# Exact Solutions for Some Fractional Differential Equations

Authors: Abdullah Sonmezoglu
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

The extended Jacobi elliptic function expansion method is used for solving fractional differential equations
in the sense of Jumarie’s modified Riemann-Liouville derivative. By means of this approach, a few fractional differential
equations are successfully solved. As a result, some new Jacobi elliptic function solutions including solitary wave solutions
and trigonometric function solutions are established. The proposed method can also be applied to other fractional differential
equations.


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## Body

## 1. Introduction

Fractional differential equations attracted attention in physics, biology, engineering, signal processing, systems identification, control theory, finance, and fractional dynamics [1–3]. Also, they are employed in social sciences such as food supplement, climate, finance, and economics.Finding approximate and exact solutions to fractional differential equations is an important task. Various analytical and numerical methods have been introduced to obtain solutions of fractional differential equations, such as the Adomian decomposition method [4, 5], the variational iteration method [6–8], the homotopy analysis method [9–12], the homotopy perturbation method [13–15], the Lagrange characteristic method [16], the finite difference method [17], the finite element method [18], the differential transformation method [19], the fractional subequation method [20–24], the first integral method [25], the (G′/G)-expansion method [26–29], the fractional complex transform method [30], and the modified simple equation method [31–33].In [34], Jumarie proposed a modified Riemann-Liouville derivative. With this kind of fractional derivative and some useful formulas, we can convert fractional differential equations into integer-order differential equations by variable transformation.In this paper, we used extended Jacobi elliptic function expansion method [35–37] to establish exact solutions for three nonlinear space-time fractional differential equations in the sense of Jumarie’s modified Riemann-Liouville derivative, namely, the space-time fractional generalized reaction duffing equation, the space-time fractional bidirectional wave equations, and the space-time fractional symmetric regularized long wave (SRLW) equation. Also, we included figures to show the properties of some Jacobi elliptic function solutions of these fractional differential equations.


## 2. Jumarie’s Modified Riemann-Liouville Derivative and the Extended Jacobi Elliptic Function Expansion Method

In this section, we first give the definition and some properties of the modified Riemann-Liouville derivative which are used further in this paper.The Jumarie modified Riemann-Liouville derivative of orderα is defined by the expression [34](1)Dxαfx=1Γ-α∫0xx-ξ-α-1fξ-f0dξ,α<0,1Γ1-αddx∫0xx-ξ-αfξ-f0dξ,0<α<1,fnxα-n,n≤α<n+1,n≥1,where f:R→R, x→f(x) denote a continuous (but not necessarily differentiable) function.Some properties of the fractional modified Riemann-Liouville derivative were summarized and three useful formulas of them are [34](2)Dxαxr=Γ1+rΓ1+r-αxr-α,Dxαfxgx=gxDxαfx+fxDxαgx,Dxαfgx=fg′gxDxαgx=Dgαfgxg′xα.Next, let us consider nonlinear partial fractional differential equation(3)Pu,Dtαu,Dxβu,Dyγu,DtαDtαu,DtαDxβu,DxβDxβu,DxβDyγu,DyγDyγu,…=0,0<α,β,γ≤1,where u is an unknown function and P is a polynomial of u. In this equation, the partial fractional derivatives involving the highest order derivatives and the nonlinear terms are included.Li and He [38] presented a fractional complex transform to convert fractional differential equations into ordinary differential equations (ODEs), so all analytical methods devoted to the advanced calculus can be easily applied to the fractional calculus. By using the traveling wave variable(4)ux,y,t=Uξ,ξ=δxβΓ1+β+ζyγΓ1+γ+λtαΓ1+α,where δ, ζ are nonzero arbitrary constants and λ is the wave speed, we can rewrite (3) as the following nonlinear ODE:(5)QU,U′,U′′,U′′′,…=0,where the prime denotes the derivation with respect to ξ. If possible, we should integrate (5) term by term one or more times.Our main goal is to derive exact or at least approximate solutions, if possible, for this ODE. For this purpose, using the extended Jacobi elliptic function expansion method,U(ξ) can be expressed as a finite series of Jacobi elliptic functions, snξ, that is, the ansatz:(6)ux,y,t=Uξ=∑j=0najsnjξ+∑j=1nbjsn-jξ.The parameter n is determined by balancing the linear term(s) of highest order with the nonlinear one(s). And(7)cn2ξ=1-sn2ξ,dn2ξ=1-m2sn2ξ,ddξsnξ=cnξdnξ,ddξcnξ=-snξdnξ,ddξdnξ=-m2snξcnξ,where cnξ and dnξ are the Jacobi elliptic cosine function and the Jacobi elliptic function of the third kind, respectively, with the modulus m(0<m<1). Therefore, the highest degree of dpU/dξp is taken as(8)OdpUdξp=n+p,p=1,2,3,…,OUqdpUdξp=q+1n+p,q=0,1,2,…,p=1,2,3,….Substituting (6)–(8) into (5) and comparing the coefficients of each power of snξ in both sides, we get an overdetermined system of nonlinear algebraic equations with respect to λ, aj(j=0,1,…,n), and bj(j=1,2,…,n). Solving this system, with the aid of Mathematica, then λ, aj(j=0,1,…,n), and bj(j=1,2,…,n) can be determined. Substituting these results into (6), then some new Jacobi elliptic function solutions of (3) can be obtained. We can get other kinds of Jacobi doubly periodic wave solutions.Since(9)limm→1⁡snξ=tanh⁡ξ,limm→1⁡cnξ=sech⁡ξ,limm→1⁡dnξ=sech⁡ξ,limm→0⁡snξ=sin⁡ξ,limm→0⁡cnξ=cos⁡ξ,limm→0⁡dnξ=1,u degenerates, respectively, as the following form. (1)
Solitary wave solutions:(10)ux,y,t=∑j=0najtanhj⁡ξ+∑j=1nbjcothj⁡ξ.(2)
Triangular function formal solution:(11)ux,y,t=∑j=0najsinj⁡ξ+∑j=1nbjcscj⁡ξ.


