# Solutions of Fractional Konopelchenko-Dubrovsky and Nizhnik-Novikov-Veselov Equations Using a Generalized Fractional Subequation Method

Authors: Yanqin Liu; Limei Yan
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A new generalized fractional subequation method based on the relationship of fractional coupled equations is proposed. This method is applied to the space-time fractional coupled Konopelchenko-Dubrovsky equations and Nizhnik-Novikov-Veselov equations. As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions, and rational solutions. It is observed that the proposed approach provides a simple and reliable tool for solving many other fractional coupled differential equations.


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## Body

## 1. Introduction

Fractional calculus is one of the generalizations of ordinary calculus. Generally speaking, there are two kinds of fractional derivatives. One is nonlocal fractional derivative [1, 2], that is, Caputo derivative and Riemann-Liouville derivative which have been used successfully in various fields of science and engineering. The other one is the local fractional derivative, that is, Kolwankar-Gangal (K-G) derivative [3, 4], Chen’s fractal derivative [5, 6], Cresson’s derivative [7], and Jumarie’s modified Riemann-Liouville derivative [8]. At the same time, fractional differential equations have attracted much attention in a variety of applied sciences. However, we have difficulty in finding exact analytical solutions [9–12] of fractional differential equations that appear more and more frequently in different research areas and engineering applications. So, numerical methods have been used to handle these equations, and some semianalytical techniques [13–16] have also largely been used to solve these equations.Based on homogeneous balance principle [17], Jumarie’s modified Riemann-Liouville derivative [8], and symbolic computation, S. Zhang and H.-Q. Zhang proposed a fractional subequation method to search for explicit solutions of FDEs. By using this method, S. Zhang and H.-Q. Zhang successfully obtained some exact solutions of space-time fractional biological population model and fractional Fokas equation [18]. Jafari et al. have given some solutions of the fractional Cahn-Hilliard and Klein-Gordon equations [19]. Tang et al. [20] proposed a generalized fractional subequation method for fractional differential equations with variable coefficients. Guo et al. [21] and Zhao et al. [22] both improved the fractional subequation and applied to space-time fractional coupled differential equations; in their paper, they choose two or three appropriate ansätz. However, for some coupled equations [23, 24], even some fractional coupled equations, we can get the relationship of the functions. So, we propose a new generalized fractional subequation which chooses only one appropriate ansätz and use this method to solve the following two NFDEs.(1) The space-time fractional coupled Konopelchenko-Dubrovsky (KD) equations in the form(1)Dtαu-Dx3αu-6buDxαu+32a2u2Dxαu-3Dyαv+3aDxαuv=0,Dyαu=Dxαv,
which is a transformed generalization of the KD equations [25], where a and b are real constants. Equation (1) is a fractional evolution equation on two spatial dimensions and one temporal, where x and y are the running coordinates, t is the time, and u=u(x,y,t) and v=v(x,y,t) are the amplitudes of the relevant waves. Dtα(·) and Dxα(·) are Jumarie’s modified Riemann-Liouville derivative of order α defined in Section 2, 0<α≤1. The Jumarie’s modified Riemann-Liouville derivative has many interesting properties. The KD equations can be used to describe the ocean dynamics, fluid mechanics, and plasma physics, and the Gardner, KP, modified KP, and KD equations are all the special cases of (1). When α=1, uy=0, (1) is the Gardner equation (combined KdV and modified equation). When α=1, a=0, (1) is the well-known Kadomtsev-Petviashvili (KP) equation, and modified KP equation reads from (1) for α=1, b=0.(2) The space-time fractional coupled Nizhnik-Novikov-Veselov (NNV) equation in the form(2)Dtαu=ADx3αu+BDy3αu-3AuDxαv-3AvDxαu-3BuDyαw-3BwDyαu,Dxαu=Dyαv,Dyαu=Dxαw,
where 0<α≤1, A and B are given constants satisfying A+B≠0, and u, v, and w are the functions of (x,y,t), the case when α→1 was studied in [26]. NNV equations have been studied over several areas of physics including condense matter physics, fluid mechanics, plasma physics, and optics.The rest of this paper is organized as follows. In Section2, some basic definitions of Jumarie’s modified Riemann-Liouville derivative and the main steps of the generalized fractional subequation method are given. In Section 3, we construct the exact solutions of above space-time fractional coupled equations via this new generalized method. Some conclusions and discussions are shown in Section 4.


