term of the lowest order in (14): (18)U=+d1exp[(q+d)ξ]+d2exp[2qξ],U2=+d3exp[2dξ]+d4exp[2qξ], where di are determined coefficients only for simplicity. From (18), we have (19)q+d=2d, and this gives (20)q=d.

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

Substituting (21) into (14) and by the help of symbolic computation, we have (22)1A[R2exp(2ξ)+R1exp(ξ)+R0+R-1exp(-ξ)+R-2exp(-2ξ)]=0, where (23)A=(b-1exp(-ξ)+b0+b1exp(ξ))2,R2=ha12-hrb12,R1=ca1b0-ca0b1+2ha1a0-2hrb1b0,R0=-2ca-1b1+2ca1b-1-2hrb1b-1-hrb02+2ha1a-1+ha02,R-1=-ca-1b0+ca0b-1+2ha0a-1-2hrb0b-1,R-2=ha-12-hrb-12.

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

Case 1.

Consider (24)a1=rb02-a024rb-1,a0=a0,a-1=-rb-1,b1=rb02-a024rb-1,b0=b0,b-1=b-1,c=2hr, where a0,b0, and b-1 are free parameters which exist provided that b-10. Substituting these results into (21), we get the following exact solution: (25)u(x,y,t)=×(+b0+b-1exp(-(νx+iνy-2hrtαΓ(1+α))))-1rb02-a024rb-1exp(νx+iνy-2hrtαΓ(1+α))+a0-rb-1exp(-(νx+iνy-2hrtαΓ(1+α)))×(+b0+b-1exp(-(νx+iνy-2hrtαΓ(1+α)))rb02-a024rb-1exp(νx+iνy-2hrtαΓ(1+α))+b0+b-1exp(-(νx+iνy-2hrtαΓ(1+α))))-1.

Case 2.

Consider (26)a1=a024rb-1,a0=a0,a-1=-rb-1,b1=-a024rb-1,b0=0,b-1=b-1,c=2hr, where a0 and b-1 are free parameters, which exist provided that b-10. Substituting these results into (21), we obtain the following exact solution: (27)u(x,y,t)=a024rb-1exp(νx+iνy-2hrtαΓ(1+α))+a0-rb-1exp(-(νx+iνy-2hrtαΓ(1+α)))×(+b-1exp(-(νx+iνy-2hrtαΓ(1+α)))-a024rb-1exp(νx+iνy-2hrtαΓ(1+α))+b-1exp(-(νx+iνy-2hrtαΓ(1+α))))-1.

5. Time Fractional Burgers Equation

We consider the one-dimensional time fractional Burgers equation with the value problem  (28)αutα+εuux-ν2ux2=0,t>0,0<α1,(29)u(x,0)=g(x), 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: (30)u(x,t)=U(ξ),ξ=λx-ctαΓ(1+α), where λ and c are nonzero constants.

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

Integrating (31) with respect to ξ yields (32)-cU+λε2U2-λ2νU+ξ0=0, 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 (33)U=c1exp[-(c+p)ξ]+c2exp[-2pξ]+,U2=c3exp[-2cξ]+c4exp[-2pξ]+, where ci are determined coefficients only for simplicity. Balancing the highest order of exp-function in (33), we have (34)-(p+c)=-2c, which leads to the result that (35)p=c. Similarly, to determine the values of d and q, we balance the linear term of the lowest order in (32): (36)U=+d1exp[(q+d)ξ]+d2exp[2qξ],U2=+d3exp[2dξ]+d4exp[2qξ], where di are determined coefficients only for simplicity. From (36), we obtain (37)q+d=2d, and this gives (38)q=d.

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

Substituting (39) into (32) and by the help of symbolic computation, we have (40)1A[R2exp(2ξ)+R1exp(ξ)+R0+R-1exp(-ξ)+R-2exp(-2ξ)]=0, where (41)A=(b-1exp(-ξ)+b0+b1exp(ξ))2,R2=ξ0b12-ca1b1+12λεa12,R1=-λ2νa1b0+λεa1a0+λ2νa0b1-ca0b1-ca1b0+2ξ0b1b0,R0=-2λ2νa1b-1+2λ2νa-1b1-ca0b0+λεa1a-1+ξ0b02-ca1b-1-ca-1b1+12λεa02+2ξ0b1b-1,R-1=λεa0a-1+λ2νa-1b0-λ2νa0b-1+2ξ0b0b-1-ca0b-1-ca-1b0,R-2=ξ0b-12-ca-1b-1+12λεa-12

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

Case 1.

