Global Journal of Pure and Applied Mathematics.ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 2629-2658© Research India Publicationshttp://www.ripublication.com/gjpam.htm

More Explicit Solitary Solutions of the Space-TimeFractional Fifth Order Nonlinear Sawada-Kotera

Equation via the Improved GeneralizedRiccati Equation Mapping Method

Sanoe Koonprasert, Sekson Sirisubtawee, and Sivaporn Ampun

King Mongkut’s University Technology North Bangkok,Applied Sciences Faculty, Department of Mathematics,

Bangkok, Thailand.

Abstract

In this paper, a new construction of exact solutions based on the improved general-ized Riccati equation mapping method with modified Reimann-Luiviile fractionalderivative and symbolic computation is proposed for seeking abundant solutionsof the space-time fractional fifth-order nonlinear Sawada-Kotera equation. Theproposed method is very simple, direct, effective and convenient for obtaining dif-ferent types of exact solutions of the space-time fractional fifth-order nonlinearSawada-Kotera equation. We can obtain new general exact solutions in variousforms including trigonometric function solutions, hyperbolic function solutions,and rational function solutions of this equation from the method with the aid of themathematical software Maple. Moreover, this method is a powerful mathematicaltool for generating more solutions for solving other fractional differential equationsand systems of nonlinear fractional differential equations.

AMS subject classification:Keywords: Exact solutions, Soliton solutions, Time-space fractional differentialequations, Local fractional derivative, Generalized Riccati equation mapping method,Sawada-Kotera equation.

2 Sanoe Koonprasert, Sekson Sirisubtawee, and Sivaporn Ampun

1. Introduction

During the past three decades, the subject of fractional derivatives and integrals ofany arbitrary real or complex order has gained considerable popularity in diverse andwidespread fields. In particular, fractional order partial differential equations (FPDEs)have been found in many important roles due to their numerous applications in a widerange of areas such as engineering and physics phenomena in viscoelasticity of fluidmechanics [2] and fractional dynamics [4], signal processing [1], electromagnetic, elec-trochemistry [5], control theory. The rapid development and efficient methods that areof great help in tackling real world problems of highly complex nature are needed, but itis difficult to get exact solutions of fractional differential equations (FPDEs). Therefore,many researchers have been interested in studying the fractional calculus and finding ac-curate and efficient methods for solving fractional partial differential equations (FPDEs).Among the investigations for FPDEs, seeking exact solutions of FPDEs is an importanttask for many researchers.

Nowadays, many efficient methods have been developed to obtain exact solutions ofFPDEs, such as the fractional sub-sequence G′/G method[11]-[13], the expo-functionmethod [10], Homotopy perturbation method [14]-[15], the simplest equation method[17], the Riccati equation [22]-[23], the tan(φ(ξ)/2)-expansion method [24]-[25], theKudryashov method [26]-[27], the modified trial equation method [28]-[29], the firstintegral method [30]-[31], the fractional G′/G ⣓ expansion method [32]-[36] , thefractional exp-function method [37]-[39], fractional sub-equation method [40]-[42], thefractional functional variable method [43], the fractional modified trial equation method[44]-[45], and etc.

According to the nonlinear ordinary differential equation φ′(ξ) = r + pφ(ξ) +qφ2(ξ), a generalized Riccati equation mapping method was introduced for solvingsome nonlinear evolution equations. The method was developed from the G′/G methodby using the cole-Hopf transformation and has constructed more solutions than the G′/G

method. Several advantages of the generalized Riccati equation mapping method arethat it is firstly a simple and straightforward method. Secondly, it is convenient forthe homogeneous balancing order and for computing a system of algebraic equations.Lastly, the method can generate a variety of structures of exact solutions of FPDEsthat gives the twenty seven types of solutions including: a soliton solution, a periodicsolution and a rational solution. In 2012, Hasibun et al. [47], applied this method tosolve the Modified Benjamin-Bona-Mahony partial differential equation. In the sameyear, Hafez[48]-[51] proposed the novel (G′/G) expansion method with the Riccattiequation to find new travelling wave solutions of the cubic nonlinear Schrodinger partialdifferential equation.

The classical Sawada-Kotera equation is an important mathematical model used inmany different physical contexts to describe the motion of long waves in shallow waterunder the gravity, quantum mechanics and nonlinear optics. It is also applied to wavephenomenon of the plasma media and fluid dynamics that are modelled by kink shapedtanh solitons, or the bell shaped sech solutions, and are used in modelling waves thatpropagate in opposite directions.

Explicit Solitary Solutions of the Space-Time Fractional Fifth Order... 3

In this paper, we further extend the space-time factional fifth order Sawada-Koteraequation and construct new and more solitary solutions of the space-time factional fifthorder Sawada-Kotera equation by using an improved generalized Riccati equation map-ping method that gives different forms of solutions. Consider the following space-timefractional (1 + 1) dimensional generalized fifth order Sawada-Kotera equation [2015Gupta Ray]:

Dαt u(x, t) + au2(x, t)Dβ

x u(x, t) + bDβx u(x, t)D2β

x u(x, t) + cu(x, t)D3βx u(x, t)

+ dD5βx u(x, t) = 0, t > 0, x > 0, (1.1)

where a, b, c and d are constants, u(x, t) is a field variable, and the parameters 0 <

α, β ≤ 1 are describing the order of the fractional time and space derivative which Dβx u

and Dαt u are the nonlocal fractional derivatives of u with respect to t of order α, x of

order β respectively. The fractional derivative is considered in the Reimann-Liouvillederivative sense[59]. We present the basic definitions and some important properties ofthe modified Riemann-Liouville derivative of order α as follows [53]-[58]:

Dαt f (t) =

⎧⎨⎩

1

�(1 − α)

d

dt

∫ t

0(t − ξ)−α(f (ξ) − f (0))dξ, 0 < α < 1

(f (n)(t))(α−n), n ≤ α < n + 1, n ≥ 1.

where f : R −→ R, denotes a continuous (but not necessarily first order-differentiable)function. Some used formulas are as follows:

Dαt tr = �(1 + r)

�(1 + r − α)tr−α, (1.2)

Dαt (cf (t)) = cDα

t f (t), where c is constant. (1.3)

Dαt c = 0, where c is constant. (1.4)

Dαt

[af (t) + bg(t)

]= aDα

t f (t) + bDαt g(t), (1.5)

Dαt (f (t)g(t)) = g(t)Dα

t f (t) + f (t)Dαt g(t), (1.6)

Dαt f (g(t)) = f

′g (g(t))Dα

t g(t) = Dαg f (g(t))(g

′(t))α. (1.7)

2. The improved generalized Riccati equation mapping method

Suppose that a general fractional order nonlinear evolution equation in the independentvariables x, t is given by

F(u, Dαt u, Dβ

x u, D2βx u, D3β

x u, D4βx u, D5β

x u, . . .) = 0, 0 < α, β ≤ 1, (2.1)

where Dαt u, Dβ

x u, D2βx u, D3β

x u, D4βx u, D5β

x u and Dαt u are the nonlocal fractional order

α, β derivatives of u with respect to x and t respectively. F is a polynomial of u =u(x, t) and its various partial derivatives that the highest-fractional order derivatives andnonlinear terms are involved.

