# A Table Lookup Method for Exact Analytical Solutions of Nonlinear Fractional Partial Differential Equations

Authors: Ji Juan-Juan; Guo Ye-Cai; Zhang Lan-Fang; Zhang Chao-Long
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

A table lookup method for solving nonlinear fractional partial differential equations (fPDEs) is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the(2+1)-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.


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## Body

## 1. Introduction

Fractional partial differential equations (fPDEs) are the generalized form of the integer order differential equations. fPDEs can more accurately describe the complex physical phenomena occurring in fluid dynamics, high-energy physics, plasma physics, elastic media, optical fibers, chemical kinematics, chemical physics, acoustic waves, biomathematics, and many other areas [1, 2]. In recent years, many researchers have shown great interest in the search for exact solutions to nonlinear fPDEs. At present, several methods for finding the exact solutions of fPDEs have been presented, for example, the Adomian decomposition method [3–6], variational iteration method [7–9], homotopy perturbation method [10–13], homotopy analysis method [14, 15], differential transform method [16], spectral methods [17, 18], discontinuous Galerkin method [19], Kansa method [20], fractional subequation method [21], generalized fractional subequation method [22], fractional projective Riccati expansion method [23], exp-function method [24, 25], (G′/G)-expansion method [26–28], functional variable method [29, 30], and first integral method [31, 32].However, the above methods either are relatively complicated or have large computational cost. We propose a table lookup method in this paper. This method is straightforward and has small computational cost. We apply it to solve nonlinear fractional order partial differential equations with using the fractional complex transform and the modified Riemann-Liouville derivative defined by Jumarie [33]. Jumarie’s modified Riemann-Liouville derivative of order α is defined by the following expression [34]:(1)
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,where f(x) denotes a continuous function and Γ(·) is the Gamma function.Moreover, the modified Riemann-Liouville derivative has various useful properties, including the following:(2)Dxαxγ=Γγ+1Γγ+1-αxγ-α,γ>0(3)Dxαcfx=cDxαfx,(4)Dxαfxgx=gxDxαfx+fxDxαgx,(5)Dxαfxgx=fg′gxDxαgx=Dgαgxgx′α.The rest of this paper is organized as follows. In Section2, the basic ideas and main steps of the table lookup method are given. In Section 3, the exact analytical solutions of the time fractional simplified MCH equation, the space-time combined KdV-mKdV equation, the (2+1)-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation are constructed using the proposed method. In Section 4, some conclusions are obtained.


## 2. Basic Idea of the Table Lookup Method

In this section, we outline the main steps of the table lookup method for solving nonlinear fPDEs. Let us consider a fractional order partial differential equation in the following form:(6)Hu,Dtαu,Dxβu,Dyγu,DtαDtαu,DtαDxβu,…=0,0<α,β,where Dtαu, Dxβu, and Dyγu denote modified Riemann-Liouville derivatives of u(x,y,t), H represents a polynomial in u(x,y,t) and its various partial derivatives, and x, y, and t are variables. In the following, we give the main steps of our proposed method.Step 1.
First, we use the fractional complex transformation as follows:(7)ux,y,t=uξ,ξ=yγΓ1+γ+xβΓ1+β-ctαΓ1+α,where c is a nonzero constant. Thus, (6) can be transformed into the following nonlinear ordinary differential equation (ODE) of integer order with respect to ξ in sense of the properties of Jumarie’s modified Riemann-Liouville derivative given in (2)–(5):(8)H~u,u′,u′′,u′′′,…=0,where H~ is a polynomial in u(ξ) and its various derivatives u′,u′′,u′′′,… and u′=du/dξ,u′′=d2u/dξ2,….Step 2.
Integrating (8) once or several times with respect to ξ, setting the integration constant to zero if possible, multiplying both sides of the equation by u′, and then integrating once again, we obtain different types of the auxiliary equation (φ′)2=a0+a1φ+a2φ2+a3φ3+a4φ4, where ai(i=0,1,…,4) are constant coefficients that can be determined in this step.Step 3.
According to the equation type obtained in Step2, we create the tables of solutions to the corresponding type of auxiliary equation.Step 4.
Looking up the tables and determining the corresponding coefficients and existence conditions of the exact solutions, we successfully obtain the exact analytical solutions to (6).


## 3. Applications of the Proposed Method

In this section, we apply the table lookup method to construct exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time combined KdV-MKdV equation, the(2+1)-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation, which are very important nonlinear fPDEs in mathematical physics and have received much attention from researchers.


