The ($G' /G, 1/G$)-expansion method for solving nonlinear space–time fractional differential equations

DOI 10.1007/s12043-016-1225-7

The

G

/G, 1 /G

-expansion method for solving nonlinear space–time fractional differential equations

EMRULLAH YA ¸SAR^∗and ˙ILKER BURAK GIRESUNLU

Department of Mathematics, Faculty of Arts and Sciences, Uludag University, 16059, Bursa, Turkey

∗Corresponding author. E-mail: eyasar@uludag.edu.tr

MS received 4 July 2015; revised 10 September 2015; accepted 1 October 2015; published online 6 July 2016 Abstract. In this work, we present

G/G,1/G

-expansion method for solving fractional differential equations based on a fractional complex transform. We apply this method for solving space–time fractional Cahn–Allen equation and space–time fractional Klein–Gordon equation. The fractional derivatives are described in the sense of modified Riemann–Lioville. As a result of some exact solution in the form of hyperbolic, trigonometric and rational solutions are deduced. The obtained solutions may be used for explaining of some physical problems.

The

G/G,1/G

-expansion method has a wider applicability for nonlinear equations. We have verified all the obtained solutions with the aid ofMaple.

Keywords. Exact solution; modified Riemann–Liouville fractional derivative; space–time Cahn–Allen equation; space–time Klein–Gordon equation;

G/G,1/G

-expansion method.

PACS Nos 02.30.Jr; 02.70.Wz; 04.20.Jb 1. Introduction

Fractional calculus can be viewed as one of the exten- sions of classical ordinary calculus. In fact, the roots of fractional calculus dates back to three hundred years ago. There have been many contributions in this area.

Oldham and Spanier [1] first considered the fractional differential equations (FDEs). The study of the exact solutions of nonlinear fractional evolution equations (NLFEEs) plays an important role in understanding the nonlinear physical phenomena which are described by these equations. For example, the nonlinear oscillation of an earthquake can be modelled with the fractional derivatives. In reality, a physical phenomenon may depend not only on the time instant but also on the previous time history, which can be successfully mod- elled by using the theory of derivatives and integrals of fractional order [2–4]. As fractional partial differential equations (FPDEs) appear frequently in diverse fields such as physics, biology, rheology, viscoelasticity con- trol theory, signal processing, systems identification and electrochemistry, they attract considerable inter- est and recently there has been significant theoretical development [5]. In recent years, several powerful and

efficient methods have been developed for finding ana- lytic solutions of NLFEEs. Some of the most important methods found in literature include the exp-function method [6], Adomian decomposition method [7], tanh–

sech function method [8], the first integral method [9]

and the(G/G)-expansion method [10] which can be used to construct exact solutions for some time and space FPDEs. Based on these methods, a variety of FPDEs have been investigated and solved.

Very recently, Jumarie [11] suggested a modified Riemann–Liouville derivative. With the help of this fractional derivative and some important formulas, one can convert FDEs into integer-order differential equa- tions by fractional complex transformation [12,13].

The main aim of this work is to extend the applica- tion of the

G/G,1/G

-expansion method [14,15] to obtain some exact travelling wave solutions to some space–time FDEs.

The first model considered is the space–time frac- tional Cahn–Allen equation:

D_t^αu−D^α_x D_x^αu

+u³−u=0. (1) Cahn–Allen equation arises in many scientific applications such as mathematical biology, quantum mechanics and plasma physics [16,17].

1

The second model studied is the space–time frac- tional Klein–Gordon equation:

D^α_t D_t^αu

−D_x^α D^α_xu

+u³−u=0. (2) The nonlinear fractional Klein–Gordon equation models many types of nonlinearities. It plays a sig- nificant role in several real-world applications, for example, in solid-state physics, nonlinear optics and quantum field theory [18].

The rest of this work is organized as follows: In

§2, we present some basic properties of the modified Riemann–Liouville derivative operator. The main steps of the

G/G,1/G

-expansion method are given in §3.

