# The Extended Fractional(DξαG/G)-Expansion Method and Its Applications to a Space-Time Fractional Fokas Equation

Authors: Yunmei Zhao; Yinghui He
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Based on a fractional subequation and the properties of the modified Riemann-Liouville fractional derivative, we propose a new analytical method named extended fractional(DξαG/G)-expansion method for seeking traveling wave solutions of fractional partial differential equations. To illustrate the effectiveness of the method, we discuss a space-time fractional Fokas equation, many types of exact analytical solutions are obtained, and the solutions include hyperbolic function and trigonometric and negative exponential solutions.


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## Body

## 1. Introduction

The nonlinear fractional partial differential equations (FPDEs) have attracted much attention because of their potential applications in various fields of science, such as fluid mechanics (especially in viscoelastic flow and theory of viscoplasticity), medical (human tissue under mechanical load model), electrical engineering (ultrasound transmission), biochemistry (polymer and protein model), material diffusion (including normal diffusion and anomalous diffusion), signal processing, and control systems.The exact solutions of FPDEs can facilitate illustrating the structural information about the complex physics phenomena and help better understand the physical interpretation. Thus, it is an important and significant task to find more exact solutions of different forms for the FPDEs. In recent decades, many mathematicians and physicists have made significant achievements and also presented some effective methods, for example, the fractional subequation method [1–3], the Jacobi elliptic equation method [4], the fractional mapping method [5], the (G′/G)-expansion method [6], the extended fractional Riccati expansion method [7], the first integral method [8], and the fractional complex transform [9]. Due to these methods, various exact solutions or numerical solutions of FPDEs have been established successfully.Searching for exact solutions of nonlinear ODEs plays an important role in the study of physical phenomena and gradually becomes one of the most important and significant tasks. In the past several decades, both mathematicians and physicists have made many significant works in this direction and presented some effective methods, such as the global error minimization method [10].Recently, Feng [11] introduced a new method which is called the improved fractional (DξαG/G) method to show traveling wave solutions of nonlinear FDEs. The method is based on the homogeneous balance principle and Jumarie’s modified Riemann-Liouville derivative. By using the fractional Riccati equation AGDξ2αG-BGDξαG-C(DξαG)2-EG2=0, Feng obtained traveling wave solutions of the (2+1)-dimensional space-time fractional Nizhnik-Novikov-Veselov system and the space-time fractional KP-BBM equation.In order to get as many results as possible, we propose a new analytical method named extended fractional(DξαG/G)-expansion method which adds in negative power exponent to seek more general traveling wave solutions. In this paper, using the proposed method, we present some new exact solutions of a space-time fractional Fokas equation. The results suggest that the method not only is simple, effective, and straightforward, but also can be used for many other nonlinear FPDEs.The organization of the paper is as follows. In Section2, some basic properties of Jumarie’s modified Riemann-Liouville derivative are shown. In Section 3, we describe the extended fractional (DξαG/G)-expansion method for finding traveling wave solutions of the proposed FPDEs. In Section 4, we apply this method to obtain exact traveling wave solutions for a space-time fractional Fokas equation. The conclusion part is in Section 5.


## 2. Preliminaries

Jumarie’s modified Riemann-Liouville derivative of orderα is defined by the expression [1](1)Dxαfx=1Γ1-α∫0xx-ξ-α-1fξ-f0dξ,α<0,1Γ1-αddx∫0xx-ξ-αfξ-f0dξ,0<α<1,fα-nxn,n≤α<n+1,n≥1.The property for the proposed modified Riemann-Liouville derivatives is listed in [1] as follows:(2)Dxαxγ=Γγ+1Γγ+1-αxγ-α,γ>0.The above equations play an important role in fractional calculus in the following sections.