## 3. Applications of the Method

In this section, we present three examples to demonstrate the effectiveness of our approach to solve nonlinear fractional partial differential equations.


### 3.1. Space-Time Fractional Generalized Reaction Duffing Equation

We have applied the extended Jacobi elliptic function expansion method to construct the exact solutions of space-time fractional generalized reaction duffing equation [39, 40] in the form(12)∂2αu∂t2α+p∂2αu∂x2α+qu+ru2+su3=0,0<α<1,where p, q, r, and s are all constants. Equation (12) reduces many well-known nonlinear fractional wave equations such as the following.(i)
Fractional Klein-Gordon equation:(13)∂2αu∂t2α-∂2αu∂x2α-au-bu3=0,t>0,0<α<1.(ii)
Fractional Landau-Ginzburg-Higgs equation:(14)∂2αu∂t2α-∂2αu∂x2α-m2u+g2u3=0,t>0,0<α<1.(iii)
Fractionalφ4 equation:(15)∂2αu∂t2α-∂2αu∂x2α+u-u3=0,t>0,0<α<1.(iv)
Fractional duffing equation:(16)∂2αu∂t2α+au+bu3=0,t>0,0<α<1.(v)
Fractional Sine-Gordon equation:(17)∂2αu∂t2α-∂2αu∂x2α+u-16u3=0,t>0,0<α<1.For our purpose, we introduce the following transformations:(18)ux,t=Uξ,ξ=lxαΓ1+α-λtαΓ1+α,where ξ is a wave variable and l and λ are constants; all of them are to be determined. Substituting (18) into (12), (12) is reduced into an ODE:(19)U′′ξ+qλ2+pl2Uξ+rλ2+pl2U2ξ+sλ2+pl2U3ξ=0,where U′=dU/dξ. Suppose that the solution of (19) can be expressed by(20)Uξ=∑j=0najsnjξ+∑j=1nbjsn-jξ.Considering the homogeneous balance between the highest order derivative U′′ and the highest order nonlinear term U3 in (19), we obtain n=1. So(21)Uξ=a0+a1snξ+b1sn-1ξ.Substituting (21) into (19) and comparing the coefficients of each power of snξ in both sides, we get an overdetermined system of nonlinear algebraic equations with respect to λ, a0, a1, and b1. Solving this system with Mathematica, we get the following results.Case 1.
Consider(22)a0=-r3s,a1=±2mr3s21+m2,b1=0,λ=±ir2+9l2ps1+m23s1+m2,q=2r29s.Case 2.
Consider(23)a0=-r3s,a1=0,b1=±2r3s21+m2,λ=±ir2+9l2ps1+m23s1+m2,q=2r29s.Case 3.
Consider(24)a0=-r3s,a1=±2mr3s1+m6+m,b1=±2r3s1+m6+m,λ=-ir2+9l2ps1+m6+m3s1+m6+m,q=2r29s.Thus, we obtain the following solutions of (12).Solution 1.
See Figure1:(25)u1=-r3s±2mr3s21+m2·snlxαΓ1+α∓ir2+9l2ps1+m23Γ1+αs1+m2tα.Figure 1
Profiles of|u| in (25) corresponding to the values m=0.1, α=0.9, p=r=s=l=1; m=0.9, α=0.2, p=r=s=l=1; and m=α=0.5, p=l=3, r=s=-4 from (a) to (c).
(a)
(b)
(c)Solution 2.
See Figure2:(26)u2=-r3s±2r3s21+m2·sn-1lxαΓ1+α∓ir2+9l2ps1+m23Γ1+αs1+m2tα.Figure 2
Profiles of|u| in (26) corresponding to the values m=0.5, α=0.9, p=r=s=l=1; m=α=0.5, p=r=s=l=1; and m=0.4, α=0.1, l=1, p=r=s=2 from (a) to (c).
(a)
(b)
(c)Solution 3.
Consider(27)u3=-r3s±2r3s1+m6+m×msnlxαΓ1+α+ir2+9l2ps1+m6+m3Γ1+αs1+m6+mtα+sn-1lxαΓ1+α+ir2+9l2ps1+m6+m3Γ1+αs1+m6+mtα.