## 2. Jumarie’s Modified Riemann-Liouville Derivative and Generalized Fractional Subequation Method

The Jumarie’s modified Riemann-Liouville derivative [8] of order α time-fractional derivative operator of order α>0 is defined as
(3)Dtαf={1Γ(1-α)∫0t‍(t-ξ)-α-1(f(ξ)-f(0))dξ,α<0,1Γ(1-α)ddt∫0t‍(t-ξ)-α(f(ξ)-f(0))dξ,0<α<1,(f(n)(t))α-n,n≤α<n+1,n≥1.Some properties for the proposed modified Riemann-Liouville derivative are listed in [8] as follows:
(4)Dtαtδ=Γ(1+δ)Γ(1+δ-α)tδ-α,δ>0,(5)Dtα(f(t)g(t))=g(t)Dtαf(t)+f(t)Dtαg(t),(6)Dtαf[g(t)]=fg′[g(t)]Dtαg(t),(7)Dtαf[g(t)]=Dgαf[g(t)](g′(t))α.
The above equations play an important role in fractional calculus in the following sections.we propose a generalized fractional subequation method; the essential steps of this method are described as follows.Step 1.
Suppose that NFDEs with independent variablesX=(x1,x2,x3,…,xm,t) are given by
(8)P(u,v,ut,ux1,…,vt,vx1,…,Dtαu,Dx1αu,…,Dtαv,Dx1αv,…)=0,0<α≤1,Q(u,v,ut,ux1,…,vt,vx1,…,Dtαu,Dx1αu,…,Dtαv,Dx1αv,…)=0,0<α≤1,
where Dtα(·) and Dx1α(·) are Jumarie’s modified Riemann-Liouville derivative with respect to t and x1, u=u(x,t), v=v(x,t) are unknown functions, P is a polynomial in u, v, and their various partial derivatives, Q is a polynomial in u,v, and their various partial derivatives, and the highest order derivatives and nonlinear terms are involved.Step 2.
By using the traveling wave transformations(9)u(x1,…,xm,t)=u(ξ),v(x1,…,xm,t)=v(ξ),ξ=k1x1+⋯+kmxm+ct,
where c is a constant to be determined later, the NFDE (7) is reduced to the following nonlinear fractional ordinary differential equation (ODE) for u(ξ) and v(ξ):
(10)P(u,v,cu′,k1u′,…,cv′,k1v′,…,cαDξαu,k1αDξαu,…,cαDξαv,k1αDξαv,…)=0,Q(u,v,cu′,k1u′,…,cv′,k1v′,…,cαDξαu,k1αDξαu,…,cαDξαv,k1αDξαv,…)=0,Step 3.
For some coupled equations, we get the relationship(11)v=f(u),
and substituting into (8), one has
(12)Q(u,v,cu′,k1u′,…,cv′,k1v′,…,cαDξαu,k1αDξαu,…,cαDξαv,k1αDξαv,…)=0.Step 4.
We suppose that (12) has the following solution:
(13)u(ξ)=∑i=-ni=n‍aiφi,
where ai(i=-n,-n+1,…,n-1,n) are constants to be determined later, n is a positive integer determined by balancing the highest order derivatives and nonlinear terms in (12) (see [17] for details), and φ=φ(ξ) satisfies the following fractional Riccati equation:
(14)Dξαφ(ξ)=σ+φ2(ξ).
By using the generalized Exp-function method via Mittag-Leffler functions, S. Zhang and H.-Q. Zhang first obtained generalized hyperbolic and trigonometric functions of fractional Riccati equation [18], and the obtained five solutions of (14) are
(15)φ(ξ)={--σtanhα(-σξ),σ<0,--σcothα(-σξ),σ<0,σtanα(σξ),σ>0,-σcotα(σξ),σ>0,-Γ(1+α)ξα+ω,ω=const.,σ=0,
where tanhα,cothα,tanα,andcotα are generalized hyperbolic and trigonometric functions defined in [18] as
(16)tanhα(x)=sinhα(x)coshα(x),cothα(x)=coshα(x)sinhα(x),sinhα(x)=Eα(xα)-Eα(-xα)2,coshα(x)=Eα(xα)+Eα(-xα)2,tanα(x)=sinα(x)cosα(x),cotα(x)=cosα(x)sinα(x),sinα(x)=Eα(ixα)-Eα(-ixα)2i,cosα(x)=Eα(ixα)+Eα(-ixα)2,
where Eα(z)=∑k=0∞(zk/Γ(1+kα)) is the Mittag-Leffler function.Step 5.
Substituting (13) into (12) along with (14) and using the properties of Jumarie’s modified Riemann-Liouville derivative (4)–(7), we can get a polynomial in φ(ξ). Setting all the coefficients of φk(k=0,1,2,…,-1,-2,…) to zero yields a set of overdetermined nonlinear algebraic equations for c, ki(i=1,2,…,m), aj(j=-n,-n+1,…,n-1,n).Step 6.
Take advantage of the known solutions of (14) to get the solutions of the fractional coupled NPDEs in concern.