Consider (42)a1=b1(εa-1-4λνb-1)εb-1,a0=0,a-1=a-1,b1=b1,b0=0,b-1=b-1,λ=λ,ε=ε,ν=ν,ξ0=λa-1(εa-1-4λνb-1)2b-12,c=λ(εa-1-2λνb-1)b-1, where a-1 and b-1 are free parameters which exist provided that b-10  and εa-1-2λνb-10. Substituting these results into (39), we obtain the following exact solution: (43)u(x,t)=b1(εa-1-4λνb-1)εb-1exp(λx-ctαΓ(1+α))+a-1exp(-(λx-ctαΓ(1+α)))×(+b-1exp(-(νx+iνy-2hrtαΓ(1+α)))b1exp(λx-ctαΓ(1+α))+b-1exp(-(λx-ctαΓ(1+α))))-1.

Case 2.

Consider (44)a1=-(εa-12b02-2a-1b02λνb-1-2εa0b0b-1a-1+2λνa0b0b-12+εa02b-12)×(εa-1-λνb-1)×(4b-12λ2ν2)-1,b1=-ε(εa-12b02-2a-1b02λνb-1-2εa0b0b-1a-1+2λνa0b0b-12+εa02b-12)×(4b-13λ2ν2)-1,a0=a0,a-1=a-1,b0=b0,b-1=b-1,λ=λ,ε=ε,ν=ν,ξ0=λa-1(εa-1-2λνb-1)2b-12,c=λ(εa-1-λνb-1)b-1, where a-1,  a0,  b-1, and b0 are free parameters which exist provided that b-10  and εa-1-2λνb-10. Substituting these results into (39), we get the following exact solution: (45)u(x,t)=-(εa-12b02-2a-1b02λνb-1-2εa0b0b-1a-1+2λνa0b0b-12+εa02b-12)(εa-1-λνb-1)×(4b-12λ2ν2)-1×exp(λx-ctαΓ(1+α))+a0+a-1exp(-(λx-ctαΓ(1+α)))×(+b-1exp(-(λx-ctαΓ(1+α)))-ε(εa-12b02-2a-1b02λνb-1-2εa0b0b-1a-1+2λνa0b0b-12+εa02b-12)×(4b-13λ2ν2)-1×exp(λx-ctαΓ(1+α))+b0+b-1exp(-(λx-ctαΓ(1+α))))-1.

6. Space-Time Fractional Cahn-Hilliard Equation

We consider the space-time fractional Cahn-Hilliard equation  (46)Dtαu-γDxαu-6u(Dxαu)2-(3u2-1)Dx2αu+Dx4αu=0, where 0<α1 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 . Nevertheless we notice that this equation is very difficult to be solved and several articles investigated it .

Firstly, we consider the following transformations: (47)u(x,t)=U(ξ),ξ=xαΓ(1+α)-ctαΓ(1+α), where c is a nonzero constant.

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

Integrating (48) with respect to ξ yields (49)-cU-γU-3U2U+U+U′′′+ξ0=0, 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  to determine the values of p,q,c, and d. By simple calculation, we have (50)U′′′=c1exp[(-7p-c)ξ]+c2exp[-8pξ]+=c1exp[-(7p+c)ξ]+c2exp[-8pξ]+,U2U=c3exp[(-p-3c)ξ]+c4exp[-4pξ]+=c3exp[-(5p+3c)ξ]+c4exp[-8pξ]+, where ci are determined coefficients only for simplicity. Balancing the highest order of exp-function in (50), we have (51)-(7p+c)=-(5c+3p), which leads to the result that (52)p=c. Similarly, to determine the values of d and q, we balance the linear term of lowest order in (49) (53)U′′′=+d1exp[(7q+d)ξ]+d2exp[8qξ],U2U=+d3exp[(2d+6q)ξ]+d4exp[8qξ], where di are determined coefficients only for simplicity. From (53), we obtain (54)(6q+2d)=(d+7q), and this gives (55)q=d. For simplicity, we set p=c=1 and q=d=1, so (11) reduces to (56)u(ξ)=a1exp(ξ)+a0+a-1exp(-ξ)b1exp(ξ)+b0+b-1exp(-ξ).