#### 3.1.1. Soliton Solutions

When the modulusm approaches to 1 in (25), (26), and (27), we can obtain solitary wave solutions of space-time fractional generalized reaction duffing equation, respectively:(28)u4=-r3s±r3stanh⁡lxαΓ1+α∓ir2+18l2ps32sΓ1+αtα,u5=-r3s±r3scoth⁡lxαΓ1+α∓ir2+18l2ps32sΓ1+αtα,u6=-r3s±r3scoth⁡2lxαΓ1+α+ir2+72l2ps38sΓ1+αtα.


#### 3.1.2. Triangular Periodic Solutions

When the modulusm approaches to zero in (26), (27), we can obtain trigonometric function solutions of space-time fractional generalized reaction duffing equation, respectively: (29)u7=-r3s±2r3scsc⁡lxαΓ1+α∓ir2+9l2ps3sΓ1+αtα,u8=-r3s±2r3s+csc⁡lxαΓ1+α+ir2+9l2ps3sΓ1+αtα.


### 3.2. Space-Time Fractional Bidirectional Wave Equations

Let us apply our method to the space-time fractional bidirectional wave equations in the form [41, 42](30)Dtαv+Dxαu+uDxαv+vDxαu+aDxαDxαDxαu-bDxαDxαDtαv=0,Dtαu+Dxαv+uDxαu+cDxαDxαDxαv-dDxαDxαDtαu=0,0<α≤1,where x represents the distance along the channel, t is the elapsed time, the variable u(x,t) is the dimensionless horizontal velocity, v(x,t) is the dimensionless deviation of the water surface from its undisturbed position, and a, b, c, and d are real constants. When α=1, (30) is the generalization of bidirectional wave equations, which can be used as a model equation for the propagation of long waves on the surface of water with a small amplitude by Bona and Chen [43].For our purpose, we use the following transformation:(31)ux,t=Uξ,vx,t=Vξ,ξ=RxαΓ1+α+StαΓ1+α,where R and S are nonzero constants. Substituting (31) into (30), we obtain(32)SV′+RU′+RUV′+RVU′+aR3U′′′-bR2SV′′′=0,SU′+RV′+RUU′+cR3V′′′-dR2SU′′′=0,where U′=dU/dξ. Suppose that the solutions of (32) can be expressed by(33)Uξ=∑j=0n1ajsnjξ+∑j=1n1bjsn-jξ,Vξ=∑j=0n2cjsnjξ+∑j=1n2djsn-jξ.Balancing the highest order derivative terms and nonlinear terms in (32), we can obtain n1=n2=2. So we have(34)Uξ=a0+a1snξ+a2sn2ξ+b1sn-1ξ+b2sn-2ξ,Vξ=c0+c1snξ+c2sn2ξ+d1sn-1ξ+d2sn-2ξ.Proceeding as in the previous case, we get the following results.Case 1.
Consider(35)a1=b1=b2=c1=d1=d2=0,c2=c2,a0=i2c+dc2+6am2R2-bc21-4cR21+m22R3bcm2bc2-2dc2+6am2R2,a2=-i23cbc2m2Rbm2bc2-2dc2+6am2R2,c0=-bc2+4cR2c2+m23+c2-2dc2+6am2R212bcm2R2,S=-icc2+6am2R23bm2bc2-2dc2+6am2R2.Case 2.
Consider(36)a1=b1=c1=d1=0,c2=c2,a0=i2c+dc2+6am2R2-bc21-4cR21+m22R3bcm2bc2-2dc2+6am2R2,a2=-i23cbc2m2Rbm2bc2-2dc2+6am2R2,c0=-bc2+4cR2c2+m23+c2-2dc2+6am2R212bcm2R2,b2=-i23cbc2Rbm2bc2-2dc2+6am2R2,S=-icc2+6am2R23bm2bc2-2dc2+6am2R2,d2=c2m2.Thus, we obtain the following solutions of (30).Solution 1.
Consider(37)u1=i2c+dc2+6am2R2-bc21-4cR21+m22R3bcm2bc2-2dc2+6am2R2-i23cbc2m2Rbm2bc2-2dc2+6am2R2·sn2RxαΓ1+α+SΓ1+αtα,v1=-bc2+4cR2c2+m23+c2-2dc2+6am2R212bcm2R2+c2sn2RxαΓ1+α+SΓ1+αtα.Solution 2.
Consider(38)u2=i2c+dc2+6am2R2-bc21-4cR21+m22R3bcm2bc2-2dc2+6am2R2-i23cbc2Rbm2bc2-2dc2+6am2R2m2sn2RxαΓ1+α+SΓ1+αtα+sn-2RxαΓ1+α+SΓ1+αtα,v2=-bc2+4cR2c2+m23+c2-2dc2+6am2R212bcm2R2+c2sn2RxαΓ1+α+SΓ1+αtα+c2m2sn-2RxαΓ1+α+SΓ1+αtα,where(39)S=-icc2+6am2R23bm2bc2-2dc2+6am2R2.