4 Sanoe Koonprasert, Sekson Sirisubtawee, and Sivaporn Ampun

Next, we illustrate the description of the improved generalized Riccati equation map-ping method for seeking exact solutions of FPDEs in the following steps [28].

Step 1: Using a nonlinear fractional complex travelling wave transformation

ξ = xβ

�(1 + β)− ωtα

�(1 + α). (2.2)

where ω is a real constant, we obtain u(x, t) = U(ξ). By the first equality of thechain rule in Eq.(1.7), then Eq. (2.1) can be reduced to the following nonlinear ordinarydifferential equation with a variable ξ :

Q(U, U ′, U ′′, U ′′′, . . .) = 0, (2.3)

where ′ denotes the derivation with respect to ξ . We should integrate Eq. (2.3) term byterm if possible.

Step 2: Assume that the solution of Eq.(2.3) can be expressed by a polynomial in φ(ξ)

as follows:

U(ξ) =n∑

i=−n

aiφ(ξ)i, (2.4)

where coefficients ai (i = 0, ±1, ±2, ... ± n) are constants to be determined later, anda−n or an may be zero, but both of them cannot be zero simultaneously. The functionφ(ξ) satisfies the Riccati differential equation as: [46]

φ′(ξ) = r + pφ(ξ) + qφ(ξ)2, (2.5)

where p, q, r are real constants.The positive integer n can be easily determined by considering the homogeneous

balance between the highest order derivatives and the highest order nonlinear term ofderivative appearing in Eq. (2.3) with the formulas of the degree of the expressions asfollows:

D [U(ξ)] = n, D

[(dpU(ξ)

dξp

)]= n + p, D

[Up

(dqU(ξ)

dξq

)s]= np + s(n + q).

Step 3: Substituting Eq. (2.4) along with Eq. (2.5) into Eq. (2.3), after collecting all theterms with the same order φi together. It obtains the polynomial equation in φi and φ−i ,(i = 0, 1, 2, . . . , n). Equating each coefficient of the resulting polynomial to be zero,then an over-determined set of algebraic equations for ai (i = 0, ±1, ±2, . . . , ±n) isobtained.

Step 4: Solving the system of algebraic equations obtained in step 3 with using the aidof algebraic software Maple, we obtain values for ai (i = 0, ±1, ±2, . . . , ±n). Then by

Explicit Solitary Solutions of the Space-Time Fractional Fifth Order... 5

substituting these values in Eq. (2.4) along with the solutions of Eq. (2.5) and using thetransformation in Eq. (2.2), we can construct a variety of exact solutions of Eq. (2.1).

In order to obtain the general solutions of equation Eq.(1.1), the generalizing Riccatiequation mapping method can generate twenty seven solutions of the Riccati equationin Eq. (2.5) including the four different families as follows [34]:

Family 1: When p2 − 4qr > 0 and pq �= 0 (or qr �= 0), the hyperbolic functionsolutions of Eq. (2.5) are:

φ1(ξ) = − 1

2q

[p +

√p2 − 4qr tanh

(√p2 − 4qr

2ξ

)],

φ2(ξ) = − 1

2q

[p +

√p2 − 4qr coth

(√p2 − 4qr

2ξ

)],

φ3(ξ) = − 1

2q

[p +

√p2 − 4qr

(tanh

(√p2 − 4qrξ

)± isech

(√p2 − 4qrξ

))],

φ4(ξ) = − 1

2q

[p +

√p2 − 4qr

(coth

(√p2 − 4qrξ

)± csch

(√p2 − 4qrξ

))],

φ5(ξ) = − 1

2q

[2p +

√p2 − 4qr

(tanh

(√p2 − 4qr

4ξ

)+ coth

(√p2 − 4qr

4ξ

))],

φ6(ξ) = 1

2q

[−p +

√(A2 + B2)(p2 − 4qr) − A

√p2 − 4qr cosh(

√p2 − 4qrξ)

A sinh(√

p2 − 4qrξ) + B

],

φ7(ξ) = 1

2q

[−p −

√(B2 − A2)(p2 − 4qr) + A

√p2 − 4qr sinh(

√p2 − 4qrξ)

A cosh(√

p2 − 4qrξ) + B

],

where A and B are two non-zero real constants and satisfy the condition B2 − A2 > 0,

φ8(ξ) =2r cosh

(√p2−4qr

2 ξ

)√

p2 − 4qr sinh

(√p2−4qr

2 ξ

)− p cosh

(√p2−4qr

2 ξ

) ,

φ9(ξ) =−2r sinh

(√p2−4qr

2 ξ

)

p sinh

(√p2−4qr

2 ξ

)− √

p2 − 4qr cosh

(√p2−4qr

2 ξ

) ,

φ10(ξ) =2r cosh

(√p2−4qr

2 ξ

)√

p2 − 4qr sinh(√

p2 − 4qrξ)

− p cosh(√

p2 − 4qrξ)

± i√

p2 − 4qr,

6 Sanoe Koonprasert, Sekson Sirisubtawee, and Sivaporn Ampun

φ11(ξ) =2r sinh

(√p2−4qr

2 ξ

)

−p sinh(√

p2 − 4qrξ)

+ √p2 − 4qr cosh

(√p2 − 4qrξ

)±√p2 − 4qr

,

φ12(ξ) =4r sinh

(√p2−4qr

4 ξ

)cosh

(√p2−4qr

4 ξ

)

−2p sinh

(√p2 − 4qr

4ξ

)cosh

(√p2 − 4qr

4ξ

)

+ 2√

p2 − 4qr cosh2

(√p2 − 4qr

4ξ

)−√

p2 − 4qr

.

Family 2: If p2 − 4qr < 0 and pq �= 0 (or qr �= 0), then the trigonometric solutionsof Eq. (2.5) are as follows:

φ13(ξ) = 1

2q

[−p +

√4qr − p2 tan

(√4qr − p2

2ξ

)],

φ14(ξ) = − 1

2q

[p +

√4qr − p2 cot

(√4qr − p2

2ξ

)],

φ15(ξ) = 1

2q

[−p +

√4qr − p2

(tan(√

4qr − p2ξ)

± sec(√

4qr − p2ξ))]

,

φ16(ξ) = − 1

2q

[p +

√4qr − p2

(cot(√

4qr − p2ξ)

± csc(√

4qr − p2ξ))]

,

φ17(ξ) = 1

4q

[−2p +

√4qr − p2

(tan

(√4qr − p2

4ξ

)− cot

(√4qr − p2

4ξ

))],

φ18(ξ) = 1

2q

[−p + ±√(A2 − B2)(4qr − p2) − A

√4qr − p2 cos(

√4qr − p2ξ)

A sin(√

4qr − p2ξ) + B

],

φ19(ξ) = 1

2q

[−p + ±√(A2 − B2)(4qr − p2) + A

√4qr − p2 cos(

√4qr − p2ξ)

A sin(√

4qr − p2ξ) + B

],

where A and B are two non-zero real constants and satisfy the condition A2 − B2 > 0.