Substituting (56) into (49) and by the help of symbolic computation, we obtain (57)  1A[R4exp(4ξ)+R3exp(3ξ)+R2exp(2ξ)+R1exp(ξ)+R0+R-1exp(-ξ)+R-2exp(-2ξ)+R-3exp(-3ξ)+R-4exp(-4ξ)]=0, where (58)A=(b-1exp(-ξ)+b0+b1exp(ξ))4,R4=ξ0b14-ca1b13-γa1b13,R3=-3a13b0-2a0b13-ca0b13-γa0b13+2a1b12b0+3a12a0b1+4ξ0b13b0-3γa1b12b0-3ca1b12b0,R2=-10a-1b13-6a13b-1-ca-1b13-γa-1b13+10a1b12b-1+6a12a-1b1-2a1b1b02+4ξ0b13b-1+2a0b12b0-6a12a0b0+6a1a02b1+6ξ0b12b02-3ca1b1b02-3ca0b12b0-3γa1b1b02-3γa0b12b0-3ca1b12b-1-3γa1b12b-1,R1=-3ca0b1b02-3γa0b1b02-2a0b1b02-ca1b03-γa1b03-3a02a1b0+4ξ0b1b03+2a1b03+3a03b1-6ca1b1b0b-1-6γa1b1b0b-1+22a0b12b-1-10a-1b12b0-3a12a-1b0-15a12a0b-1-12a1b1b0b-1-3ca0b12b-1-3ca-1b12b0-3γa0b12b-1-3γa-1b12b0+18a1b1a0a-1+12ξ0b12b0b-1,R0=-3ca-1b1b02-6γa0b1b0b-1+ξ0b04-ca0b03+28a-1b12b-1-28a1b-12b1+8a1b-1b02-6ca0b1b0b-1-8a-1b1b02-3ca1b-12b1-3ca1b02b-1-3ca-1b12b-1-γa0b03-3γa1b-12b1-3γa1b02b-1-3γa-1b12b-1-3γa-1b02b1+12ξ0b1b02b-1-12a1a02b-1-12a12b-1a-1+12a1a-12b1+12b1a02a-1,R-1=-3ca0b-1b02-3γa0b-1b02+2a0b-1b02-ca-1b03-γa-1b03+3a02a-1b0+4ξ0b03b-1-2a-1b03-3a03b-1-6ca-1b1b0b-1-6γa-1b1b0b-1+10a1b-12b0-22a0b-12b1+3a1a-12b0+15a0a-12b1-12a-1b1b0b-1-3ca1b0b-12-3ca0b1b-12-3γa1b-12b0-3γa0b1b-12-18a1a-1a0b-1+12ξ0b-12b0b1,R-2=10a1b-13+6a-13b1-ca1b-13-γa1b-13-10a-1b1b-12-6a1a-12b-1+4ξ0b-13b1-2a0b0b-12+2a-1b02b-1+6a0a-12b0-6a-1a02b-1+6ξ0b-12b02-3ca0b0b-12-3ca-1b-1b02-3γa0b0b-12-3γa-1b02b-1-3ca-1b-12b1-3γa-1b-12b1,R-3=2a0b-13+3a-13b0-ca0b-13-γa0b-13-2a-1b0b-12-3a-12a0b-1+4ξ0b0b-13-3ca-1b0b-12-3γa-1b0b-12,R-4=ξ0b-14-ca-1b-13-γa-1b-13. Solving this system of algebraic equations by using symbolic computation, we obtain the following results: (59)a1=a0268b-1,a0=a0,a-1=b-123,b1=3a028b-1,b0=-a06,b-1=b-1,c=c,γ=-c,ξ0=0, where a0 and b-10 are free parameters.

From (59), substituting these results into (56), we obtain the following exact solution: (60)u(x,t)=a0268b-1exp(xαΓ(1+α)-ctαΓ(1+α))+a0+b-123exp(-(xαΓ(1+α)-ctαΓ(1+α)))×(+b-1exp(-(νx+iνy-2hrtαΓ(1+α)))3a028b-1exp(xαΓ(1+α)-ctαΓ(1+α))-a06+b-1exp(-(xαΓ(1+α)-ctαΓ(1+α))))-1.

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.

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