#### 3.2.1. Soliton Solutions

When the modulusm approaches to 1 in (37), (38), we can obtain solitary wave solutions of the space-time fractional bidirectional wave equations, respectively:(40)u3=i2c+dc2+6aR2-bc21-8cR22R3bcbc2-2dc2+6aR2-i·23cbc2Rbbc2-2dc2+6aR2·tanh2⁡RxαΓ1+α+SΓ1+αtα,v3=-bc2+4cR23+2c2-2dc2+6aR212bcR2+c2·tanh2⁡RxαΓ1+α+SΓ1+αtα,u4=i2c+dc2+6aR2-bc21-8cR22R3bcbc2-2dc2+6aR2-i·43cbc2Rbbc2-2dc2+6aR21+2csch2⁡2RxαΓ1+α+SΓ1+αtα,v4=-bc2+4cR23+2c2-2dc2+6aR212bcm2R2+2c21+2csch2⁡2RxαΓ1+α+SΓ1+αtα,where(41)S=-icc2+6aR23bbc2-2dc2+6aR2.


### 3.3. The Space-Time Nonlinear Fractional SRLW Equation

We consider the space-time nonlinear fractional SRLW equation [44, 45](42)Dt2αu+Dx2αu+uDtαDxαu+DtαuDxαu+Dt2αDx2αu=0,0<α≤1,which arises in several physical applications including ion sound waves in plasma. For our purpose, we use the following transformation:(43)ux,t=Uξ,ξ=kxαΓ1+α+ctαΓ1+α+ξ0,where k, c, and ξ0 are constants with k, c≠0. Substituting (43) into (42), we obtain(44)2k2c2U′′+2k2+c2U+kcU2=0.Suppose that the solutions of (44) can be expressed by(45)Uξ=∑j=0najsnjξ+∑j=1nbjsn-jξ.Considering the homogeneous balance between the highest order derivative U′′ and the highest order nonlinear term U2 in (44), we obtain n=2. So we have(46)Uξ=a0+a1snξ+a2sn2ξ+b1sn-1ξ+b2sn-2ξ.Proceeding as in the previous cases, we get the following results.Case 1.
Consider(47)a1=b1=b2=0,a2=-12ckm2,a0=4ck1+m2+1-m2+m4,where(48)c=ik21+4k21-m2+m4.Case 2.
Consider(49)a1=a2=b1=0,b2=-12ck,a0=4ck1+m2-1-m2+m4,where(50)c=ik21-4k21-m2+m4.Case 3.
Consider(51)a1=b1=0,a2=-12ckm2,b2=-12ck,a0=4ck1+m2+1+14m2+m4,where(52)c=ik21+4k21+14m2+m4.Thus, we obtain the following solutions of (42).Solution 1.
Consider(53)u1=4ck1+m2+1-m2+m4-12ckm2sn2kxαΓ1+α+iΓ1+αk21+4k21-m2+m4tα+ξ0.Solution 2.
Consider(54)u2=4ck1+m2-1-m2+m4-12cksn-2kxαΓ1+α+iΓ1+αk21-4k21-m2+m4tα+ξ0.Solution 3.
Consider(55)u3=4ck1+m2+1+14m2+m4-12ckm2sn2kxαΓ1+α+iΓ1+αk21+4k21+14m2+m4tα+ξ0+sn-2kxαΓ1+α+iΓ1+αk21+4k21+14m2+m4tα+ξ0.