### 3.1. The Time Fractional Simplified Modified Camassa-Holm (MCH) Equation

We now consider the following time fractional simplified modified Camassa-Holm (MCH) equation:(9)Dtαu+2kux-Dtαuxx+βu2ux=0,where k∈R, β>0, and 0<α<1. Equation (9) is a variation of the following equation: (10)ut+2kux-uxxt+βu2ux=0.Using the fractional complex transformationu(x,t)=u(ξ), ξ=x-ctα/Γ(1+α) with (9), we can obtain the following nonlinear ODE:(11)-cu′+2ku′+cu′′′+βu2u′=0.Integrating (11) once with respect to ξ and setting the integral constant to zero, we obtain(12)2k-cu+cu′′+β3u3=0.Multiplying (12) by u′ and then integrating once again, we obtain (13)u′2=a0+a2u2+a4u4,where a2=-2k-c/c, a4=-β/6c, and a0 is an integration constant.According to the solutions of the auxiliary equation presented in [35], the exact solutions of (14) are listed in Table 1:(14)φ′2=a0+a2φ2+a4φ4.Table 1
Solutions of (14), ε=±1.
Number φ 1 φ = - a 2 a 4 sech ⁡ a 2 ξ, a2>0, a4<0, a0=0  2 φ = - a 2 a 4 csch ⁡ a 2 ξ, a2>0, a4<0, a0=0  3 φ = - a 2 a 4 sec ⁡ - a 2 ξ, a2<0, a4>0, a0=0  4 φ = - a 2 a 4 csc ⁡ - a 2 ξ, a2<0, a4>0, a0=0  5 φ = ε - a 2 2 a 4 tanh ⁡ - a 2 2 ξ, a2<0, a4>0, a0=a224a4  6 φ = ε - a 2 2 a 4 coth ⁡ - a 2 2 ξ, a2<0, a4>0, a0=a224a4  7 φ = ε - a 2 2 a 4 tan ⁡ a 2 2 ξ, a2>0, a4>0, a0=a224a4  8 φ = ε - a 2 2 a 4 cot ⁡ a 2 2 ξ, a2>0, a4>0, a0=a224a4  9 φ = - a 2 m 2 a 4 2 m 2 - 1 cn ⁡ a 2 2 m 2 - 1 ξ, a2>0, a4<0, a0=1-m2a22m2-12  10 φ = ε - a 2 m 2 a 4 m 2 + 1 sn ⁡ - a 2 m 2 + 1 ξ, a2<0, a4>0, a0=a22m22a4m2+1Equations (13) and (14) have the same form. Thus, looking up Table 1 and determining the coefficients and the existence condition of the exact solutions, we can obtain the following solutions of (9):(1)
Whenc>2k&c>0, a0=0, and ε=±1, we obtain (15)u1x,t=-62k-cβsech⁡-2k-ccx-ctαΓ1+α,(16)u2x,t=-62k-cβcsch⁡-2k-ccx-ctαΓ1+α.(2)
When2k<c<0, a0=0, and ε=±1, we obtain (17)u3x,t=-62k-cβsec⁡2k-ccx-ctαΓ1+α,(18)u4x,t=-62k-cβcsc⁡2k-ccx-ctαΓ1+α.(3)
When2k<c<0, a0=-3(2k-c)2/2βc, and ε=±1, we obtain (19)u5x,t=ε-32k-cβtanh⁡2k-c2cx-ctαΓ1+α,(20)u6x,t=ε-32k-cβcoth⁡2k-c2cx-ctαΓ1+α.(4)
Whenc<2k&c<0, a0=-3(2k-c)2/2βc, and ε=±1, we obtain (21)u7x,t=ε32k-cβtan⁡-2k-c2cx-ctαΓ1+α,(22)u8x,t=ε32k-cβcot⁡-2k-c2cx-ctαΓ1+α.(5)
Whenc>2k&c>0, a0=-(2k-c)(1-m2)/c(2m2-1)2, and ε=±1, we obtain (23)u9x,t=-62k-cm2β2m2-1cn⁡-2k-c2c2m2-1x-ctαΓ1+α.(6)
Whenc>2k&c<0, a0=-3(2k-c)2m2/2βc(m2+1), and ε=±1, we obtain (24)u10x,t=ε-62k-cm2βm2+1sn⁡2k-ccm2+1x-ctαΓ1+α.The evolution of exact solution for (15)–(24) is shown in Figures 1–6.Figure 1
Exact solutionsu1(x,t) for (15) and its position at t=1.5, when the parameters k=1, β=1, c=3, and α=0.5.Figure 2
Exact solutionsu2(x,t) for (16) and its position at t=2, when the parameters k=1, β=1, c=3, and α=0.5.Figure 3
Exact solutionsu3(x,t) for (17) and its position at t=0, when the parameters k=-1, β=1, c=-1, and α=0.6.Figure 4
Exact solutionsu5(x,t) for (19) and its position at t=2, when the parameters k=-3, β=2, c=-1, ε=1, and α=0.6.Figure 5
Exact solutionsu7(x,t) for (21) and its position at t=0, when the parameters k=5, β=3, c=-3, and α=0.2.Figure 6
Exact solutionsu9(x,t) for (23) and its position at t=0, when the parameters k=1, β=1.2, c=7, and α=0.8.It can be observed from (15)–(24) and Figures 1–6 that triangular periodic solution, bell-shaped solitary wave solution, kink-shaped solitary wave solution, and Jacobi elliptic function solution of the time fractional simplified MCH are obtained.Whenα=1, (9) becomes (10), namely, simplified MCH equation, and ξ=x-ct, obviously, (10) still have triangular periodic solution, bell-shaped solitary wave solution, kink-shaped solitary wave solution, and Jacobi elliptic function four types of solutions. Reference [36] only obtained triangular periodic solutions, kink-shaped solitary wave solution, and singular solution of simplified MCH equation. Thus, our table lookup method is more powerful.