## 3. Solutions of Fractional Coupled KD Equation and NNV Equations

In this section, we apply the generalized fractional subequation method for solving the NPDEs (1) and (2).Example 1.
The space-time fractional KD equations. By considering the traveling wave transformationsu=u(ξ),v=v(ξ),andξ=lx+my+nt, (1) can be reduced to the following nonlinear fractional ODEs:
(17)nαDξαu-l3αDξ3αu-6blαuDξαu+32a2lαu2Dξαu-3mαDξαv+3alαvDξαu=0,(18)mαDξαu=lαDξαv.
From (18) and using the definition of Jumarie’s modified Riemann-Liouville derivative, one gets
(19)v=mαlαu+c,
where c is the arbitrary constant. Substituting (19) into (17), one obtains
(20)(nα-6blαu+32a2lαu2-3m2αlα+3amαu+3alαc)×Dξαu-l3αDξ3αu=0.
By balancing the highest order derivative terms and nonlinear terms in (20), we suppose that (20) has the following formal solution:
(21)u(ξ)=a0+a1φ(ξ)+a2φ(ξ).
Substituting (21) into (20) along with (14) and collecting the coefficients of φi and setting them to be zero, we can get a set of algebraic equation about l, m, n, c, a0, a1, and a2. Solving the algebraic equations by Mathematica, we have the following.Case 1.
One has(22)a0=2b-amαl-αa2,a1=0,a2=-2lασa,n=((2a2)-1(-12abmα+9a2m2αl-α+12b2l6α-6ba3clα+4a3l3ασ)×(2a2)-1)1/α.Case 2.
One has(23)a0=2b-amαl-αa2,a1=0,a2=2lασa,n=((2a2)-1(-12abmα+9a2m2αl-α+12b2l6α-6ba3clα+4a3l3ασ)×(2a2)-1)1/α.Case 3.
One has(24)a0=2b-amαl-αa2,a1=2lαa,a2=0,n=((2a2)-1(-12abmα+9a2m2αl-α+12b2l6α-6ba3clα+4a3l3ασ)×(2a2)-1)1/α.Case 4.
One has(25)a0=2b-amαl-αa2,a1=-2lαa,a2=0,n=((2a2)-1(-12abmα+9a2m2αl-α+12b2l6α-6ba3clα+4a3l3ασ)×(2a2)-1)1/α.Case 5.
One has(26)a0=2b-amαl-αa2,a1=-2lαa,a2=-2lασa,n=((2a2)-1(-12abmα+9a2m2αl-α+12b2l6α-6ba3clα-8a3l3ασ)×(2a2)-1)1/α.Case 6.
One has(27)a0=2b-amαl-αa2,a1=2lαa,a2=-2lασa,n=((2a2)-1(-12abmα+9a2m2αl-α+12b2l6α-6ba3clα+16a3l3ασ)×(2a2)-1)1/α.Case 7.
One has(28)a0=2b-amαl-αa2,a1=-2lαa,a2=2lασa,n=((2a2)-1(-12abmα+9a2m2αl-α+12b2l6α-6ba3clα+16a3l3ασ)×(2a2)-1)1/α.Case 8.
One has(29)a0=2b-amαl-αa2,a1=2lαa,a2=2lασa,n=((2a2)-1(-12abmα+9a2m2αl-α+12b2l6α-6ba3clα-9a3l3ασ)×(2a2)-1)1/α.
Using Case1, (21), and the solutions of (14), we can find the following exact solutions of NFDEs (1):
(30)u1=-2-σlαatanhα(--σξ)+2blα-amαa2lα,v1=-2-σmαatanhα(--σξ)+2bmαlα-am2αa2l2α+c,
whereσ<0, ξ=lx+my+nt,
(31)u2=-2-σlαacothα(--σξ)+2blα-amαa2lα,v2=-2-σmαacothα(--σξ)+2bmαlα-am2αa2l2α+c,
whereσ<0, ξ=lx+my+nt,
(32)u3=-2σlαatanα(σξ)+2blα-amαa2lα,v3=-2σmαatanα(σξ)+2bmαlα-am2αa2l2α+c,
where σ>0, ξ=lx+my+nt,
(33)u4=2σlαacotα(σξ)+2blα-amαa2lα,v4=2σmαacotα(σξ)+2bmαlα-am2αa2l2α+c,