φ20(ξ) = −2r cos

(√4qr−p2

2 ξ

)√

4qr − p2 sin

(√4qr−p2

2 ξ

)+ p cos

(√4qr−p2

2 ξ

) ,

Explicit Solitary Solutions of the Space-Time Fractional Fifth Order... 7

φ21(ξ) =2r sin

(√4qr−p2

2 ξ

)

−p sin

(√4qr−p2

2 ξ

)+ √

4qr − p2ξ cos

(√4qr−p2

2 ξ

) ,

φ22(ξ) = −2r cos

(√4qr−p2

2 ξ

)√

4qr − p2 sin(√

4qr − p2ξ)

+ p cos(√

4qr − p2ξ)

±√4qr − p2,

φ23(ξ) =2r sin

(√4qr − p2ξ

)−p sin

(√4qr − p2ξ

)+ √

4qr − p2 cos(√

4qr − p2ξ)

±√4qr − p2,

φ24(ξ) =4r sin

(√4qr−p2

4 ξ

)cos

(√4qr−p2

4 ξ

)

−2p sin

(√4qr − p2

4ξ

)cos

(√4qr − p2

4ξ

)

+ 2√

4qr − p2 cos2

(√4qr − p2

4ξ

)−√

4qr − p2

.

Family 3: When r = 0 and pq �= 0, the hyperbolic function solutions of Eq. (2.5) are

φ25(ξ) = −pd

q (d + cosh(pξ) − sinh(pξ)),

φ26(ξ) = −p (cosh(pξ) + sinh(pξ))

q (d + cosh(pξ) + sinh(pξ)),

where d is an arbitrary constant.

Family 4: When q �= 0 and r = p = 0 , the rational solution of Eq. (2.5) is

φ27(ξ) = − 1

qξ + c1,

where c1 is an arbitrary constant.

3. Exact solutions of the space-time fractional ordernonlinear Sawada-Kotera equation

In this section, we apply the improved generalized Riccati equation mapping methodto the space-time fractional (1+1) dimensional fifth order Sawada-Kotera equation witha = 45, b = c = 15, and d = 1 [30]:

Dαt u(x, t) + 45u2(x, t)Dβ

x u(x, t) + 15Dβx u(x, t)D2β

x u(x, t)

+15u(x, t)D3βx u(x, t) + D5β

x u(x, t) = 0, 0 < α, β ≤ 1, (3.1)

8 Sanoe Koonprasert, Sekson Sirisubtawee, and Sivaporn Ampun

where u denotes a solution. Using the nonlinear fractional order traveling wave trans-formation

ξ = xβ

�(1 + β)− ωtα

�(1 + α)(3.2)

where ω is a non-zero arbitrary constant, we have u(x, y, z, t) = U(ξ) and then employ-ing the chain rule in Eq. (1.7) and the formula in Eq. (1.2) the fractional order derivativesof u can be written in terms of the ordinary derivatives of U as follows:

Dαt u = U

′Dα

t (ξ) = −ωU ′, Dβx u = U ′Dβ

x (ξ) = U ′, D2βx u = U ′′,

D3βx u = U ′′′, D5β

x u = U(5). (3.3)

Substituting the above results in Eq. (3.3) into Eq. (3.1), we obtain the integrable nonlineardifferential equation

−wU ′(ξ) + 45U2(ξ)U ′(ξ) + 15U ′(ξ)U ′′(ξ) + 15U(ξ)U(3)(ξ) + U(5)(ξ) = 0. (3.4)

Suppose that the solution of Eq. (3.4) can be expressed by

U(ξ) =n∑

i=−n

aiφ(ξ)i, (3.5)

where φ(ξ) satisfies the Riccati equation in Eq.(2.5). Considering the degree of U(5) :D

(d5U(ξ)

dξ5

)= n+5, while the degree of U2U

′′ : D

[U2(

d2U(ξ)

dξ2

)]= 2n+(n+2),

we obtain from balancing the order between U(5) and U2U′′

in Eq. (3.4) that n = 2.Then the solution of Eq.(3.4) is given by

U(ξ) = a−2φ(ξ)−2 + a−1φ(ξ)−1 + a0 + a1φ(ξ) + a2φ(ξ)2. (3.6)

Explicit Solitary Solutions of the Space-Time Fractional Fifth Order... 9

Using the Riccati equation (2.5), we can derive the higher derivative of U in terms of apolynomial φ(ξ) as

U ′(ξ) = −2a−2

(r

φ(ξ)3+ p

φ2(ξ)+ q

φ(ξ)

)− a−1

(r

φ(ξ)2+ p

φ(ξ)+ q

)

+a1

(+ r + pφ(ξ) + qφ2(ξ)

)

+2a2

(rφ(ξ) + pφ2(ξ) + qφ3(ξ)

),

U ′′(ξ) = 2a−2

(3r2

φ(ξ)4+ 5pr

φ(ξ)3+ 2p2 + 4qr

φ2(ξ)+ 3pq

φ(ξ)+ q2

)

+a−1

(2r2

φ(ξ)3+ 3pr

φ(ξ)2+ p2 + 2qr

φ(ξ)+ pq

)

+a1

(pr + (p2 + 2qr)φ(ξ) + 3pqφ2(ξ) + 2q2φ3(ξ)

)

+2a2

(r2 + 3prφ(ξ) + 2(p2 + 2qr)φ2(ξ)

+5pqφ3(ξ) + 3q2φ4(ξ)

),

... (3.7)

Substituting U(n)(ξ) (n = 1, 2, 3, 5) into Eq. (3.4), after collecting all terms with thesame order of φi(ξ), i = −7, −6, −5, . . . , 5 and equating each coefficient to zero, weobtain a system of nonlinear algebraic equations as follows:

φ−7(ξ) : −720a2r4 − 540a2

2r2 − 90a32 = 0,

φ−6(ξ) : −120a1r4 − 1680a2pr3 − 600a1a2r

2 − 750a22pr − 225a1a

22 = 0,

φ−5(ξ) : −240a1pr3 − 1320a2p2r2 − 960a2qr3 − 360a0a2r

2 − 120a21r2

−780a1a2pr − 240a22p2 − 480a2

2qr − 180a0a22 − 180a2

1a2 = 0,

φ−4(ξ) : −150a1p2r2 − 120a1qr3 − 390a2p

3r − 1320a2pqr2 − 90a0a1r2

−450a0a2pr − 270a1a2r2 − 135a2

1pr − 225a1a2p2 − 450a1a2qr

−270a22pq − 270a0a1a2 − 135a1a2

2 − 45a31 = 0,

φ−3(ξ) : −30a1p3r − 120a1pqr2 − 32a2p

4 − 464a2p2qr − 272a2q

2r2

−90a0a1pr − 120a0a2p2 − 240a0a2qr − 60a1a1r

2 − 330a1a2pr

−240a2a2r2 − 30a2

1p2 − 60a21qr − 210a1a2pq − 60a2

2q2 − 90a02a2

−90a0a21 − 180a1a1a2 − 90a2a2

2 + 2a2ω = 0,

10 Sanoe Koonprasert, Sekson Sirisubtawee, and Sivaporn Ampun

φ−2(ξ) : −a1p4 − 22a1p

2qr − 16a1q2r2 − 30a2p

3q − 120a2pq2r − 15a0a1p2

−30a0a1qr − 90a0a2pq − 60a1a1pr − 75a1a2p2 − 150a1a2qr

−60a2a1r2 − 240a2a2pr − 15a2

1pq − 30a1a2q2 − 45a2

0a1

−90a0a1a2 − 45a1a21 − 90a2a1a2 + a1ω = 0

φ1(ξ) : 30a1p3q + 120a1pq2r + 32a2p

4 + 464a2p2qr + 272a2q

2r2 + 90a0a1pq,

+120a0a2p2 + 240a0a2qr + 30a2

1p2 + 60a21qr + 210a1a2pr

+60a1a1q2 + 60a2

2r2 + 330a2a1pq + 240a2a2q2 + 90a2

0a2

+90a0a21 + 180a1a2a1 + 90a2

2a2 − 2a2ω = 0,

φ2(ξ) : 150a1p2q2 + 120a1q

3r + 390a2p3q + 1320a2pq2r + 90a0a1q

2

+450a0a2pq + 135a21pq + 225a1a2p

2 + 450a1a2qr + 270a22pr

+270a2a1q2 + 270a0a1a2 + 45a3

1 + 135a22a1 = 0,

φ3(ξ) : 240a1pq3 + 1320a2p2q2 + 960a2q

3r + 360a0a2q2 + 120a2

1q2 + 780a1a2p

−q + 240a22p2 + 480a2

2qr + 180a0a22 + 180a2

1a2 = 0,

φ4(ξ) : 120a1q4 + 1680a2pq3 + 600a1a2q

2 + 750a22pq + 225a1a

22 = 0,

φ5(ξ) : 720a2q4 + 540a2

2q2 + 90a32 = 0.

Solving these nonlinear algebraic equations with the aid of the mathematical softwareMaple, it yields three cases of the value a−2, a−1, a0, a1, a2 and ω in terms of p, q

and r as follows:

Case 1:

a0 = a0, a1 = 0, a2 = 0, a−1 = −2pr, a−2 = −2r2,

ω = p4 + 22p2qr + 76q2r2 + 15a0p2 + 120a0qr + 45a2

0,

Case 2:

a0 = a0, a1 = −2pq, a2 = −2q2, a−1 = 0, a−2 = 0,

ω = p4 + 22p2qr + 76q2r2 + 15a0p2,

Case 3:

a0 = −1

3p2 − 8

3qr, a1 = −2pq, a2 = −2q2,

a−1 = −2pr, a−2 = −2r2, ω = p4 − 8p2qr + 16q2r2.

Substituting the results from above case into Eq. (3.6) along with the mentionedRiccati solutions, we can obtain abundant exact solutions including solitary wave equa-tions, periodic wave solutions, and rational solutions as follows:

In Case 1, substituting the values of

a0 = a0, a1 = 0, a2 = 0, a−1 = −2pr, a−2 = −2r2,

Explicit Solitary Solutions of the Space-Time Fractional Fifth Order... 11

and

ω = p4 + 22p2qr + 76q2r2 + 15a0p2 + 120a0qr + 45a2

0

and using

ξ = xβ

�(1 + β)− ωtα

�(1 + α),

we obtain the exact solutions u(x, t) of Eq. (3.1) that classify by four families as follow-ings:

Family 1: when p2 −4qr > 0 and pq �= 0 (or qr �= 0), we have the hyperbolic functionsolutions of Eq. (3.1) as:

u1,1(x, t) = − 8q2r2(p +√p2 − 4qr tanh

(12

√p2 − 4qrξ

))2

+ 4pqr

p +√p2 − 4qr tanh(

12

√p2 − 4qrξ

) + a0,

u1,2(x, t) = − 8q2r2(p +√p2 − 4qr coth

(12

√p2 − 4qrξ

))2

+ 4pqr

p +√p2 − 4qr coth(

12

√p2 − 4qrξ

) + a0,

u1,3(x, t) = −8q2r2(

p +√

p2 − 4qr(

tanh(√

p2 − 4qrξ)

± isech(√

p2 − 4qrξ)))−2

+ 4pqr

p +√p2 − 4qr(

tanh(√

p2 − 4qrξ)

± isech(√

p2 − 4qrξ)) + a0,

u1,4(x, t) = −8q2r2(

p +√

p2 − 4qr(

coth(√

p2 − 4qrξ)

± csch(√

p2 − 4qrξ)))−2

+ 4pqr

p +√p2 − 4qr(

coth(√

p2 − 4qrξ)

± csch(√

p2 − 4qrξ)) + a0,

12 Sanoe Koonprasert, Sekson Sirisubtawee, and Sivaporn Ampun

u1,5(x, t) = −32q2r2(

2p +√

p2 − 4qr(

tanh(1

4

√p2 − 4qrξ

)

+ coth(1

4

√p2 − 4qrξ

)))−2

+ 8pqr

2p +√p2 − 4qr(

tanh(

14

√p2 − 4qrξ

)+ coth

(14

√p2 − 4qrξ

)) + a0,

u1,6(x, t) = −8q2r2

⎡⎢⎢⎣−p +

√(A2 + B2

)(p2 − 4qr

)− A

√p2 − 4qr cosh

(√p2 − 4qrξ

)A sinh

(√p2 − 4qrξ

)+ B

⎤⎥⎥⎦

−2

− 4pqr

⎡⎢⎢⎣−p +

√(A2 + B2

)(p2 − 4qr

)− A

√p2 − 4qr cosh

(√p2 − 4qrξ

)A sinh

(√p2 − 4qrξ

)+ B

⎤⎥⎥⎦

−1

+ a0,

u1,7(x, t) = −8q2r2

⎡⎢⎢⎣−p −

√(B2 − A2

)(p2 − 4qr

)+ A

√p2 − 4qr sinh

(√p2 − 4qrξ

)A cosh

(√p2 − 4qrξ

)+ B

⎤⎥⎥⎦

−2

− 4pqr

⎡⎢⎢⎣−p −

√(B2 − A2

)(p2 − 4qr

)+ A

√p2 − 4qr sinh

(√p2 − 4qrξ

)A cosh

(√p2 − 4qrξ

)+ B

⎤⎥⎥⎦

−1

+ a0,

where A and B are two non-zero real constants and satisfies B2 − A2 > 0,

u1,8(x, t) = −1

2

(√p2 − 4qr sinh

(1

2

√p2 − 4qrξ

)

− p cosh(1

2

√p2 − 4qrξ

))2

sech2(1

2

√p2 − 4qrξ

)

− p

(√p2 − 4qr sinh

(1

2

√p2 − 4qrξ

)

− p cosh(1

2

√p2 − 4qrξ

))sech

(1

2

√p2 − 4qrξ

)+ a0,

Explicit Solitary Solutions of the Space-Time Fractional Fifth Order... 13

u1,9(x, t) = −1

2

(p sinh

(1

2

√p2 − 4qrξ

)

−√

p2 − 4qr cosh(1

2

√p2 − 4qrξ

))2

csch2(1

2

√p2 − 4qrξ

)

+ p

(p sinh

(1

2

√p2 − 4qrξ

)