#### 3.3.1. Soliton Solutions

When the modulusm approaches to 1 in (53), (54), and (55), we can obtain solitary wave solutions of the space-time nonlinear fractional SRLW equation, respectively:(56)u4=12cksech2⁡kxαΓ1+α+iΓ1+αk21+4k2tα+ξ0,u5=-8ck-12ckcsch2⁡kxαΓ1+α+iΓ1+αk21-4k2tα+ξ0,u6=-48ckcsch2⁡2kxαΓ1+α+iΓ1+αk21+16k2tα+ξ0.


#### 3.3.2. Triangular Periodic Solutions

We can obtain trigonometric function solutions of the space-time nonlinear fractional SRLW equation, when the modulusm approaches to zero; for example, (54), (55) give the same solution:(57)u7=-12ck·csc2⁡kxαΓ1+α+iΓ1+αk21-4k2tα+ξ0,u8=8ck-12ck·csc2⁡kxαΓ1+α+iΓ1+αk21+4k2tα+ξ0.


## 3.1. Space-Time Fractional Generalized Reaction Duffing Equation

We have applied the extended Jacobi elliptic function expansion method to construct the exact solutions of space-time fractional generalized reaction duffing equation [39, 40] in the form(12)∂2αu∂t2α+p∂2αu∂x2α+qu+ru2+su3=0,0<α<1,where p, q, r, and s are all constants. Equation (12) reduces many well-known nonlinear fractional wave equations such as the following.(i)
Fractional Klein-Gordon equation:(13)∂2αu∂t2α-∂2αu∂x2α-au-bu3=0,t>0,0<α<1.(ii)
Fractional Landau-Ginzburg-Higgs equation:(14)∂2αu∂t2α-∂2αu∂x2α-m2u+g2u3=0,t>0,0<α<1.(iii)
Fractionalφ4 equation:(15)∂2αu∂t2α-∂2αu∂x2α+u-u3=0,t>0,0<α<1.(iv)
Fractional duffing equation:(16)∂2αu∂t2α+au+bu3=0,t>0,0<α<1.(v)
Fractional Sine-Gordon equation:(17)∂2αu∂t2α-∂2αu∂x2α+u-16u3=0,t>0,0<α<1.For our purpose, we introduce the following transformations:(18)ux,t=Uξ,ξ=lxαΓ1+α-λtαΓ1+α,where ξ is a wave variable and l and λ are constants; all of them are to be determined. Substituting (18) into (12), (12) is reduced into an ODE:(19)U′′ξ+qλ2+pl2Uξ+rλ2+pl2U2ξ+sλ2+pl2U3ξ=0,where U′=dU/dξ. Suppose that the solution of (19) can be expressed by(20)Uξ=∑j=0najsnjξ+∑j=1nbjsn-jξ.Considering the homogeneous balance between the highest order derivative U′′ and the highest order nonlinear term U3 in (19), we obtain n=1. So(21)Uξ=a0+a1snξ+b1sn-1ξ.Substituting (21) into (19) and comparing the coefficients of each power of snξ in both sides, we get an overdetermined system of nonlinear algebraic equations with respect to λ, a0, a1, and b1. Solving this system with Mathematica, we get the following results.Case 1.
Consider(22)a0=-r3s,a1=±2mr3s21+m2,b1=0,λ=±ir2+9l2ps1+m23s1+m2,q=2r29s.Case 2.
Consider(23)a0=-r3s,a1=0,b1=±2r3s21+m2,λ=±ir2+9l2ps1+m23s1+m2,q=2r29s.Case 3.
Consider(24)a0=-r3s,a1=±2mr3s1+m6+m,b1=±2r3s1+m6+m,λ=-ir2+9l2ps1+m6+m3s1+m6+m,q=2r29s.Thus, we obtain the following solutions of (12).Solution 1.
See Figure1:(25)u1=-r3s±2mr3s21+m2·snlxαΓ1+α∓ir2+9l2ps1+m23Γ1+αs1+m2tα.Figure 1
Profiles of|u| in (25) corresponding to the values m=0.1, α=0.9, p=r=s=l=1; m=0.9, α=0.2, p=r=s=l=1; and m=α=0.5, p=l=3, r=s=-4 from (a) to (c).
(a)
(b)
(c)Solution 2.
See Figure2:(26)u2=-r3s±2r3s21+m2·sn-1lxαΓ1+α∓ir2+9l2ps1+m23Γ1+αs1+m2tα.Figure 2
Profiles of|u| in (26) corresponding to the values m=0.5, α=0.9, p=r=s=l=1; m=α=0.5, p=r=s=l=1; and m=0.4, α=0.1, l=1, p=r=s=2 from (a) to (c).
(a)
(b)
(c)Solution 3.
Consider(27)u3=-r3s±2r3s1+m6+m×msnlxαΓ1+α+ir2+9l2ps1+m6+m3Γ1+αs1+m6+mtα+sn-1lxαΓ1+α+ir2+9l2ps1+m6+m3Γ1+αs1+m6+mtα.