### 3.2. The Space-Time Fractional Combined KdV-mKdV Equation

In this section, we will apply the table lookup method to the following space-time fractional combined KdV-mKdV equation [37]: (25)Dtαu+μuDxαu+δu2Dxαu+Dx3αu=0,where μ and δ are nonzero constants. This equation may describe the wave propagation of a bound particle, sound wave, or thermal pulse.Now, we use the fractional complex transformationu(x,t)=u(ξ), ξ=xα/Γ(1+α)-ctα/Γ(1+α) with (25), which is reduced to the following ODE of integer order:(26)-cu′+μuu′+δu2u′+u′′′=0.Integrating (26) once with the integral constant of zero yields(27)-cu+μ2u2+δ3u3+u′′=0.Multiplying (27) by u′, integrating once again, and setting the integral constant to zero, we obtain(28)u′2=a2u2+a3u3+a4u4,where a2=c, a3=-μ/3, and a4=-δ/6.By virtue of the solutions of the auxiliary equation given in [38], the exact solutions of (29) are shown in Table 2:(29)φ′2=a2φ2+a3φ3+a4φ4.Table 2
Solutions of (29), Δ=a32-4a2a4, ε=±1.
Number φ 1 - a 2 a 3 sech 2 ⁡ a 2 / 2 ξ a 3 - a 2 a 4 1 + ε tanh ⁡ a 2 / 2 ξ 2, a2>0  2 - a 2 a 3 csch 2 ⁡ a 2 / 2 ξ a 3 - a 2 a 4 1 + ε coth ⁡ a 2 / 2 ξ 2, a2>0  3 2 a 2 sech ⁡ a 2 ξ ε Δ - a 3 sech ⁡ a 2 ξ, a2>0, Δ>0  4 2 a 2 sec ⁡ - a 2 ξ ε Δ - a 3 sec ⁡ - a 2 ξ, a2<0, Δ>0  5 2 a 2 csc ⁡ - a 2 ξ ε - Δ - a 3 csc ⁡ - a 2 ξ, a2<0, Δ>0  6 2 a 2 csch ⁡ a 2 ξ ε - Δ - a 3 csch ⁡ a 2 ξ, a2>0, Δ<0  7 - a 2 sech 2 ⁡ a 2 / 2 ξ a 3 + 2 ε a 2 a 4 tanh ⁡ a 2 / 2 ξ, a2>0, a4>0  8 a 2 a 3 csch 2 ⁡ a 2 / 2 ξ a 3 2 - a 2 a 4 1 + ε coth ⁡ a 2 / 2 ξ, a2>0, a4>0  9 - a 2 sech 2 ⁡ - a 2 / 2 ξ a 3 + 2 ε - a 2 a 4 tan ⁡ - a 2 / 2 ξ, a2<0, a4>0  10 - a 2 csc 2 ⁡ - a 2 / 2 ξ a 3 + 2 ε - a 2 a 4 cot ⁡ - a 2 / 2 ξ, a2<0, a4>0  11 - a 2 a 3 1 + ε tanh ⁡ a 2 2 ξ, a2>0, Δ=0  12 - a 2 a 3 1 + ε coth ⁡ a 2 2 ξ, a2>0, Δ=0Equations (28) and (29) have the same form. Thus, looking up Table 2 and determining the coefficients and existence conditions of the exact solutions, we can obtain the following solutions of (25):(1)
Ifc>0 and ε=±1, we obtain (30)u1x,t=cμ/3sech2⁡c/2xα/Γ1+α-ctα/Γ1+αμ/3+δc/61+εtanh⁡c/2xα/Γ1+α-ctα/Γ1+α2,(31)u2x,t=cμ/3csch2⁡c/2xα/Γ1+α-ctα/Γ1+αμ/3+δc/61+εcoth⁡c/2xα/Γ1+α-ctα/Γ1+α2.(2)
Ifc>0, μ2>-6δc, and ε=±1, we obtain (32)u3x,t=6csech⁡cxα/Γ1+α-ctα/Γ1+αεμ2+6δc+μsech⁡cxα/Γ1+α-ctα/Γ1+α.(3)
Ifc<0, μ2>-6δc, and ε=±1, we obtain (33)u4x,t=6csec⁡-cxα/Γ1+α-ctα/Γ1+αεμ2+6δc+μsec⁡-cxα/Γ1+α-ctα/Γ1+α,(34)u5x,t=6ccsc⁡-cxα/Γ1+α-ctα/Γ1+αεμ2+6δc+μcsc⁡-cxα/Γ1+α-ctα/Γ1+α.(4)
Ifc>0, μ2<-6δc, and ε=±1, we obtain (35)u6x,t=6ccsch⁡cxα/Γ1+α-ctα/Γ1+αε-μ2+6δc+μcsch⁡cxα/Γ1+α-ctα/Γ1+α.(5)
Ifc>0, δ<0, and ε=±1, we obtain (36)u7x,t=-csech2⁡c/2xα/Γ1+α-ctα/Γ1+α-μ/3+2ε-δc/6tanh⁡c/2xα/Γ1+α-ctα/Γ1+α,(37)u8x,t=cμ/3csch2⁡c/2xα/Γ1+α-ctα/Γ1+αμ2/9+δc/61+εcoth⁡c/2xα/Γ1+α-ctα/Γ1+α.(6)
Ifc<0, δ<0, and ε=±1, we obtain (38)u9x,t=-csec2⁡-c/2xα/Γ1+α-ctα/Γ1+α-μ/3+2εδc/6tan⁡-c/2xα/Γ1+α-ctα/Γ1+α,(39)u10x,t=-ccsc2⁡-c/2xα/Γ1+α-ctα/Γ1+α-μ/3+2εδc/6cot⁡-c/2xα/Γ1+α-ctα/Γ1+α.(7)
Ifc>0, μ2=-6δc, and ε=±1, we obtain (40)u11x,t=3cμ1+εtanh⁡c2xαΓ1+α-ctαΓ1+α,(41)u12x,t=3cμ1+εcoth⁡c2xαΓ1+α-ctαΓ1+α.The evolution of exact solution for (30)–(41) is described in Figures 7–13.Figure 7
Exact solutionsu1(x,t) for (30) and its position at t=1.2, when the parameters μ=2, δ=1, c=3, and α=0.5.Figure 8
Exact solutionsu3(x,t) for (32) and its position at t=1.2, when the parameters μ=2, δ=1, c=3, ε=1, and α=0.7.Figure 9
Exact solutionsu4(x,t) for (33) and its position at t=0, when the parameters μ=6, δ=1, c=-3, ε=1, and α=0.5.Figure 10
Exact solutionsu6(x,t) for (35) and its position at t=0, when the parameters μ=4, δ=-1, c=6, ε=1, and α=0.8.Figure 11
Exact solutionsu7(x,t) for (36) and its position at t=3, when the parameters μ=2, δ=-15, c=3, ε=1, and α=0.8.Figure 12
Exact solutionsu9(x,t) for (38) and its position at t=0, when the parameters μ=2, δ=-10, c=-7, ε=1, and α=0.8.Figure 13
Exact solutionsu11(x,t) for (40) and its position at t=0, when the parameters μ=6, δ=-2, c=3, ε=1, and α=0.8.It can be observed from (30)–(41) that we have successfully obtained twelve exact analytical solutions of the space-time fractional combined KdV-mKdV equation. In comparison, [37] using the fractional mapping method only obtained five analytical solutions; thus, the proposed lookup table method is more concise and more effective.