−√

p2 − 4qr cosh(1

2

√p2 − 4qrξ

))csch

(1

2

√p2 − 4qrξ

)+ a0,

u1,10(x, t) = −1

2

(√p2 − 4qr sinh

(√p2 − 4qrξ

)

− p cosh(√

p2 − 4qrξ)

+ I√

p2 − 4qr

)2

sech2(√

p2 − 4qrξ)

− p

(√p2 − 4qr sinh

(√p2 − 4qrξ

)− p cosh

(√p2 − 4qrξ

)

+ I√

p2 − 4qr

)sech

(√p2 − 4qrξ

)+ a0,

u1,11(x, t) = −1

2

(− p sinh

(√p2 − 4qrξ

)+√

p2 − 4qr cosh(√

p2 − 4qrξ)

−√

p2 − 4qr

)2

csch2(√

p2 − 4qrξ)

− p

(− p sinh

(√p2 − 4qrξ

)

+√

p2 − 4qr cosh(√

p2 − 4qrξ)

−√

p2 − 4qr

)csch

(√p2 − 4qrξ

)+ a0,

u1,12(x, t) = −

(− 2p sinh

(1

4

√p2 − 4qrξ

)cosh

(1

4

√p2 − 4qrξ

)

+2√

p2 − 4qr cosh2(1

4

√p2 − 4qrξ

)−√

p2 − 4qr

)2

8 sinh2(

14

√p2 − 4qrξ

)cosh2

(14

√p2 − 4qrξ

)

−

p

(− 2p sinh

(1

4

√p2 − 4qrξ

)cosh

(1

4

√p2 − 4qrξ

)+2√

p2 − 4qr cosh2(1

4

√p2 − 4qrξ

)−√

4qr − p2

)

2 sinh(

14

√p2 − 4qrξ

)cosh

(14

√p2 − 4qrξ

)+ a0.

14 Sanoe Koonprasert, Sekson Sirisubtawee, and Sivaporn Ampun

Family 2: If p2−4qr < 0 and pq �= 0 (or qr �= 0), we have the following trigonometricfunction solutions

u1,13(x, t) = − 8q2r2(− p +√4qr − p2 tan

(12

√4qr − p2ξ

))2

− 4pqr

−p +√4qr − p2 tan(

12

√4qr − p2ξ

) + a0,

u1,14(x, t) = − 8q2r2(p +√4qr − p2 cot

(12

√4qr − p2ξ

))2

+ 4pqr

p +√4qr − p2 cot(

12

√4qr − p2ξ

) + a0,

u1,15(x, t) = −8q2r2(

− p +√

4qr − p2

(tan(√

4qr − p2ξ)

+ sec(√

4qr − p2ξ)))−2

− 4pqr

−p +√4qr − p2

(tan(√

4qr − p2ξ)

+ sec(√

4qr − p2ξ)) + a0,

u1,16(x, t) = −8q2r2(

p +√

4qr − p2

(cot(√

4qr − p2ξ)

− csc(√

4qr − p2ξ)))−2

+ 4pqr

p +√4qr − p2

(cot(√

4qr − p2ξ)

− csc(√

4qr − p2ξ)) + a0,

u1,17(x, t) = −32q2r2(

− 2p +√

4qr − p2

(tan(1

4

√4qr − p2ξ

)− cot

(1

4

√4qr − p2ξ

)))−2

− 8pqr

−2p +√4qr − p2

(tan(

14

√4qr − p2ξ

)− cot

(14

√4qr − p2ξ

)) + a0,

u1,18(x, t) = −8q2r2(

− p +

√(A2 − B2

)(4qr − p2

)− A

√4qr − p2 cos

(√4qr − p2ξ

)A sin

(√4qr − p2ξ

)+ B

)−2

− 4pqr

(− p

√(A2 − B2

)(4qr − p2

)− A

√4qr − p2 cos

(√4qr − p2ξ

)A sin

(√4qr − p2ξ

)+ B

)−1

+ a0,

Explicit Solitary Solutions of the Space-Time Fractional Fifth Order... 15

u1,19(x, t) = −8q2r2

⎛⎜⎜⎝−p −

√(A2 − B2

)(4qr − p2

)+ A

√4qr − p2 cos

(√4qr − p2ξ

)A sin

(√4qr − p2ξ

)+ B

⎞⎟⎟⎠

−2

− 4pqr

⎛⎜⎜⎝−p −

√(A2 − B2

)(4qr − p2

)+ A

√4qr − p2 cos

(√4qr − p2ξ

)A sin

(√4qr − p2ξ

)+ B

⎞⎟⎟⎠

−1

+ a0,

where A and B are two non-zero real constants and satisfies A2 − B2 > 0,

u2,20(x, t) = −

(√4qr − p2 sin

(12

√4qr − p2ξ

)+ p cos

(12

√4qr − p2ξ

))2

2 cos(

12

√4qr − p2ξ

)2

+p

(√4qr − p2 sin

(12

√4qr − p2ξ

)+ p cos

(12

√4qr − p2ξ

))

cos(

12

√4qr − p2ξ

) + a0,

u2,21(x, t) = −

(− p sin

(12

√4qr − p2ξ

)+√

4qr − p2 cos(

12

√4qr − p2ξ

))2

2 sin(

12

√4qr − p2ξ

)2

−p

(− p sin

(12

√4qr − p2ξ

)−√

4qr − p2 cos(

12

√4qr − p2ξ

))

sin(

12

√4qr − p2ξ

) + a0,

u1,22(x, t) = −

(√4qr − p2 sin

(√4qr − p2ξ

)+ p cos

(√4qr − p2ξ

)+√

4qr − p2)2

2 cos2(√

4qr − p2ξ)

+p

(√4qr − p2 sin

(√4qr − p2ξ

)+ p cos

(√4qr − p2ξ

)+√

4qr − p2)

cos(√

4qr − p2ξ) + a0,

u2,23(x, t) = −

(− p sin

(√4qr − p2ξ

)+√

4qr − p2 cos(√

4qr − p2ξ)

+√

4qr − p2)2

2 sin(√

4qr − p2ξ)2

−p

(− p sin

(√4qr − p2ξ

)+√

4qr − p2 cos(√

4qr − p2ξ)

+√

4qr − p2)

sin(√

4qr − p2ξ) + a0,

16 Sanoe Koonprasert, Sekson Sirisubtawee, and Sivaporn Ampun

u1,24(x, t) = −

(− 2p sin

(1

4

√4qr − p2ξ

)cos(1

4

√4qr − p2ξ

)

+2√

4qr − p2 cos2(1

4

√4qr − p2ξ

)−√

4qr − p2)2

8 sin2(

14

√4qr − p2ξ

)cos2

(14

√4qr − p2ξ

)

−

p

(− 2p sin

(1

4

√4qr − p2ξ

)cos(1

4

√4qr − p2ξ

)+2√

4qr − p2 cos2(1

4

√4qr − p2ξ

)−√

4qr − p2)

2 sin(

14

√4qr − p2ξ

)cos(

14

√4qr − p2ξ

)+a0.