## 3. Description of the Extended Fractional(DξαG/G)-Expansion Method

In this section, the main steps of the extended fractional(DξαG/G)-expansion method are described as follows.Step 1.
Suppose that a fractional partial differential equation in the variablesx, y, t is given by(3)Pu,ut,ux,uy,∂αu∂xα,∂αu∂yα,∂αu∂tα,∂2αu∂x2α,∂2αu∂y2α,∂2αu∂t2α,…=0,0<α≤1,where ∂αu/∂xα, ∂αu/∂yα, and ∂αu/∂tα are Jumarie’s modified Riemann-Liouville derivatives of u with respect to x, y, and t, u=u(x,y,t) is an unknown function, and P is a polynomial in u and its various partial derivatives in which the highest order derivatives and nonlinear terms are involved.Step 2.
By using the traveling wave transformation,(4)ux,y,t=uξ,ξ=kx+ly+ωt,where k, l, and ω are constants to be determined later, the FDE (3) is reduced to the following nonlinear fractional ordinary differential equation (ODE) for u=u(ξ):(5)Pu,ωu′,ku′,lu′,kαDξαu,lαDξαu,ωαDξαu,…=0.Step 3.
We suppose the solutionu of (5) can be expressed in the following form:(6)u=∑i=0naiDξαGξGξi+∑j=1nbjDξαGξGξ-j,where ai, bj(i=0,1,2,…,n;j=1,2,…,n) are arbitrary constants to be determined later, the positive integer n can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (5), and G=G(ξ) satisfies the following fractional ordinary differential equation:(7)AGξDξ2αGξ-BGξDξαGξ-CDξαGξ2-EG2ξ=0,where DξαG(ξ) denotes the modified Riemann-Liouville derivative of order α for G(ξ) with respect to ξ.Step 4.
Substituting (6) into (5) and using (7) and collecting all terms with the same order of (DξαG/G) together, the left-hand side of (5) is converted into another polynomial in (DξαG/G). Equating each coefficient of this polynomial to zero yields a set of algebraic equations for ai, bj,(i=0,1,…,n,j=1,2,…,n), k, l, and ω.Step 5.
Solving the equations system in Step4 and using the general expressions for (DξαG/G), we can construct a variety of exact solutions for (3).
In order to obtain the general expressions for(DξαG/G) in (7), we suppose(8)Gξ=Hη,η=ξαΓ1+α.Then by use of (2) one has Dξαη=1 and furthermore, DξαG(ξ)=DξαH(η)=H′(η)Dξαη=H′(η), so (7) can be turned into the following second ordinary differential equation:(9)AHηH′′η-BHηH′η-CH′η2-EH2η=0.By the general expressions forH′(η)/H(η) in [12], one can obtain the following expressions for DξαG(ξ)/G(ξ):(1)
WhenB≠0, Ψ=A-C, and Ω=B2+4EΨ>0,(10)DξαGξGξ=B2Ψ+Ω2ΨC1sinh⁡Ω/2AΓ1+αξα+C2cosh⁡Ω/2AΓ1+αξαC1cosh⁡Ω/2AΓ1+αξα+C2sinh⁡Ω/2AΓ1+αξα.(2)
WhenB≠0, Ψ=A-C, and Ω=B2+4EΨ<0,(11)DξαGξGξ=B2Ψ+-Ω2Ψ-C1sin⁡-Ω/2AΓ1+αξα+C2cos⁡-Ω/2AΓ1+αξαC1cos⁡-Ω/2AΓ1+αξα+C2sin⁡-Ω/2AΓ1+αξα.(3)
WhenB≠0, Ψ=A-C, and Ω=B2+4EΨ=0,(12)DξαGξGξ=B2Ψ-AC2Γ1+αC1Γ1+α-C2ξαΨ.(4)
WhenB=0, Ψ=A-C, and Δ=ΨE>0,(13)DξαGξGξ=ΔΨC1sinh⁡Δ/AΓ1+αξα+C2cosh⁡Δ/AΓ1+αξαC1cosh⁡Δ/AΓ1+αξα+C2sinh⁡Δ/AΓ1+αξα.(5)
WhenB=0, Ψ=A-C, and Δ=ΨE<0,(14)DξαGξGξ=-ΔΨ-C1sin⁡-Δ/AΓ1+αξα+C2cos⁡-Δ/AΓ1+αξαC1cos⁡-Δ/AΓ1+αξα+C2sin⁡-Δ/AΓ1+αξα.