### 3.3. The(2+1)-Dimensional Time Fractional Zoomeron Equation

Now, we use the proposed methods to construct the exact solutions of the following nonlinear(2+1)-dimensional time fractional Zoomeron equation [39]:(42)Dtt2αuxyu-uxyuxx+Dtα2u2x=0,0<α≤1,where u(x,y,t) is the amplitude of the relative mode.We perform the fractional complex transformation:(43)ux,y,t=uξ,ξ=x+y-ctαΓ1+α.Substituting (43) into (42) and utilizing (2), we can reduce (42) into an ODE of integer order as follows:(44)c2u′′u′′-u′′u′′+2cu2′′=0.Integrating (44) twice with respect to ξ, we obtain(45)c2-1u′′-2cu3+Ru=0,where R is the first integral constant (R≠0) and the second integral constant is set to zero.Multiplying (45) by u′ and then integrating once again, we obtain(46)u′2=a0+a2u2+a4u4,where a2=c/c2-1, a4=-R/c2-1, and a0 is an integral constant.According to the characteristics of (46), looking up Table 1 and determining the corresponding coefficients and existence conditions of the exact solutions, we obtain several types of solutions to (42) as follows:(1)
Whenc>1, R>0, a0=0, and ε=±1, we obtain (47)u1x,y,t=cRsech⁡cc2-1x+y-ctαΓ1+α,u2x,y,t=cRcsch⁡cc2-1x+y-ctαΓ1+α.(2)
Whenc<-1, R<0, a0=0, and ε=±1, we obtain (48)u3x,y,t=cRsec⁡-cc2-1x+y-ctαΓ1+α,u4x,y,t=cRcsc⁡-cc2-1x+y-ctαΓ1+α.(3)
Whenc<-1, R<0, a0=-c2/4R(c2-1), and ε=±1, we obtain (49)u5x,y,t=εc2Rtanh⁡-c2c2-1x+y-ctαΓ1+α,u6x,y,t=εc2Rcoth⁡-c2c2-1x+y-ctαΓ1+α.(4)
Whenc>1, R<0, a0=-c2/4R(c2-1), and ε=±1, we obtain (50)u7x,y,t=ε-cRtan⁡c2c2-1x+y-ctαΓ1+α,u8x,y,t=ε-cRcot⁡c2c2-1x+y-ctαΓ1+α.(5)
Whenc>1, R>0, a0=(1-m2)c/(2m2-1)2(c2-1), and ε=±1, we obtain (51)u9x,y,t=cm2R2m2-1cn⁡cc2-12m2-1x+y-ctαΓ1+α.(6)
Whenc<-1, R<0, a0=-c2m2/2R(c2-1)(m2+1), and ε=±1, we obtain (52)u10x,y,t=εcm2Rm2+1sn⁡-cc2-1m2+1x+y-ctαΓ1+α.By the way, when takingy=0, (42) and (9) have the same type solutions, so that we no longer give figures of the obtained solutions of (42) in this section.If we takek=ε=1, σ=c/(2(c2-1)), A=c, and R=-c, then the expression of u5(x,y,t) is the same as that of u1(x,y,t) for Eq. (27) in [39], the expression of u6(x,y,t) is the same as that of u2(x,y,t) for Eq. (27) in [39], and they can be, respectively, expressed as follows: (53)ux,y,t=12tanh⁡-c2c2-1x+y-ctαΓ1+α,ux,y,t=12coth⁡-c2c2-1x+y-ctαΓ1+α.If we takek=ε=1, σ=c/(2(c2-1)), A=c, and R=-2c, then the expression of u7(x,y,t) is the same as that of u3(x,y,t) for Eq. (27) in [39], the expression of u8(x,y,t) is the same as that of u4(x,y,t) for Eq. (27) in [39], and they can be, respectively, expressed as follows:(54)ux,y,t=-12tan⁡-c2c2-1x+y-ctαΓ1+α,ux,y,t=12cot⁡-c2c2-1x+y-ctαΓ1+α.However, we also obtain the bell-shaped solitary wave solution and Jacobi elliptic function solution of the(2+1)-dimension time fractional Zoomeron equation.