### 3.1.1. Soliton Solutions

When the modulusm approaches to 1 in (25), (26), and (27), we can obtain solitary wave solutions of space-time fractional generalized reaction duffing equation, respectively:(28)u4=-r3s±r3stanh⁡lxαΓ1+α∓ir2+18l2ps32sΓ1+αtα,u5=-r3s±r3scoth⁡lxαΓ1+α∓ir2+18l2ps32sΓ1+αtα,u6=-r3s±r3scoth⁡2lxαΓ1+α+ir2+72l2ps38sΓ1+αtα.


### 3.1.2. Triangular Periodic Solutions

When the modulusm approaches to zero in (26), (27), we can obtain trigonometric function solutions of space-time fractional generalized reaction duffing equation, respectively: (29)u7=-r3s±2r3scsc⁡lxαΓ1+α∓ir2+9l2ps3sΓ1+αtα,u8=-r3s±2r3s+csc⁡lxαΓ1+α+ir2+9l2ps3sΓ1+αtα.


## 3.1.1. Soliton Solutions

When the modulusm approaches to 1 in (25), (26), and (27), we can obtain solitary wave solutions of space-time fractional generalized reaction duffing equation, respectively:(28)u4=-r3s±r3stanh⁡lxαΓ1+α∓ir2+18l2ps32sΓ1+αtα,u5=-r3s±r3scoth⁡lxαΓ1+α∓ir2+18l2ps32sΓ1+αtα,u6=-r3s±r3scoth⁡2lxαΓ1+α+ir2+72l2ps38sΓ1+αtα.


## 3.1.2. Triangular Periodic Solutions

When the modulusm approaches to zero in (26), (27), we can obtain trigonometric function solutions of space-time fractional generalized reaction duffing equation, respectively: (29)u7=-r3s±2r3scsc⁡lxαΓ1+α∓ir2+9l2ps3sΓ1+αtα,u8=-r3s±2r3s+csc⁡lxαΓ1+α+ir2+9l2ps3sΓ1+αtα.