## 4. Applications of the Extended Fractional(DξαG/G) Method to the Space-Time Fractional Fokas Equation

We consider a space-time fractional Fokas equation [13]:(15)4∂2αq∂tα∂x1α-∂4αq∂x13α∂x2α+∂4αq∂x23α∂x1α+12∂αq∂x1α∂αq∂x2α+12q∂2αq∂x1α∂x2α-6∂2αq∂y1α∂y2α=0,0<α≤1,which is a transformed generalization of the (4+1)-dimensional Fokas equation(16)∂2q∂t∂x1=14∂4q∂x13∂x2-14∂4q∂x13∂x2-32∂2q2∂x1∂x2+32∂2q∂y1∂y2.Equation (16) is one of the new high-dimensional nonlinear wave Fokas equations recently obtained by extending the integrable KP equation and DS equation.To solve (15), we take the following traveling wave transformation:(17)q=qξ,ξ=k1x1+k2x2+l1y1+l2y2+ωt;then (15) is reduced into a nonlinear fractional ODE in the form(18)k1αk23α-k13αk2αDξ4αq+4ωαk1α-6l1αl2αDξ2αq+12k1αk2αDξαq2+12k1αk2αqDξ2αq=0.BalancingDξ4αq and qDξ2αq, we have n+4=2n+2⇒n=2. So we have(19)qξ=a0+a1DξαGξGξ+a2DξαGξGξ2+b1DξαGξGξ-1+b2DξαGξGξ-2,where a0, a1, a2, b1, and b2 are constants to be determined later, and function G(ξ) satisfies (7).Substituting (19) together with (7) into (18), the left-hand side of (18) is converted into polynomials in (DξαG/G)i, (i=…,-2,-1,0,1,2,…). We collect each coefficient of these resulting polynomials to zero, which yields a set of simultaneous algebraic equations for a0, a1, a2, b1, b2, k1, k2, l1, l2, and ω. Solving this system of algebraic equations, with the aid of Maple, we obtain (20)1a0=k12α-k22αB2-8EΨ12A2+3l1αl2α-2ωαk1α6k1αk2α,a1=k22α-k12αΨBA2,a2=Ψ2k12α-k22αA2,b1=0,b2=0.(21)2a0=k12α-k22αB2-8EΨ12A2+3l1αl2α-2ωαk1α6k1αk2α,a1=0,a2=0,b1=k12α-k22αBEA2,b2=E2k12α-k22αA2,where Ψ=A-C, A, B, C, E, k1, k2, l1, l2, and ω are free parameters.Substituting (20) into (19) and combining with (10)–(14), we can obtain the following exact traveling wave solutions to (15).(1.1) When B≠0, Ψ=A-C, and Ω=B2+4EΨ>0, (22)q1ξ=k12α-k22αB2-8EΨ12A2+3l1αl2α-2ωαk1α6k1αk2α+k22α-k12αBA2B2+Ω2C1sinh⁡Ω/2AΓ1+αξα+C2cosh⁡Ω/2AΓ1+αξαC1cosh⁡Ω/2AΓ1+αξα+C2sinh⁡Ω/2AΓ1+αξα+k12α-k22αA2B2+Ω2C1sinh⁡Ω/2AΓ1+αξα+C2cosh⁡Ω/2AΓ1+αξαC1cosh⁡Ω/2AΓ1+αξα+C2sinh⁡Ω/2AΓ1+αξα2.(1.2) When B≠0, Ψ=A-C, and Ω=B2+4EΨ<0, (23)q2ξ=k12α-k22αB2-8EΨ12A2+3l1αl2α-2ωαk1α6k1αk2α+k22α-k12αBA2B2+-Ω2-C1sin⁡-Ω/2AΓ1+αξα+C2cos⁡-Ω/2AΓ1+αξαC1cos⁡-Ω/2AΓ1+αξα+C2sin⁡-Ω/2AΓ1+αξα+k12α-k22αA2B2+-Ω2-C1sin⁡-Ω/2AΓ1+αξα+C2cos⁡-Ω/2AΓ1+αξαC1cos⁡-Ω/2AΓ1+αξα+C2sin⁡-Ω/2AΓ1+αξα2.(1.3) When B≠0, Ψ=A-C, and Ω=B2+4EΨ=0,(24)q3ξ=3l1αl2α-2ωαk1α6k1αk2α+C22k12α-k22αΓ1+α2C1Γ1+α-C2ξα2.(1.4) When B=0, Ψ=A-C, and Δ=ΨE>0, (25)q4ξ=-2k12α-k22αΔ3A2+3l1αl2α-2ωαk1α6k1αk2α+k12α-k22αΔA2C1sinh⁡Δ/AΓ1+αξα+C2cosh⁡Δ/AΓ1+αξαC1cosh⁡Δ/AΓ1+αξα+C2sinh⁡Δ/AΓ1+αξα2.(1.5) When B=0, Ψ=A-C, and Δ=ΨE<0, (26)q5ξ=-2k12α-k22αΔ3A2+3l1αl2α-2ωαk1α6k1αk2α-k12α-k22αΔA2-C1sin⁡-Δ/AΓ1+αξα+C2cos⁡-Δ/AΓ1+αξαC1cos⁡-Δ/AΓ1+αξα+C2sin⁡-Δ/AΓ1+αξα2,where ξ=k1x1+k2x2+l1y1+l2y2+ωt, k1, k2, l1, l2, and ω are arbitrary constants.Substituting (21) into (19) and combining with (10)–(14), we can obtain the following exact traveling wave solutions to (15).