### 3.4. The Space-Time Fractional ZKBBM Equation

Next, we solve the exact solution of the following space-time fractional nonlinear Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation [40]:(55)∂αu∂tα+∂βu∂xβ-2au∂βu∂xβ-b∂α∂tα∂2βu∂x2β=0,t>0,α>0,β≤1,where a and b are arbitrary nonzero constants.We first use the fractional complex transformu(x,t)=u(ξ), ξ=xβ/Γ(1+β)-ctα/Γ(1+α), and (55) becomes an ODE of integer order: (56)1-cu′-2auu′+bcu′′′=0.Integrating (56) once with an integral constant of zero yields(57)1-cu-au2+bcu′′=0.Multiplying (57) by u′ and integrating once again with an integral constant of zero, we obtain(58)u′2=a2u2+a3u3,where a2=c-1/bc and a3=2a/3bc.According to the solutions of the auxiliary equation presented in [35], the exact solutions of the (59) are listed in Table 3:(59)φ′2=a2φ2+a3φ3.Table 3
Solutions of (59).
Number φ 1 φ = - a 2 a 3 sech 2 ⁡ a 2 2 ξ, a2>0, a0=0, a1=0  2 φ = - a 2 a 3 sec 2 ⁡ - a 2 2 ξ, a2<0, a0=0, a1=0  3 φ = 1 a 3 ξ 2, a2=0, a0=0, a1=0Equations (58) and (59) have the same form. Thus, looking up Table 3 and determining the coefficients and existence conditions, we can obtain the following solutions of (55):(1)
Ifc>1 and b>0, we obtain (60)u1x,t=-3c-12asech2⁡12c-1bcxβΓ1+β-ctαΓ1+α.(2)
Ifc>1 and b<0, we obtain(61)u2x,t=-3c-12asec2⁡12-c-1bcxβΓ1+β-ctαΓ1+α.(3)
Ifc=1, we obtain (62)u3x,t=2a3bcxβ/Γ1+β-ctα/Γ1+α2.The asymptotic behavior for the obtained exact solution (60)–(62) is shown in Figures 14–16.Figure 14
Exact solutionsu1(x,t) for (60) and its position at t=0, when the parameters a=-4, b=2, c=3, α=0.8, and β=0.5.Figure 15
Exact solutionsu2(x,t) for (61) and its position at t=5, when the parameters a=-4, b=-0.5, c=10, α=0.7, and β=0.9.Figure 16
Exact solutionsu3(x,t) for (62): (a) α=0.5, β=0.2; (b) α=β=0.5, when the parameters a=-4, b=-0.5, and c=10.
(a) (b)It can be observed from Figures14–16 that we obtained single solitary wave and multiple solitary wave solutions of fractional ZKBBM equation, and when α=β, (62) is singular wave solution, and when α≠β, (62) is multiple solitary wave solution. Consequently, the exact solution of fractional differential equation can better describe the complex wave propagation. Reference [40] only obtained Jacobi elliptic doubly periodic solution of fractional ZKBBM equation, but we obtain three types of solutions using our proposed method.


## 3.1. The Time Fractional Simplified Modified Camassa-Holm (MCH) Equation

We now consider the following time fractional simplified modified Camassa-Holm (MCH) equation:(9)Dtαu+2kux-Dtαuxx+βu2ux=0,where k∈R, β>0, and 0<α<1. Equation (9) is a variation of the following equation: (10)ut+2kux-uxxt+βu2ux=0.Using the fractional complex transformationu(x,t)=u(ξ), ξ=x-ctα/Γ(1+α) with (9), we can obtain the following nonlinear ODE:(11)-cu′+2ku′+cu′′′+βu2u′=0.Integrating (11) once with respect to ξ and setting the integral constant to zero, we obtain(12)2k-cu+cu′′+β3u3=0.Multiplying (12) by u′ and then integrating once again, we obtain (13)u′2=a0+a2u2+a4u4,where a2=-2k-c/c, a4=-β/6c, and a0 is an integration constant.According to the solutions of the auxiliary equation presented in [35], the exact solutions of (14) are listed in Table 1:(14)φ′2=a0+a2φ2+a4φ4.Table 1