where σ>0, ξ=lx+my+nt,
(34)u5=2σlα(ω+ξα)aΓ(1+a)+2blα-amαa2lα,v5=2σmα(ω+ξα)aΓ(1+a)+2bmαlα-am2αa2l2α+c,
where σ=0, ξ=lx+my+nt. And n=((-12abmα+9a2m2αl-α+12b2l6α-6ba3clα+4a3l3ασ)/2a2)1/α,  l, m, c, and ω are arbitrary constants.
From Cases2, 3, 4, 5, 6, 7, and 8, we obtain many other exact solutions of (1). Here, we omit them for simplicity.
Forα=1, generalized hyperbolic function solutions and generalized trigonometric function solutions degrade into hyperbolic function solutions and trigonometric function solutions. We stress on the fact that when α→1 these obtained exact solutions including solitary solutions and rational solutions give the ones of the standard form equation of the space-time fractional KD equation (1).Example 2 (The space-time fractional NNV equations).
By considering the traveling wave transformationsu=u(ξ), v=v(ξ),  and ξ=lx+my+nt, (2) can be reduced to the following nonlinear fractional ODEs:
(35)nαDξαu=Al3αDξ3αu+Bm3αuDξ3αu-3AlαuDξαv-3AlαvDξαu-3BmαuDξαw-3BmαwDξαu,(36)lαDξαu=mαDξαv,(37)mαDξαu=lαDξαw.
From (36)-(37) and using the definition of Jumarie’s modified Riemann-Liouville derivative, one gets
(38)v=lαmαu+c1,w=mαlαu+c2,
where c1 and c2 are arbitrary constants. Substituting (38) into (35), one obtains
(39)(nα+6Al2αmαu+3Alαc1+6bm2αlαu+3Bmαc2)×Dξαu-(Al3α+Bm3α)Dξ3αu=0.
By balancing the highest order derivative terms and nonlinear terms in (39), we suppose that (39) has the following formal solution:
(40)u(ξ)=b0+b1φ(ξ)+b2φ2(ξ)+b3φ(ξ)+b4φ2(ξ).
Substituting (40) into (39) along with (14), and collecting the coefficients of φi and setting them to be zero, we can get a set of algebraic equation about l, m, n, c1, c2, b0, b1, b2, b3, and b4. Solving the algebraic equations by Mathematica, we haveCase 1.
One has(41)b0=mαlα(8Al3ασ+8Bm3ασ-3Ac1lα-3Bc2mα-nα)6Al3α+6Bm3α,b1=0,b2=-2mαlα,b3=0,b4=2mαlασ2.Case 2.
One has(42)b0=mαlα(8Al3ασ+8Bm3ασ-3Ac1lα-3Bc2mα-nα)6Al3α+6Bm3α,b1=0,b2=0,b3=0,b4=2mαlασ2.Case 3.
One has(43)b0=mαlα(8Al3ασ+8Bm3ασ-3Ac1lα-3Bc2mα-nα)6Al3α+6Bm3α,b1=0,b2=2mαlα,b3=0,b4=0.
Using Case1, (40), and the solutions of (14), we can find the following exact solutions of NFDEs (2):
(44)u1=-2mαlασtanhα2(-σξ)-2mαlασtanhα2(-σξ)+mαlα(8Al3ασ+8Bm3ασ-3Bc2mα-3Ac1lα-nα)6Al3α+6Bm3α,v1=-2l2ασtanhα2(-σξ)-2l2ασtanhα2(-σξ)+(mαlα(8Al5ασ+8Bl2αm3ασ-3Bc2l2αmα-3Ac1l3α-l2αnα)×(6Al3α+6Bm3α)-1+c1),w1=-2m2ασtanhα2(-σξ)-2m2ασtanhα2(-σξ)+((6Al3α+6Bm3α)-1mαlα(8Al3αm2ασ+8Bm5ασ-3Bc2m3αmα-3Ac1lαm2α-m2αnα)×(6Al3α+6Bm3α)-1)+c2,
where σ<0, ξ=lx+my+nt,