In case 2, substituting the values of

a0 = a0, a1 = −2pq, a2 = −2q2, a−1 = 0, a−2 = 0,

and

ω = p4 + 22p2qr + 76q2r2 + 15a0p2

and the Riccati solutions into Eq. (3.6), then using

ξ = xβ

�(1 + β)− ωtα

�(1 + α),

we obtain the following exact solutions of Eq. (3.1).

Family 1: When p2 − 4qr > 0 and pq �= 0 (or qr �= 0), the hyperbolic solutions ofEq. (3.1) are expressed below:

u2,1(x, t) = a0 + p

(p +

√p2 − 4qr tanh

(1

2

√p2 − 4qrξ

))

− 1

2

(p +

√p2 − 4qr tanh

(1

2

√p2 − 4qr

))2

,

u2,2(x, t) = a0 + p

(p +

√p2 − 4qr coth

(1

2

√p2 − 4qrξ

))

− 1

2

(p +

√p2 − 4qr coth

(1

2

√p2 − 4qr

))2

,

Explicit Solitary Solutions of the Space-Time Fractional Fifth Order... 17

u2,3(x, t) = a0 + p

(p +

√p2 − 4qr

(tanh

(√p2 − 4qrξ

)± isech

(√p2 − 4qrξ

)))

− 1

2

(p +

√p2 − 4qr

(tanh

(√p2 − 4qrξ

)± isech

(√p2 − 4qr

)))2

,

u2,4(x, t) = a0 + p

(p +

√p2 − 4qr

(coth

(√p2 − 4qrξ

)± csch

(√p2 − 4qrξ

)))

− 1

2

(p +

√p2 − 4qr

(coth

(√p2 − 4qrξ

)± csch

(√p2 − 4qr

)))2

,

u2,5(x, t) = a0 + 1

2p

(2p +

√p2 − 4qr

(tanh

(1

4

√p2 − 4qrξ

)+ coth

(1

4

√p2 − 4qrξ

)))

− 1

8

(2p +

√p2 − 4qr

(tanh

(1

4

√p2 − 4qrξ

)+ coth

(1

4

√p2 − 4qr

)))2

,

u2,6(x, t) = a0 − p

[− p +

√(A2 + B2)(p2 − 4qr) − A

√p2 − 4qr cosh

(√p2 − 4qrξ

)A sinh

(√p2 − 4qrξ

)+ B

]

− 1

2

[− p +

√(A2 + B2)(p2 − 4qr) − A

√p2 − 4qr cosh

(√p2 − 4qrξ

)A sinh

(√p2 − 4qrξ

)+ B

]2

,

u2,7(x, t) = a0 − p

[− p −

√(B2 − A2)(p2 − 4qr) + A

√p2 − 4qr sinh

(√p2 − 4qrξ

)A cosh

(√p2 − 4qrξ

)+ B

]

− 1

2

[− p −

√(B2 − A2)(p2 − 4qr) + A

√p2 − 4qr sinh

(√p2 − 4qrξ

)A cosh

(√p2 − 4qrξ

)+ B

]2

,

where A and B are two non-zero real constants and satisfies B2 − A2 > 0,

u2,8(x, t) = a0 −4pqr cosh

(12

√p2 − 4qrξ

)√

p2 − 4qr sinh

(12

√p2 − 4qrξ

)− p cosh

(12

√p2 − 4qrξ

)

−8q2r2 cosh2

(12

√p2 − 4qrξ

)(√

p2 − 4qr sinh

(12

√p2 − 4qrξ

)− p cosh

(12

√p2 − 4qrξ

))2 ,

18 Sanoe Koonprasert, Sekson Sirisubtawee, and Sivaporn Ampun

u2,9(x, t) = a0 +4pqr sinh

(12

√p2 − 4qrξ

)

p sinh

(12

√p2 − 4qrξ

)−√p2 − 4qr cosh

(12

√p2 − 4qrξ

)

−8q2r2 sinh2

(12

√p2 − 4qrξ

)(

p sinh

(12

√p2 − 4qrξ

)−√p2 − 4qr cosh

(12

√p2 − 4qrξ

))2 ,

u2,10(x, t) = a0 −4pqr cosh

(√p2 − 4qrξ

)√

p2 − 4qr sinh(√

p2 − 4qrξ)

− p cosh(√

p2 − 4qrξ)

+ i√

p2 − 4qr

−8q2r2 cosh2

(√p2 − 4qrξ

)(√

p2 − 4qr sinh(√

p2 − 4qrξ)

− p cosh(√

p2 − 4qrξ)

+ i√

p2 − 4qr

)2 ,

u2,11(x, t) = a0 −4pqr sinh

(√p2 − 4qrξ

)−p sinh

(√p2 − 4qrξ

)+√p2 − 4qr cosh

(√p2 − 4qrξ

)−√p2 − 4qr

−8q2r2 cosh2

(√p2 − 4qrξ

)(

− p sinh(√

p2 − 4qrξ)

+√p2 − 4qrξ cosh(√

p2 − 4qrξ)

−√p2 − 4qr

)2 ,

u2,12(x, t) = a0 −8pqr sinh

(14

√p2 − 4qrξ

)cosh

(14

√p2 − 4qrξ

)

−2p sinh

(1

4

√p2 − 4qrξ

)cosh

(1

4

√p2 − 4qrξ

)

+2√

p2 − 4qr cosh2(

1

4

√p2 − 4qrξ

)−√

p2 − 4qr

−32q2r2 sinh2

(14

√p2 − 4qrξ

)cosh2

(14

√p2 − 4qrξ

)(

− 2p sinh

(1

4

√p2 − 4qrξ

)cosh

(1

4

√p2 − 4qrξ

)

+2√

p2 − 4qr cosh2(

1

4

√p2 − 4qrξ

)−√

p2 − 4qr

)2

.

Family 2: If p2−4qr < 0 and pq �= 0 (or qr �= 0), we have the following trigonometricfunction solutions of Eq. (3.1).

u2,13(x, t) = a0 − p

(− p +

√4qr − p2 tan

(1

2

√4qr − p2

))

− 1

2

(− p +

√4qr − p2 tan

(1

2

√4qr − p2

))2

,

Explicit Solitary Solutions of the Space-Time Fractional Fifth Order... 19

u2,14(x, t) = a0 + p

(p +

√4qr − p2 cot

(1

2

√4qr − p2

))

− 1

2

(p +

√4qr − p2 cot

(1

2

√4qr − p2

))2

,

u2,15(x, t) = a0 − p

(− p +

√4qr − p2

(tan(√

4qr − p2ξ)

+ sec(√

4qr − p2ξ)))

− 1

2

(− p +

√4qr − p2

(tan(√

4qr − p2ξ)

+ sec(√

4qr − p2)))2

,

u2,16(x, t) = a0 + p

(p +

√4qr − p2

(cot(√

4qr − p2ξ)

− csc(√

4qr − p2ξ)))

− 1

2

(p +

√4qr − p2

(cot(√

4qr − p2ξ)

− csc(√

4qr − p2)))2

,

u2,17(x, t) = a0 − 1

2p

(− 2p +

√4qr − p2

(tan

(1

4

√4qr − p2ξ

)− cot

(1

4

√4qr − p2ξ

)))