## 3.2. Space-Time Fractional Bidirectional Wave Equations

Let us apply our method to the space-time fractional bidirectional wave equations in the form [41, 42](30)Dtαv+Dxαu+uDxαv+vDxαu+aDxαDxαDxαu-bDxαDxαDtαv=0,Dtαu+Dxαv+uDxαu+cDxαDxαDxαv-dDxαDxαDtαu=0,0<α≤1,where x represents the distance along the channel, t is the elapsed time, the variable u(x,t) is the dimensionless horizontal velocity, v(x,t) is the dimensionless deviation of the water surface from its undisturbed position, and a, b, c, and d are real constants. When α=1, (30) is the generalization of bidirectional wave equations, which can be used as a model equation for the propagation of long waves on the surface of water with a small amplitude by Bona and Chen [43].For our purpose, we use the following transformation:(31)ux,t=Uξ,vx,t=Vξ,ξ=RxαΓ1+α+StαΓ1+α,where R and S are nonzero constants. Substituting (31) into (30), we obtain(32)SV′+RU′+RUV′+RVU′+aR3U′′′-bR2SV′′′=0,SU′+RV′+RUU′+cR3V′′′-dR2SU′′′=0,where U′=dU/dξ. Suppose that the solutions of (32) can be expressed by(33)Uξ=∑j=0n1ajsnjξ+∑j=1n1bjsn-jξ,Vξ=∑j=0n2cjsnjξ+∑j=1n2djsn-jξ.Balancing the highest order derivative terms and nonlinear terms in (32), we can obtain n1=n2=2. So we have(34)Uξ=a0+a1snξ+a2sn2ξ+b1sn-1ξ+b2sn-2ξ,Vξ=c0+c1snξ+c2sn2ξ+d1sn-1ξ+d2sn-2ξ.Proceeding as in the previous case, we get the following results.Case 1.
Consider(35)a1=b1=b2=c1=d1=d2=0,c2=c2,a0=i2c+dc2+6am2R2-bc21-4cR21+m22R3bcm2bc2-2dc2+6am2R2,a2=-i23cbc2m2Rbm2bc2-2dc2+6am2R2,c0=-bc2+4cR2c2+m23+c2-2dc2+6am2R212bcm2R2,S=-icc2+6am2R23bm2bc2-2dc2+6am2R2.Case 2.
Consider(36)a1=b1=c1=d1=0,c2=c2,a0=i2c+dc2+6am2R2-bc21-4cR21+m22R3bcm2bc2-2dc2+6am2R2,a2=-i23cbc2m2Rbm2bc2-2dc2+6am2R2,c0=-bc2+4cR2c2+m23+c2-2dc2+6am2R212bcm2R2,b2=-i23cbc2Rbm2bc2-2dc2+6am2R2,S=-icc2+6am2R23bm2bc2-2dc2+6am2R2,d2=c2m2.Thus, we obtain the following solutions of (30).Solution 1.
Consider(37)u1=i2c+dc2+6am2R2-bc21-4cR21+m22R3bcm2bc2-2dc2+6am2R2-i23cbc2m2Rbm2bc2-2dc2+6am2R2·sn2RxαΓ1+α+SΓ1+αtα,v1=-bc2+4cR2c2+m23+c2-2dc2+6am2R212bcm2R2+c2sn2RxαΓ1+α+SΓ1+αtα.Solution 2.
Consider(38)u2=i2c+dc2+6am2R2-bc21-4cR21+m22R3bcm2bc2-2dc2+6am2R2-i23cbc2Rbm2bc2-2dc2+6am2R2m2sn2RxαΓ1+α+SΓ1+αtα+sn-2RxαΓ1+α+SΓ1+αtα,v2=-bc2+4cR2c2+m23+c2-2dc2+6am2R212bcm2R2+c2sn2RxαΓ1+α+SΓ1+αtα+c2m2sn-2RxαΓ1+α+SΓ1+αtα,where(39)S=-icc2+6am2R23bm2bc2-2dc2+6am2R2.


### 3.2.1. Soliton Solutions

When the modulusm approaches to 1 in (37), (38), we can obtain solitary wave solutions of the space-time fractional bidirectional wave equations, respectively:(40)u3=i2c+dc2+6aR2-bc21-8cR22R3bcbc2-2dc2+6aR2-i·23cbc2Rbbc2-2dc2+6aR2·tanh2⁡RxαΓ1+α+SΓ1+αtα,v3=-bc2+4cR23+2c2-2dc2+6aR212bcR2+c2·tanh2⁡RxαΓ1+α+SΓ1+αtα,u4=i2c+dc2+6aR2-bc21-8cR22R3bcbc2-2dc2+6aR2-i·43cbc2Rbbc2-2dc2+6aR21+2csch2⁡2RxαΓ1+α+SΓ1+αtα,v4=-bc2+4cR23+2c2-2dc2+6aR212bcm2R2+2c21+2csch2⁡2RxαΓ1+α+SΓ1+αtα,where(41)S=-icc2+6aR23bbc2-2dc2+6aR2.


## 3.2.1. Soliton Solutions

When the modulusm approaches to 1 in (37), (38), we can obtain solitary wave solutions of the space-time fractional bidirectional wave equations, respectively:(40)u3=i2c+dc2+6aR2-bc21-8cR22R3bcbc2-2dc2+6aR2-i·23cbc2Rbbc2-2dc2+6aR2·tanh2⁡RxαΓ1+α+SΓ1+αtα,v3=-bc2+4cR23+2c2-2dc2+6aR212bcR2+c2·tanh2⁡RxαΓ1+α+SΓ1+αtα,u4=i2c+dc2+6aR2-bc21-8cR22R3bcbc2-2dc2+6aR2-i·43cbc2Rbbc2-2dc2+6aR21+2csch2⁡2RxαΓ1+α+SΓ1+αtα,v4=-bc2+4cR23+2c2-2dc2+6aR212bcm2R2+2c21+2csch2⁡2RxαΓ1+α+SΓ1+αtα,where(41)S=-icc2+6aR23bbc2-2dc2+6aR2.