Solutions of (14), ε=±1.
Number φ 1 φ = - a 2 a 4 sech ⁡ a 2 ξ, a2>0, a4<0, a0=0  2 φ = - a 2 a 4 csch ⁡ a 2 ξ, a2>0, a4<0, a0=0  3 φ = - a 2 a 4 sec ⁡ - a 2 ξ, a2<0, a4>0, a0=0  4 φ = - a 2 a 4 csc ⁡ - a 2 ξ, a2<0, a4>0, a0=0  5 φ = ε - a 2 2 a 4 tanh ⁡ - a 2 2 ξ, a2<0, a4>0, a0=a224a4  6 φ = ε - a 2 2 a 4 coth ⁡ - a 2 2 ξ, a2<0, a4>0, a0=a224a4  7 φ = ε - a 2 2 a 4 tan ⁡ a 2 2 ξ, a2>0, a4>0, a0=a224a4  8 φ = ε - a 2 2 a 4 cot ⁡ a 2 2 ξ, a2>0, a4>0, a0=a224a4  9 φ = - a 2 m 2 a 4 2 m 2 - 1 cn ⁡ a 2 2 m 2 - 1 ξ, a2>0, a4<0, a0=1-m2a22m2-12  10 φ = ε - a 2 m 2 a 4 m 2 + 1 sn ⁡ - a 2 m 2 + 1 ξ, a2<0, a4>0, a0=a22m22a4m2+1Equations (13) and (14) have the same form. Thus, looking up Table 1 and determining the coefficients and the existence condition of the exact solutions, we can obtain the following solutions of (9):(1)
Whenc>2k&c>0, a0=0, and ε=±1, we obtain (15)u1x,t=-62k-cβsech⁡-2k-ccx-ctαΓ1+α,(16)u2x,t=-62k-cβcsch⁡-2k-ccx-ctαΓ1+α.(2)
When2k<c<0, a0=0, and ε=±1, we obtain (17)u3x,t=-62k-cβsec⁡2k-ccx-ctαΓ1+α,(18)u4x,t=-62k-cβcsc⁡2k-ccx-ctαΓ1+α.(3)
When2k<c<0, a0=-3(2k-c)2/2βc, and ε=±1, we obtain (19)u5x,t=ε-32k-cβtanh⁡2k-c2cx-ctαΓ1+α,(20)u6x,t=ε-32k-cβcoth⁡2k-c2cx-ctαΓ1+α.(4)
Whenc<2k&c<0, a0=-3(2k-c)2/2βc, and ε=±1, we obtain (21)u7x,t=ε32k-cβtan⁡-2k-c2cx-ctαΓ1+α,(22)u8x,t=ε32k-cβcot⁡-2k-c2cx-ctαΓ1+α.(5)
Whenc>2k&c>0, a0=-(2k-c)(1-m2)/c(2m2-1)2, and ε=±1, we obtain (23)u9x,t=-62k-cm2β2m2-1cn⁡-2k-c2c2m2-1x-ctαΓ1+α.(6)
Whenc>2k&c<0, a0=-3(2k-c)2m2/2βc(m2+1), and ε=±1, we obtain (24)u10x,t=ε-62k-cm2βm2+1sn⁡2k-ccm2+1x-ctαΓ1+α.The evolution of exact solution for (15)–(24) is shown in Figures 1–6.Figure 1
Exact solutionsu1(x,t) for (15) and its position at t=1.5, when the parameters k=1, β=1, c=3, and α=0.5.Figure 2
Exact solutionsu2(x,t) for (16) and its position at t=2, when the parameters k=1, β=1, c=3, and α=0.5.Figure 3
Exact solutionsu3(x,t) for (17) and its position at t=0, when the parameters k=-1, β=1, c=-1, and α=0.6.Figure 4
Exact solutionsu5(x,t) for (19) and its position at t=2, when the parameters k=-3, β=2, c=-1, ε=1, and α=0.6.Figure 5
Exact solutionsu7(x,t) for (21) and its position at t=0, when the parameters k=5, β=3, c=-3, and α=0.2.Figure 6
Exact solutionsu9(x,t) for (23) and its position at t=0, when the parameters k=1, β=1.2, c=7, and α=0.8.It can be observed from (15)–(24) and Figures 1–6 that triangular periodic solution, bell-shaped solitary wave solution, kink-shaped solitary wave solution, and Jacobi elliptic function solution of the time fractional simplified MCH are obtained.Whenα=1, (9) becomes (10), namely, simplified MCH equation, and ξ=x-ct, obviously, (10) still have triangular periodic solution, bell-shaped solitary wave solution, kink-shaped solitary wave solution, and Jacobi elliptic function four types of solutions. Reference [36] only obtained triangular periodic solutions, kink-shaped solitary wave solution, and singular solution of simplified MCH equation. Thus, our table lookup method is more powerful.