In §4 and 5, we illustrate this method in detail with space–time fractional Cahn–Allen equation and space–

time fractional Klein–Gordon equations, respectively.

In the last section, some conclusions are provided.

2. Description of Jumarie’s modified

Riemann–Liouville derivative operator and its important property

The Jumarie’s modified Riemann–Liouville (RL) derivative is defined as follows [11]:

D^α_t f (t) = 1 (1−α)

d dt

_t

0

(t−ξ)^−α(f (ξ)−f (0))dξ, 0< α <1, (3) where f: R → R, t → f (t) denotes a continuous function andf (t)−f (0)=0 fort <0. Also, iff (0) in any case is infinity, then specifically one needs to takef (t)−f (0⁺)or the finite part of(f (t)−f (0)).

As the expression in (3) is also a Caputo deriva- tive, only iff (t) is differentiable, one may state that in this Jumarie’s definition of modified RL fractional derivative with offsetting via function value at start point of fractional differentiation, the condition of differentiability off (t)is mandatory.

For the derivative, we give the following important property which is useful to solve FDEs:

D^α_t t^r = (1+r)

(1+r−α)t^r−α.

3. Algorithm of(G/G,1/G)-expansion method with fractional complex transform

Before presenting the algorithm, we need the solutions of the auxilary equation (see [14,15]).

Remark1. If we consider the second-order linear ODE:

G(ξ)+λG(ξ)=μ (4)

and setφ =G/G,ψ =1/G, then we get

φ = −φ²+μψ−λ, ψ = −φψ. (5) One can consider the general solution of (4), in two distinct subcases:

Case1. Ifλ < 0, then the general solutions of (4) has the form:

G(ξ)=A1sinh ξ√

−λ

+A2cosh ξ√

−λ +μ

λ, (6) whereA1,A2are constants. Consequently, we have ψ² = −λ

λ²σ+μ²(φ²−2μψ+λ), (7) whereσ =A²₁−A²₂.

Case2. Ifλ > 0, then the general solutions of (4) has the form

G(ξ)=A1sin ξ√

λ

+A2cos ξ√

λ +μ

λ, (8) and hence

ψ² = λ

λ²σ−μ²(φ²−2μψ+λ), (9) whereσ =A²₁+A²₂.

The main steps of this method are described as follows:

Step1. We consider the following general nonlinear FDE of the form:

F

u, D^α_t u, D_x^βu, D^α_t D_t^αu, D_t^αD^β_xu, D_x^βD^β_xu, . . .

=0, (10) where F is a polynomial in u(x, t) and its partial fractional derivatives.

The complex wave variable u(x, t)=U(ξ), ξ = kx^β

(1+β)+ ct^α

(1+α), (11) wherekandcare constants, reduces (10) to an ODE in the form:

P

U, U, U, U, . . .

=0, (12)

whereP is a polynomial ofU(ξ)and its total deriva- tives with respect toξ. If possible, we should integrate (12), term by term one or more times.

Step2. Assume that the solution of (12) can be expressed by a polynomial in the two variablesφ and ψas follows:

U(ξ)= N i=0

aiφⁱ+ N j=1

bjφ^j−1ψ, (13) whereai(i = 0,1,2, . . . , N) and bj(j = 1, . . . , N) are constants to be determined later.

Determine the positive integer N in (13) by using the homogeneous balance between the highest-order derivatives and the nonlinear terms in (12).

Step3. Substitute (13) into (12) along with (5) and (7), the left-hand side of (12) can be converted into a poly- nomial inφ and ψ, in which the degree ofψ is not greater than 1. Equating each coefficient of this poly- nomial to zero, yields a system of algebraic equations which can be solved by usingMapleto get the values ofai,bi,k,c,μ,A1,A2andλ,whereλ <0.

Step4. Similar to Step 3, substituting (13) into (12), along with (5) and (9) forλ > 0, we obtain the exact solutions of (12) expressed by trigonometric functions.