## 3.3. The Space-Time Nonlinear Fractional SRLW Equation

We consider the space-time nonlinear fractional SRLW equation [44, 45](42)Dt2αu+Dx2αu+uDtαDxαu+DtαuDxαu+Dt2αDx2αu=0,0<α≤1,which arises in several physical applications including ion sound waves in plasma. For our purpose, we use the following transformation:(43)ux,t=Uξ,ξ=kxαΓ1+α+ctαΓ1+α+ξ0,where k, c, and ξ0 are constants with k, c≠0. Substituting (43) into (42), we obtain(44)2k2c2U′′+2k2+c2U+kcU2=0.Suppose that the solutions of (44) can be expressed by(45)Uξ=∑j=0najsnjξ+∑j=1nbjsn-jξ.Considering the homogeneous balance between the highest order derivative U′′ and the highest order nonlinear term U2 in (44), we obtain n=2. So we have(46)Uξ=a0+a1snξ+a2sn2ξ+b1sn-1ξ+b2sn-2ξ.Proceeding as in the previous cases, we get the following results.Case 1.
Consider(47)a1=b1=b2=0,a2=-12ckm2,a0=4ck1+m2+1-m2+m4,where(48)c=ik21+4k21-m2+m4.Case 2.
Consider(49)a1=a2=b1=0,b2=-12ck,a0=4ck1+m2-1-m2+m4,where(50)c=ik21-4k21-m2+m4.Case 3.
Consider(51)a1=b1=0,a2=-12ckm2,b2=-12ck,a0=4ck1+m2+1+14m2+m4,where(52)c=ik21+4k21+14m2+m4.Thus, we obtain the following solutions of (42).Solution 1.
Consider(53)u1=4ck1+m2+1-m2+m4-12ckm2sn2kxαΓ1+α+iΓ1+αk21+4k21-m2+m4tα+ξ0.Solution 2.
Consider(54)u2=4ck1+m2-1-m2+m4-12cksn-2kxαΓ1+α+iΓ1+αk21-4k21-m2+m4tα+ξ0.Solution 3.
Consider(55)u3=4ck1+m2+1+14m2+m4-12ckm2sn2kxαΓ1+α+iΓ1+αk21+4k21+14m2+m4tα+ξ0+sn-2kxαΓ1+α+iΓ1+αk21+4k21+14m2+m4tα+ξ0.


### 3.3.1. Soliton Solutions

When the modulusm approaches to 1 in (53), (54), and (55), we can obtain solitary wave solutions of the space-time nonlinear fractional SRLW equation, respectively:(56)u4=12cksech2⁡kxαΓ1+α+iΓ1+αk21+4k2tα+ξ0,u5=-8ck-12ckcsch2⁡kxαΓ1+α+iΓ1+αk21-4k2tα+ξ0,u6=-48ckcsch2⁡2kxαΓ1+α+iΓ1+αk21+16k2tα+ξ0.


### 3.3.2. Triangular Periodic Solutions

We can obtain trigonometric function solutions of the space-time nonlinear fractional SRLW equation, when the modulusm approaches to zero; for example, (54), (55) give the same solution:(57)u7=-12ck·csc2⁡kxαΓ1+α+iΓ1+αk21-4k2tα+ξ0,u8=8ck-12ck·csc2⁡kxαΓ1+α+iΓ1+αk21+4k2tα+ξ0.


## 3.3.1. Soliton Solutions

When the modulusm approaches to 1 in (53), (54), and (55), we can obtain solitary wave solutions of the space-time nonlinear fractional SRLW equation, respectively:(56)u4=12cksech2⁡kxαΓ1+α+iΓ1+αk21+4k2tα+ξ0,u5=-8ck-12ckcsch2⁡kxαΓ1+α+iΓ1+αk21-4k2tα+ξ0,u6=-48ckcsch2⁡2kxαΓ1+α+iΓ1+αk21+16k2tα+ξ0.


## 3.3.2. Triangular Periodic Solutions

We can obtain trigonometric function solutions of the space-time nonlinear fractional SRLW equation, when the modulusm approaches to zero; for example, (54), (55) give the same solution:(57)u7=-12ck·csc2⁡kxαΓ1+α+iΓ1+αk21-4k2tα+ξ0,u8=8ck-12ck·csc2⁡kxαΓ1+α+iΓ1+αk21+4k2tα+ξ0.


## 4. Conclusion

In this paper, we used the extended Jacobi elliptic function expansion method for solving fractional differential equations and applied it to find exact solutions of the space-time fractional generalized reaction duffing equation, the space-time fractional bidirectional wave equations, and the space-time fractional symmetric regularized long wave (SRLW) equation. With the aid of Mathematica, we successfully obtained some new Jacobi elliptic function solutions including solitary wave solutions and trigonometric function solutions for these equations. This method is effective and can also be applied to other fractional differential equations.



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