(45)u2=-2mαlασcothα2(-σξ)-2mαlασcothα2(-σξ)+mαlα(8Al3ασ+8Bm3ασ-3Bc2mα-3Ac1lα-nα)6Al3α+6Bm3α,v2=-2l2ασcothα2(-σξ)-2l2ασcothα2(-σξ)+((6Al3α+6Bm3α)-1(8Al5ασ+8Bl2αm3ασ-3Bc2l2αmα-3Ac1l3α-l2αnα)×(6Al3α+6Bm3α)-1)+c1,w2=-2m2ασcothα2(-σξ)-2m2ασcothα2(-σξ)+((6Al3α+6Bm3α)-1(8Al3αm2ασ+8Bm5ασ-3Bc2m3αmα-3Ac1lαm2α-m2αnα)×(6Al3α+6Bm3α)-1)+c2,
where σ<0, ξ=lx+my+nt,
(46)u3=2mαlασtanα2(σξ)+2mαlασtanα2(σξ)+mαlα(8Al3ασ+8Bm3ασ-3Bc2mα-3Ac1lα-nα)6Al3α+6Bm3α,v3=2l2ασtanα2(σξ)+2l2ασtanα2(σξ)+((6Al3α+6Bm3α)-1(8Al5ασ+8Bl2αm3ασ-3Bc2l2αmα-3Ac1l3α-l2αnα)×(6Al3α+6Bm3α)-1)+c1,w3=2m2ασtanα2(σξ)+2m2ασtanα2(σξ)+((6Al3α+6Bm3α)-1(8Al3αm2ασ+8Bm5ασ-3Bc2m3αmα-3Ac1lαm2α-m2αnα)×(6Al3α+6Bm3α)-1)+c2,
where σ>0, ξ=lx+my+nt. 							(47)u4=2mαlασcotα2(σξ)+2mαlασcotα2(σξ)+mαlα(8Al3ασ+8Bm3ασ-3Bc2mα-3Ac1lα-nα)6Al3α+6Bm3α,v4=2l2ασcotα2(σξ)+2l2ασcotα2(σξ)+((6Al3α+6Bm3α)-1(8Al5ασ+8Bl2αm3ασ-3Bc2l2αmα-3Ac1l3α-l2αnα)×(6Al3α+6Bm3α)-1)+c1,w4=2m2ασcotα2(σξ)+2m2ασcotα2(σξ)+((6Al3α+6Bm3α)-1(8Al3αm2ασ+8Bm5ασ-3Bc2m3αmα-3Ac1lαm2α-m2αnα)×(6Al3α+6Bm3α)-1)+c2,
where σ>0, ξ=lx+my+nt,
(48)u5=2σ2lαmα(ω+ξα)2Γ2(1+a)+2lαmαΓ2(1+a)(ω+ξα)2+mαlα(8Al3ασ+8Bm3ασ-3Bc2mα-3Ac1lα-nα)6Al3α+6Bm3α,v5=2σ2lα(ω+ξα)2Γ2(1+a)+2l2αΓ2(1+a)(ω+ξα)2+((6Al3α+6Bm3α)-1(8Al5ασ+8Bl2αm3ασ-3Bc2mαl2α-3Ac1l3α-l2αnα)×(6Al3α+6Bm3α)-1)+c1,w5=2σ2lα(ω+ξα)2Γ2(1+a)+2l2αΓ2(1+a)(ω+ξα)2+((6Al3α+6Bm3α)-1(8Al3αm2ασ+8Bl5αm3ασ-3Bc2m3α-3Ac1lαm2α-m2αnα)×(6Al3α+6Bm3α)-1)+c2,
where σ=0, ξ=lx+my+nt. And l, m, c, and ω are arbitrary constants, tanhα, cothα, tanα, and cotα are generalized hyperbolic and trigonometric functions.
From Cases2 and 3, we obtain many other exact solutions of (2). Here, we omit them for simplicity, too.
Asα→1, solutions (44)–(48) obtained above become the ones of the standard form equation of the NNV model, and the solutions cannot be directly constructed by other methods.


## 4. Conclusion

In the paper, based on the relationship of the fractional coupled equations and the properties of the Jumarie’s modified Riemann-Liouville derivative, we proposed a new generalized fractional subequation method to construct exact solutions of space-time fractional coupled differential equations. In order to illustrate the validity and advantages of the algorithm, we apply it to space-time fractional coupled Konopelchenko-Dubrovsky equations and Nizhnik-Novikov-Veselov equations. As a results, many exact solutions are obtained. The results show that this new generalized fractional subequations method is direct, effective, and can be used for many other fractional coupled differential equations.



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