## 3.2. The Space-Time Fractional Combined KdV-mKdV Equation

In this section, we will apply the table lookup method to the following space-time fractional combined KdV-mKdV equation [37]: (25)Dtαu+μuDxαu+δu2Dxαu+Dx3αu=0,where μ and δ are nonzero constants. This equation may describe the wave propagation of a bound particle, sound wave, or thermal pulse.Now, we use the fractional complex transformationu(x,t)=u(ξ), ξ=xα/Γ(1+α)-ctα/Γ(1+α) with (25), which is reduced to the following ODE of integer order:(26)-cu′+μuu′+δu2u′+u′′′=0.Integrating (26) once with the integral constant of zero yields(27)-cu+μ2u2+δ3u3+u′′=0.Multiplying (27) by u′, integrating once again, and setting the integral constant to zero, we obtain(28)u′2=a2u2+a3u3+a4u4,where a2=c, a3=-μ/3, and a4=-δ/6.By virtue of the solutions of the auxiliary equation given in [38], the exact solutions of (29) are shown in Table 2:(29)φ′2=a2φ2+a3φ3+a4φ4.Table 2
Solutions of (29), Δ=a32-4a2a4, ε=±1.
Number φ 1 - a 2 a 3 sech 2 ⁡ a 2 / 2 ξ a 3 - a 2 a 4 1 + ε tanh ⁡ a 2 / 2 ξ 2, a2>0  2 - a 2 a 3 csch 2 ⁡ a 2 / 2 ξ a 3 - a 2 a 4 1 + ε coth ⁡ a 2 / 2 ξ 2, a2>0  3 2 a 2 sech ⁡ a 2 ξ ε Δ - a 3 sech ⁡ a 2 ξ, a2>0, Δ>0  4 2 a 2 sec ⁡ - a 2 ξ ε Δ - a 3 sec ⁡ - a 2 ξ, a2<0, Δ>0  5 2 a 2 csc ⁡ - a 2 ξ ε - Δ - a 3 csc ⁡ - a 2 ξ, a2<0, Δ>0  6 2 a 2 csch ⁡ a 2 ξ ε - Δ - a 3 csch ⁡ a 2 ξ, a2>0, Δ<0  7 - a 2 sech 2 ⁡ a 2 / 2 ξ a 3 + 2 ε a 2 a 4 tanh ⁡ a 2 / 2 ξ, a2>0, a4>0  8 a 2 a 3 csch 2 ⁡ a 2 / 2 ξ a 3 2 - a 2 a 4 1 + ε coth ⁡ a 2 / 2 ξ, a2>0, a4>0  9 - a 2 sech 2 ⁡ - a 2 / 2 ξ a 3 + 2 ε - a 2 a 4 tan ⁡ - a 2 / 2 ξ, a2<0, a4>0  10 - a 2 csc 2 ⁡ - a 2 / 2 ξ a 3 + 2 ε - a 2 a 4 cot ⁡ - a 2 / 2 ξ, a2<0, a4>0  11 - a 2 a 3 1 + ε tanh ⁡ a 2 2 ξ, a2>0, Δ=0  12 - a 2 a 3 1 + ε coth ⁡ a 2 2 ξ, a2>0, Δ=0Equations (28) and (29) have the same form. Thus, looking up Table 2 and determining the coefficients and existence conditions of the exact solutions, we can obtain the following solutions of (25):(1)
Ifc>0 and ε=±1, we obtain (30)u1x,t=cμ/3sech2⁡c/2xα/Γ1+α-ctα/Γ1+αμ/3+δc/61+εtanh⁡c/2xα/Γ1+α-ctα/Γ1+α2,(31)u2x,t=cμ/3csch2⁡c/2xα/Γ1+α-ctα/Γ1+αμ/3+δc/61+εcoth⁡c/2xα/Γ1+α-ctα/Γ1+α2.(2)
Ifc>0, μ2>-6δc, and ε=±1, we obtain (32)u3x,t=6csech⁡cxα/Γ1+α-ctα/Γ1+αεμ2+6δc+μsech⁡cxα/Γ1+α-ctα/Γ1+α.(3)
Ifc<0, μ2>-6δc, and ε=±1, we obtain (33)u4x,t=6csec⁡-cxα/Γ1+α-ctα/Γ1+αεμ2+6δc+μsec⁡-cxα/Γ1+α-ctα/Γ1+α,(34)u5x,t=6ccsc⁡-cxα/Γ1+α-ctα/Γ1+αεμ2+6δc+μcsc⁡-cxα/Γ1+α-ctα/Γ1+α.(4)
Ifc>0, μ2<-6δc, and ε=±1, we obtain (35)u6x,t=6ccsch⁡cxα/Γ1+α-ctα/Γ1+αε-μ2+6δc+μcsch⁡cxα/Γ1+α-ctα/Γ1+α.(5)
Ifc>0, δ<0, and ε=±1, we obtain (36)u7x,t=-csech2⁡c/2xα/Γ1+α-ctα/Γ1+α-μ/3+2ε-δc/6tanh⁡c/2xα/Γ1+α-ctα/Γ1+α,(37)u8x,t=cμ/3csch2⁡c/2xα/Γ1+α-ctα/Γ1+αμ2/9+δc/61+εcoth⁡c/2xα/Γ1+α-ctα/Γ1+α.(6)
Ifc<0, δ<0, and ε=±1, we obtain (38)u9x,t=-csec2⁡-c/2xα/Γ1+α-ctα/Γ1+α-μ/3+2εδc/6tan⁡-c/2xα/Γ1+α-ctα/Γ1+α,(39)u10x,t=-ccsc2⁡-c/2xα/Γ1+α-ctα/Γ1+α-μ/3+2εδc/6cot⁡-c/2xα/Γ1+α-ctα/Γ1+α.(7)
Ifc>0, μ2=-6δc, and ε=±1, we obtain (40)u11x,t=3cμ1+εtanh⁡c2xαΓ1+α-ctαΓ1+α,(41)u12x,t=3cμ1+εcoth⁡c2xαΓ1+α-ctαΓ1+α.The evolution of exact solution for (30)–(41) is described in Figures 7–13.Figure 7
Exact solutionsu1(x,t) for (30) and its position at t=1.2, when the parameters μ=2, δ=1, c=3, and α=0.5.Figure 8
Exact solutionsu3(x,t) for (32) and its position at t=1.2, when the parameters μ=2, δ=1, c=3, ε=1, and α=0.7.Figure 9
Exact solutionsu4(x,t) for (33) and its position at t=0, when the parameters μ=6, δ=1, c=-3, ε=1, and α=0.5.Figure 10
Exact solutionsu6(x,t) for (35) and its position at t=0, when the parameters μ=4, δ=-1, c=6, ε=1, and α=0.8.Figure 11
Exact solutionsu7(x,t) for (36) and its position at t=3, when the parameters μ=2, δ=-15, c=3, ε=1, and α=0.8.Figure 12
Exact solutionsu9(x,t) for (38) and its position at t=0, when the parameters μ=2, δ=-10, c=-7, ε=1, and α=0.8.Figure 13
Exact solutionsu11(x,t) for (40) and its position at t=0, when the parameters μ=6, δ=-2, c=3, ε=1, and α=0.8.It can be observed from (30)–(41) that we have successfully obtained twelve exact analytical solutions of the space-time fractional combined KdV-mKdV equation. In comparison, [37] using the fractional mapping method only obtained five analytical solutions; thus, the proposed lookup table method is more concise and more effective.