− 1

8

(− 2p +

√4qr − p2

(tan

(1

4

√4qr − p2ξ

)− cot

(1

4

√4qr − p2

)))2

,

u2,18(x, t) = a0 − p

(− p +

√(A2 − B2

)(4qr − p2

)− A

√4qr − p2 cos

(√4qr − p2ξ

)A sin

(√4qr − p2ξ

)+ B

)

− 1

2

(− p +

√(A2 − B2

)(4qr − p2

)− A

√4qr − p2 cos

(√4qr − p2ξ

)A sin

(√4qr − p2ξ

)+ B

)2

,

u2,19(x, t) = a0 − p

(− p −

√(A2 − B2

)(4qr − p2

)+ A

√4qr − p2 cos

(√4qr − p2ξ

)A sin

(√4qr − p2ξ

)+ B

)

− 1

2

(− p −

√(A2 − B2

)(4qr − p2

)+ A

√4qr − p2 cos

(√4qr − p2ξ

)A sin

(√4qr − p2ξ

)+ B

)2

,

where A and B are two non-zero real constants and satisfies A2 − B2 > 0,

u2,20(x, t) = a0 +4pqr cos

(12

√4qr − p2ξ

)√

4qr − p2 sin(

12

√4qr − p2ξ

)+ p cos

(12

√4qr − p2ξ

)

−8q2r2 cos2

(12

√4qr − p2ξ

)(√

4qr − p2 sin(

12

√4qr − p2ξ

)+ p cos

(12

√4qr − p2ξ

)2 ,

20 Sanoe Koonprasert, Sekson Sirisubtawee, and Sivaporn Ampun

u2,21(x, t) = a0 −4pqr sin

(12

√4qr − p2ξ

)−p sin

(12

√4qr − p2ξ

)+√4qr − p2 cos

(12

√4qr − p2ξ

)

−8q2r2 sin2

(12

√4qr − p2ξ

)(

− p sin(

12

√4qr − p2ξ

)+√4qr − p2 cos

(12

√4qr − p2ξ

)2 ,

u2,22(x, t) = a0 +4pqr cos

(√4qr − p2ξ

)√

4qr − p2 sin(

12

√4qr − p2ξ

)+ p cos

(12

√4qr − p2ξ

)+√4qr − p2

−8q2r2 cos2

(12

√4qr − p2ξ

)(√

4qr − p2 sin(

12

√4qr − p2ξ

)+ p cos

(12

√4qr − p2ξ

)+√4qr − p2

)2 ,

u4,23(x, t) = a0 +4pqr sin

(√4qr − p2ξ

)−p sin

(√4qr − p2ξ

)+√4qr − p2 cos

(√4qr − p2ξ

)+√4qr − p2

−8q2r2 sin2

(√4qr − p2ξ

)(

− p sin(√

4qr − p2ξ)

+√4qr − p2 cos(√

4qr − p2ξ)

+√4qr − p2

)2 ,

u2,24(x, t) = a0 −8pqr sin

(14

√4qr − p2ξ

)cos(

14

√4qr − p2ξ

)−2p sin

(1

4

√4qr − p2ξ

)cos(1

4

√4qr − p2ξ

)+2√

4qr − p2 cos2(1

4

√4qr − p2ξ

)−√

4qr − p2

−32q2r2 sin2

(14

√4qr − p2ξ

)cos2

(14

√4qr − p2

)(

− 2p sin(1

4

√4qr − p2ξ

)cos(1

4

√4qr − p2ξ

)

+2√

4qr − p2cos2(1

4

√4qr − p2ξ

)−√

4qr − p2

)2

.

Family 3: When r = 0 and pq �= 0, we have the following solutions:

u2,25(x, t) = a0 + 2p2d

d + cosh(pξ) − sinh(pξ)− 2p2d2(

d + cosh(pξ) − sinh(pξ))2 ,

Explicit Solitary Solutions of the Space-Time Fractional Fifth Order... 21

u2,26(x, t) = a0 +2p2(

cosh(pξ) + sinh(pξ))

d + cosh(pξ) + sinh(pξ)−

2p2(

cosh(pξ) + sinh(pξ))2

(d + cosh(pξ) + sinh(pξ)

)2 ,

where d is a real constant.

Family 4: When q �= 0 and r = p = 0, we have the rational solutions:

u2,27(x, t) = a0 − 2q2(qξ + c1

)2 ,

where c1 is a real constant.

In the case 3, substituting all values of

a0 = −1

3p2 − 8

3qr, a1 = −2pq, a2 = −2q2, a−1 = −2pr, a−2 = −2r2,

and ω = (p2 − 4qr)2 and the Riccati solutions into Eq. (3.6), then using

ξ = xβ

�(1 + β)− ωtα

�(1 + α),

we obtain the following exact solutions of Eq. (3.1).

Family 1: When p2 − 4qr > 0 and pq �= 0 (or qr �= 0), we have the hyperbolicfunction solutions of Eq. (3.1)

u3,1(x, t) = − 8q2r2(p +√p2 − 4qr tanh

(12

√p2 − 4qrξ

))2

+ 4pqr

p +√p2 − 4qr tanh(

12

√p2 − 4qrξ

)− p2

2tanh2

(1

2

√p2 − 4qrξ

)+ 2qr tanh2

(1

2

√p2 − 4qrξ

)− 1

3p2 − 8

3qr

22 Sanoe Koonprasert, Sekson Sirisubtawee, and Sivaporn Ampun

u3,2(x, t) = − 8q2r2(p +√p2 − 4qr coth

(12

√p2 − 4qrξ

))2

+ 4pqr

p +√p2 − 4qr coth(

12

√p2 − 4qrξ

)− p2

2coth2

(1

2

√p2 − 4qrξ

)+ 2qr coth2

(1

2

√p2 − 4qrξ

)− 1

3p2 − 8

3qr

u3,3(x, t) = −8q2r2(

p +√

p2 − 4qr(

tanh(√

p2 − 4qrξ)

± isech(√

p2 − 4qrξ)))−2

+ 4pqr

p +√p2 − 4qr(

tanh(√

p2 − 4qrξ)

± isech(√

p2 − 4qrξ))

+ 2qr +(

± (4qr − p2)I sinh(√

p2 − 4qrξ)

+ (p2 − 4qr)

)

× cosh−2(√

p2 − 4qrξ)

− 1

3p2 − 8

3qr

u3,4(x, t) = −8q2r2(

p +√

p2 − 4qr(

coth(√

p2 − 4qrξ)

± csch(√

p2 − 4qrξ)))−2

+ 4pqr

p +√p2 − 4qr(

coth(√

p2 − 4qrξ)

± csch(√

p2 − 4qrξ))

+ p

(p +

√p2 − 4qr

(coth

(√p2 − 4qrξ

)± csch

(√p2 − 4qrξ

)))

− 1

2

(p +

√p2 − 4qr

(coth

(√p2 − 4qrξ

)± csch

(√p2 − 4qr

)))2

+ a0

...

u3,12 = −

(− 2p sinh

(1

4

√p2 − 4qrξ

)cosh

(1

4

√p2 − 4qrξ

)