(2.1) When B≠0, Ψ=A-C, and Ω=B2+4EΨ>0, (27)q6ξ=k12α-k22αB2-8EΨ12A2+3l1αl2α-2ωαk1α6k1αk2α+k12α-k22αBEA2B/2Ψ+Ω/2ΨC1sinh⁡Ω/2AΓ1+αξα+C2cosh⁡Ω/2AΓ1+αξα/C1cosh⁡Ω/2AΓ1+αξα+C2sinh⁡Ω/2AΓ1+αξα+k12α-k22αE2A2B/2Ψ+Ω/2ΨC1sinh⁡Ω/2AΓ1+αξα+C2cosh⁡Ω/2AΓ1+αξα/C1cosh⁡Ω/2AΓ1+αξα+C2sinh⁡Ω/2AΓ1+αξα2.(2.2) When B≠0, Ψ=A-C, and Ω=B2+4EΨ<0, (28)q7ξ=k12α-k22αB2-8EΨ12A2+3l1αl2α-2ωαk1α6k1αk2α+k12α-k22αEBA2B/2Ψ+-Ω/2Ψ-C1sin⁡-Ω/2AΓ1+αξα+C2cos⁡-Ω/2AΓ1+αξα/C1cos⁡-Ω/2AΓ1+αξα+C2sin⁡-Ω/2AΓ1+αξα+k12α-k22αE2A2B/2Ψ+-Ω/2Ψ-C1sin⁡-Ω/2AΓ1+αξα+C2cos⁡-Ω/2AΓ1+αξα/C1cos⁡-Ω/2AΓ1+αξα+C2sin⁡-Ω/2AΓ1+αξα2.(2.3) When B≠0, Ψ=A-C, and Ω=B2+4EΨ=0,(29)q8ξ=3l1αl2α-2ωαk1α6k1αk2α+B2C22k12α-k22αΓ1+α22AC2-BC1Γ1+α+BC2ξα2.(2.4) When B=0, Ψ=A-C, and Δ=ΨE>0, (30)q9ξ=-2k12α-k22αΔ3A2+3l1αl2α-2ωαk1α6k1αk2α+k12α-k22αΔA2C1cosh⁡Δ/AΓ1+αξα+C2sinh⁡Δ/AΓ1+αξαC1sinhΔ/AΓ1+αξα+C2cosh⁡Δ/AΓ1+αξα2.(2.5) When B=0, Ψ=A-C, and Δ=ΨE<0, (31)q10ξ=-2k12α-k22αΔ3A2+3l1αl2α-2ωαk1α6k1αk2α-k12α-k22αΔA2C1cos⁡-Δ/AΓ1+αξα+C2sin⁡-Δ/AΓ1+αξαC2cos⁡-Δ/AΓ1+αξα-C1sin⁡-Δ/AΓ1+αξα2,where ξ=k1x1+k2x2+l1y1+l2y2+ωt, k1, k2, l1, l2, and ω are arbitrary constants.In particular, ifC1=0, but C2≠0, or C2=0, but C1≠0, then q(ξ) changes as follows:(3.1) When B≠0, Ψ=A-C, and Ω=B2+4EΨ>0, (32)q11ξ=k12α-k22αB2-8EΨ12A2+3l1αl2α-2ωαk1α6k1αk2α+k22α-k12αBA2B2+Ω2tanh⁡Ωξα2AΓ1+α+k12α-k22αA2B2+Ω2tanh⁡Ωξα2AΓ1+α2,(33)q12ξ=k12α-k22αB2-8EΨ12A2+3l1αl2α-2ωαk1α6k1αk2α+k22α-k12αBA2B2+Ω2coth⁡Ωξα2AΓ1+α+k12α-k22αA2B2+Ω2coth⁡Ωξα2AΓ1+α2,(34)q13ξ=k12α-k22αB2-8EΨ12A2+3l1αl2α-2ωαk1α6k1αk2α+k12α-k22αBEA2B/2Ψ+Ω/2Ψtanh⁡Ωξα/2AΓ1+α+E2k12α-k22αA2B/2Ψ+Ω/2Ψtanh⁡Ωξα/2AΓ1+α2,(35)q14ξ=k12α-k22αB2-8EΨ12A2+3l1αl2α-2ωαk1α6k1αk2α+k12α-k22αBEA2B/2Ψ+Ω/2Ψcoth⁡Ωξα/2AΓ1+α+E2k12α-k22αA2B/2Ψ+Ω/2Ψcoth⁡Ωξα/2AΓ1+α2.(3.2) When B≠0, Ψ=A-C, and Ω=B2+4EΨ<0, (36)q15ξ=k12α-k22αB2-8EΨ12A2+3l1αl2α-2ωαk1α6k1αk2α+k22α-k12αBA2B2--Ω2tan⁡-Ωξα2AΓ1+α+k12α-k22αA2B2--Ω2tan⁡-Ωξα2AΓ1+α2,q16ξ=k12α-k22αB2-8EΨ12A2+3l1αl2α-2ωαk1α6k1αk2α+k22α-k12αBA2B2+-Ω2cot⁡-Ωξα2AΓ1+α+k12α-k22αA2B2+-Ω2cot⁡-Ωξα2AΓ1+α2,q17ξ=k12α-k22αB2-8EΨ12A2+3l1αl2α-2ωαk1α6k1αk2α+k12α-k22αBEA2B/2Ψ--Ω/2Ψtan⁡-Ωξα/2AΓ1+α+E2k12α-k22αA2B/2Ψ--Ω/2Ψtan⁡-Ωξα/2AΓ1+α2,q18ξ=k12α-k22αB2-8EΨ12A2+3l1αl2α-2ωαk1α6k1αk2α+k12α-k22αBEA2B/2Ψ+-Ω/2Ψcot⁡-Ωξα/2AΓ1+α+E2k12α-k22αA2B/2Ψ+-Ω/2Ψcot⁡-Ωξα/2AΓ1+α2.(3.3) When B=0, Ψ=A-C, and Δ=ΨE>0,(37)q19ξ=3l1αl2α-2ωαk1α6k1αk2α+k12α-k22αΔA2-23+tanh2⁡ΔξαAΓ1+α,(38)q20ξ=3l1αl2α-2ωαk1α6k1αk2α+k12α-k22αΔA2-23+coth2⁡ΔξαAΓ1+α.(3.4) When B=0, Ψ=A-C, and Δ=ΨE<0,(39)q21ξ=3l1αl2α-2ωαk1α6k1αk2α-k12α-k22αΔA223+tan2⁡-ΔξαAΓ1+α,q22ξ=3l1αl2α-2ωαk1α6k1αk2α-k12α-k22αΔA223+cot2⁡-ΔξαAΓ1+α,where ξ=k1x1+k2x2+l1y1+l2y2+ωt, k1, k2, l1, l2, and ω are arbitrary constants.