## 3.3. The(2+1)-Dimensional Time Fractional Zoomeron Equation

Now, we use the proposed methods to construct the exact solutions of the following nonlinear(2+1)-dimensional time fractional Zoomeron equation [39]:(42)Dtt2αuxyu-uxyuxx+Dtα2u2x=0,0<α≤1,where u(x,y,t) is the amplitude of the relative mode.We perform the fractional complex transformation:(43)ux,y,t=uξ,ξ=x+y-ctαΓ1+α.Substituting (43) into (42) and utilizing (2), we can reduce (42) into an ODE of integer order as follows:(44)c2u′′u′′-u′′u′′+2cu2′′=0.Integrating (44) twice with respect to ξ, we obtain(45)c2-1u′′-2cu3+Ru=0,where R is the first integral constant (R≠0) and the second integral constant is set to zero.Multiplying (45) by u′ and then integrating once again, we obtain(46)u′2=a0+a2u2+a4u4,where a2=c/c2-1, a4=-R/c2-1, and a0 is an integral constant.According to the characteristics of (46), looking up Table 1 and determining the corresponding coefficients and existence conditions of the exact solutions, we obtain several types of solutions to (42) as follows:(1)
Whenc>1, R>0, a0=0, and ε=±1, we obtain (47)u1x,y,t=cRsech⁡cc2-1x+y-ctαΓ1+α,u2x,y,t=cRcsch⁡cc2-1x+y-ctαΓ1+α.(2)
Whenc<-1, R<0, a0=0, and ε=±1, we obtain (48)u3x,y,t=cRsec⁡-cc2-1x+y-ctαΓ1+α,u4x,y,t=cRcsc⁡-cc2-1x+y-ctαΓ1+α.(3)
Whenc<-1, R<0, a0=-c2/4R(c2-1), and ε=±1, we obtain (49)u5x,y,t=εc2Rtanh⁡-c2c2-1x+y-ctαΓ1+α,u6x,y,t=εc2Rcoth⁡-c2c2-1x+y-ctαΓ1+α.(4)
Whenc>1, R<0, a0=-c2/4R(c2-1), and ε=±1, we obtain (50)u7x,y,t=ε-cRtan⁡c2c2-1x+y-ctαΓ1+α,u8x,y,t=ε-cRcot⁡c2c2-1x+y-ctαΓ1+α.(5)
Whenc>1, R>0, a0=(1-m2)c/(2m2-1)2(c2-1), and ε=±1, we obtain (51)u9x,y,t=cm2R2m2-1cn⁡cc2-12m2-1x+y-ctαΓ1+α.(6)
Whenc<-1, R<0, a0=-c2m2/2R(c2-1)(m2+1), and ε=±1, we obtain (52)u10x,y,t=εcm2Rm2+1sn⁡-cc2-1m2+1x+y-ctαΓ1+α.By the way, when takingy=0, (42) and (9) have the same type solutions, so that we no longer give figures of the obtained solutions of (42) in this section.If we takek=ε=1, σ=c/(2(c2-1)), A=c, and R=-c, then the expression of u5(x,y,t) is the same as that of u1(x,y,t) for Eq. (27) in [39], the expression of u6(x,y,t) is the same as that of u2(x,y,t) for Eq. (27) in [39], and they can be, respectively, expressed as follows: (53)ux,y,t=12tanh⁡-c2c2-1x+y-ctαΓ1+α,ux,y,t=12coth⁡-c2c2-1x+y-ctαΓ1+α.If we takek=ε=1, σ=c/(2(c2-1)), A=c, and R=-2c, then the expression of u7(x,y,t) is the same as that of u3(x,y,t) for Eq. (27) in [39], the expression of u8(x,y,t) is the same as that of u4(x,y,t) for Eq. (27) in [39], and they can be, respectively, expressed as follows:(54)ux,y,t=-12tan⁡-c2c2-1x+y-ctαΓ1+α,ux,y,t=12cot⁡-c2c2-1x+y-ctαΓ1+α.However, we also obtain the bell-shaped solitary wave solution and Jacobi elliptic function solution of the(2+1)-dimension time fractional Zoomeron equation.


## 3.4. The Space-Time Fractional ZKBBM Equation

Next, we solve the exact solution of the following space-time fractional nonlinear Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation [40]:(55)∂αu∂tα+∂βu∂xβ-2au∂βu∂xβ-b∂α∂tα∂2βu∂x2β=0,t>0,α>0,β≤1,where a and b are arbitrary nonzero constants.We first use the fractional complex transformu(x,t)=u(ξ), ξ=xβ/Γ(1+β)-ctα/Γ(1+α), and (55) becomes an ODE of integer order: (56)1-cu′-2auu′+bcu′′′=0.Integrating (56) once with an integral constant of zero yields(57)1-cu-au2+bcu′′=0.Multiplying (57) by u′ and integrating once again with an integral constant of zero, we obtain(58)u′2=a2u2+a3u3,where a2=c-1/bc and a3=2a/3bc.According to the solutions of the auxiliary equation presented in [35], the exact solutions of the (59) are listed in Table 3:(59)φ′2=a2φ2+a3φ3.Table 3
Solutions of (59).
Number φ 1 φ = - a 2 a 3 sech 2 ⁡ a 2 2 ξ, a2>0, a0=0, a1=0  2 φ = - a 2 a 3 sec 2 ⁡ - a 2 2 ξ, a2<0, a0=0, a1=0  3 φ = 1 a 3 ξ 2, a2=0, a0=0, a1=0Equations (58) and (59) have the same form. Thus, looking up Table 3 and determining the coefficients and existence conditions, we can obtain the following solutions of (55):(1)
Ifc>1 and b>0, we obtain (60)u1x,t=-3c-12asech2⁡12c-1bcxβΓ1+β-ctαΓ1+α.(2)
Ifc>1 and b<0, we obtain(61)u2x,t=-3c-12asec2⁡12-c-1bcxβΓ1+β-ctαΓ1+α.(3)
Ifc=1, we obtain (62)u3x,t=2a3bcxβ/Γ1+β-ctα/Γ1+α2.The asymptotic behavior for the obtained exact solution (60)–(62) is shown in Figures 14–16.Figure 14
Exact solutionsu1(x,t) for (60) and its position at t=0, when the parameters a=-4, b=2, c=3, α=0.8, and β=0.5.Figure 15
Exact solutionsu2(x,t) for (61) and its position at t=5, when the parameters a=-4, b=-0.5, c=10, α=0.7, and β=0.9.Figure 16
Exact solutionsu3(x,t) for (62): (a) α=0.5, β=0.2; (b) α=β=0.5, when the parameters a=-4, b=-0.5, and c=10.
(a) (b)It can be observed from Figures14–16 that we obtained single solitary wave and multiple solitary wave solutions of fractional ZKBBM equation, and when α=β, (62) is singular wave solution, and when α≠β, (62) is multiple solitary wave solution. Consequently, the exact solution of fractional differential equation can better describe the complex wave propagation. Reference [40] only obtained Jacobi elliptic doubly periodic solution of fractional ZKBBM equation, but we obtain three types of solutions using our proposed method.


## 4. Conclusions

In this paper, by looking up the tables of solutions of corresponding auxiliary equations, the exact analytical solutions of the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the(2+1)-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation have been constructed directly and successfully. Compared with previous results, solutions of more types are obtained, some of which are new. Furthermore, the solving process is more concise and straightforward. Consequently, the table lookup method is concluded to be effective method and would be a powerful mathematical tool for obtaining more exact analytical solutions of a large number of nonlinear fPDEs.



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