+2√

p2 − 4qr cosh2(1

4

√p2 − 4qrξ

)−√

p2 − 4qr

)2

8 sinh2(

14

√p2 − 4qrξ

)cosh2

(14

√p2 − 4qrξ

)

Explicit Solitary Solutions of the Space-Time Fractional Fifth Order... 23

−

p

(− 2p sinh

(1

4

√p2 − 4qrξ

)cosh

(1

4

√p2 − 4qrξ

)+2√

p2 − 4qr cosh2(1

4

√p2 − 4qrξ

)−√

4qr − p2

)

2 sinh(

14

√p2 − 4qrξ

)cosh

(14

√p2 − 4qrξ

)

−8pqr sinh

(14

√p2 − 4qrξ

)cosh

(14

√p2 − 4qrξ

)−2p sinh

(1

4

√p2 − 4qrξ

)cosh

(1

4

√p2 − 4qrξ

)+2√

p2 − 4qr cosh2(1

4

√p2 − 4qrξ

)−√

p2 − 4qr

+ a0

Family 2: If p2 −4qr < 0 and pq �= 0 (or qr �= 0), we have the trigonometric functionsolutions as.

u3,13 = − 8q2r2(− p +√4qr − p2 tan

(12

√4qr − p2ξ

))2

− 4pqr

−p +√4qr − p2 tan(

12

√4qr − p2ξ

)− p

(− p +

√4qr − p2 tan

(1

2

√4qr − p2

))

− 1

2

(− p +

√4qr − p2 tan

(1

2

√4qr − p2

))2

+ a0,

u3,14 = − 8q2r2(p +√4qr − p2 cot

(12

√4qr − p2ξ

))2

+ 4pqr

p +√4qr − p2 cot(

12

√4qr − p2ξ

)− p

(− p +

√4qr − p2 tan

(1

2

√4qr − p2

))

− 1

2

(− p +

√4qr − p2 tan

(1

2

√4qr − p2

))2

+ a0,

...

24 Sanoe Koonprasert, Sekson Sirisubtawee, and Sivaporn Ampun

u3,17 = −32q2r2(

− 2p +√

4qr − p2

(tan(1

4

√4qr − p2ξ

)− cot

(1

4

√4qr − p2ξ

)))−2

− 8pqr

−2p +√4qr − p2

(tan(

14

√4qr − p2ξ

)− cot

(14

√4qr − p2ξ

))

− 1

2p

(− 2p +

√4qr − p2

(tan(1

4

√4qr − p2ξ

)− cot

(1

4

√4qr − p2ξ

)))

− 1

8

(− 2p +

√4qr − p2

(tan(1

4

√4qr − p2ξ

)− cot

(1

4

√4qr − p2

)))2

+ a0,

...

u3,24 = −

(− 2p sin

(1

4

√4qr − p2ξ

)cos(1

4

√4qr − p2ξ

)

+2√

4qr − p2 cos2(1

4

√4qr − p2ξ

)−√

4qr − p2

)2

8 sin2(

14

√4qr − p2ξ

)cos2

(14

√4qr − p2ξ

)

−

p

(− 2p sin

(1

4

√4qr − p2ξ

)cos(1

4

√4qr − p2ξ

)+2√

4qr − p2 cos2(1

4

√4qr − p2ξ

)−√

4qr − p2

)

2 sin(

14

√4qr − p2ξ

)cos(

14

√4qr − p2ξ

)

−8pqr sin

(14

√4qr − p2ξ

)cos(

14

√4qr − p2ξ

)−2p sin

(1

4

√4qr − p2ξ

)cos(1

4

√4qr − p2ξ

)+2√

4qr − p2 cos2(1

4

√4qr − p2ξ

)−√

4qr − p2

−32q2r2 sin2

(14

√4qr − p2ξ

)cos2

(14

√4qr − p2

)(

− 2p sin(1

4

√4qr − p2ξ

)cos(1

4

√4qr − p2ξ

)

+2√

4qr − p2cos2(1

4

√4qr − p2ξ

)−√

4qr − p2

)2

+ a0, (3.8)

Explicit Solitary Solutions of the Space-Time Fractional Fifth Order... 25

(a) β = 0.21 (b) β = 0.5 (c) β = 0.72

Figure 1: Graphs of solutions u3,17(x, t) of the space-time Sawada-Kotera equation (3.1)with the fractional order α = 0.5 and β = 0.21, 0.5, 0.72.

Family 3: When r = 0 and pq �= 0, we have the following hyperbolic function solutions:

u3,25 = −1

3p2 + 4p2d

d + cosh(pξ) − sinh(pξ)− 4p2d2

(d + cosh(pξ) − sinh(pξ))2

u3,26 = −1

3p2 +

4p2(

cosh(pξ) + sinh(pξ))

d + cosh(pξ) + sinh(pξ)−

4p2(

cosh(pξ) + sinh(pξ)2)

(d + cosh(pξ) + sinh(pξ)

)2 ,

where d is a real constant.

Family 4: When q �= 0 and r = p = 0, we have the solutions:

u3,27(x, t) = − 2q2(qξ + c1

)2 − 1

3p2 − 8

3qr,

where c1 is a real constant.

Remark. All exact solutions that were obtained in Case 1, Case 2, and Case 3 have beenverified that they exactly satisfy the Sawada-Kotera equation (3.1).

26 Sanoe Koonprasert, Sekson Sirisubtawee, and Sivaporn Ampun

(a) β = 0.55 (b) β = 0.8 (c) β = 0.99

Figure 2: Graphs of the solution u3,17(x, t) of the space-time Sawada-Kotera equationfor u3,26 when fractional order α = 0.5

4. Graphical illustrations of the solutions

The Sawada-Kotera equation is an important model in physical problems describing thevarious types of motion of long waves in shallow water under gravity. The graphs ofsolution u3,17(x, t) in (3.8) with the parameters p = 3, q = 1, r = 2 describing thesingular soliton solutions for the fractional order α = 0.5, β = 0.21, 0.5, 0.72 areshown in Fig. 1.

The solution u3,26(x, t) with the parameters α = 0.5, p = 2, q = 1, r = 0 isrepresented in the following Fig. 2 that depends on the fractional order β = 0.55, 0.8and β = 0.99 respectively.

5. Conclusions

In this paper, the improved generalized Riccati mapping method is proposed to obtainexact solutions of the space-time fractional (1+1) dimensional Sawada-Kotera equa-tion. To the best of our knowledge, new exact solutions of the space-time fractionalSawada-Kotera equation have not been reported in previous literatures. Using the im-proved generalized Riccati mapping method which is direct, efficient and powerful, weobtained more abundant exact solutions that include 36 trigonometric function solutions,

Explicit Solitary Solutions of the Space-Time Fractional Fifth Order... 27

38 hyperbolic function solutions, and one rational function solution. All of our resultshave been verified with the Maple 17 program by substituting them back into the originalequation in (3.1) which they are satisfied. Some graphical travelling wave solutions areillustrated by varying the values of the fractional order α, β. Moreover, the proposedmethod can be applied to obtain exact solutions of many other nonlinear fractional partialdifferential equations.

Acknowledgements

The paper was supported by the Faculty of Applied Science, King Mongkut’s Universityof Technology North Bangkok.

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