## 5. Figures of Some Exact Solutions

In this section, some typical wave figures are given as Figures1 and 2.Figure 1
2D and 3D figures of solutionq11(ξ) in (32) with k1=4, k2=0.2, l1=4, l2=1, A=1, C=-3, E=-1, ω=20, x2=0.01, y1=0.01, y2=0.01, B=5.
(a)
α = 0.45; 0.75; 0.9; 1; t=0.5
(b)
α
=
1
,
-
2
≤
t
≤
2Figure 2
2D and 3D figures of solutionq19(ξ) in (37) with k1=0.1, k2=1, l1=0.1, l2=1, A=2, C=1, E=5, ω=0.2, x2=0.3, y1=0.2, y2=0.5.
(a)
α = 0.45; 0.75; 0.9; 1; t=0.5
(b)
α
=
1
,
-
1
≤
t
≤
1


## 6. Conclusions

Finding exact traveling wave solutions of FPDEs is an important and difficult work. Based on Feng’s [11] work, by adding in negative power exponents, a new analytical method named extended fractional (DξαG/G)-expansion method is proposed. It can help to seek more general traveling wave solutions. In this paper, using this new method, we obtain more new exact solutions of the space-time fractional Fokas equation. These exact solutions include hyperbolic function and trigonometric function which are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. To the best of our knowledge, the solutions obtained in this paper have not been reported in other literatures. The traveling wave transformation used for ξ here ensures that a certain fractional partial differential equation can be converted into another fractional ordinary differential equation, and the solutions of the latter are assumed to possess forms in one certain polynomial in G(ξ), where G(ξ) satisfies a given fractional ordinary differential equation denoted by (7), and the degree of the polynomial can be determined by the homogeneous balance principle. It is worthy of further study to extend other existing methods, like the generalized fractional Jacobi elliptic equation-based subequation method [4] used for differential equations, to fractional differential equations. This is our task in the future.



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