4. Implementation of

G/G,1/G

-expansion method for the exact solutions of space–time fractional Cahn–Allen equation

We consider space–time fractional Cahn–Allen equation

D^α_t u−D_x^α D^α_xu

+u³−u=0, (14) whereu(x, t)is an unknown function.

For our goal, we present the following transformation:

u(x, t)=U(ξ), ξ = kx^α

(1+α)+ ct^α

(1+α), (15) wherecandkare constants. Then, by using (15), (14) can be turned into an ODE:

−k²U+cU+U³−U =0, (16)

whereU =dU/dξ.

Balancing the order of U and U³ in (16), we get N =1. Consequently, we get

U(ξ)=a0+a1φ+b1ψ, (17) wherea0,a1andb1are constants.

Case1. Hyperbolic function solutions (λ <0)

Ifλ <0, substituting (17) into (16) and using (5) and (7), the left-hand side of eq. (16) becomes a polynomial inφ andψ. Setting the coefficients of this polynomial to zero, yields a system of algebraic equations in a0, a1,b1,candkas follows:

φ³: μ²a₁²−2μ²k²−3b₁²λ−2k²λ²σ+a²₁λ²σ=0, φ²ψ: 3μ²a1²−2μ²k²+3a²1λ²σ−2k²λ²σ−b²1λ=0, φ²: −3a0b1²λμ²−ca1μ⁴−ca1λ⁴σ²+3a0a1²λ⁴σ²

−3a0b²₁λ³σ+6a0a²₁μ²λ²σ −k²b1λ³μσ

−2ca1μ²λ²σ −2b³₁λ²μ−k²b1λμ³ +3a0a²₁μ⁴=0,

φψ: 3k²a1μ³−μ²cb1+6μ²a0a1b1+3μk²a1λ²σ +6μa1b²1λ+6a0a1b1λ²σ−cb1λ²σ =0, φ: −μ²−2μ²k²λ+3μ²a²₀+3a₀²λ²σ−3b₁²λ²

−λ²σ −2k²λ³σ =0,

ψ: −b³₁λ⁴σ +3b₁³λ²μ²+3a²₀b1μ⁴−b1λ⁴σ² +ca1μ⁵+ca1μλ⁴σ²−2b1μ²λ²σ +6a0²b1μ²λ²σ +k²b1λμ⁴+3a²0b1λ⁴σ² +6a0b²₁λμ³+6a0b²₁λ³μσ −k²b1λ⁵σ²

−b1μ⁴+2ca1μ³λ²σ =0,

cons.: −3a0b1²λ⁴σ−3a0b1²λ²μ²−2a0λ²σμ² +2a0³λ²σμ²−ca1λμ⁴−k²b1λ²μ³−a0μ⁴

−k²b1λ⁴μσ−a0λ⁴σ²−2ca1λ³σμ²+a₀³λ⁴σ²

−ca1λ⁵σ²−2b₁³λ³μ+a₀³μ⁴ =0.

On solving the above algebraic equations using Maple, we get the following results:

a0=0, a1= ∓ 1

√−λ, b1= ∓ λ²σ +μ²

λ , c=0, k = ∓ 2

−λ

,

a0= 1

2, a1 = ∓ 1 2√

−λ, b1= ∓ λ²σ+μ²

2λ , c= ∓ 3

2√

−λ, k = ∓ 1

−2λ

,

a0= −1

2, a1= ∓ 1 2√

−λ, b1= ∓ λ²σ+μ²

2λ , c= ± 3

2√

−λ, k = ∓ 1

−2λ

.

In this case, the exact solutions of (14) are u11(x, t)= A1cosh

√ 2x^α (1+α)

+A2sinh √

2x^α (1+α)

+ √

μ²+λ²σ λ

A1sinh √

2x^α (1+α)

+A2cosh √

2x^α (1+α)

+ ^μ_λ ,

u12(x, t)= −A1cosh

−₍^√₁²_+α)^xα

+A2sinh

−₍^√₁²_+α)^xα +√

μ²+λ²σ λ

A1sinh

−₍^√₁²_+α)^xα

+A2cosh

−₍^√₁²_+α)^xα

+^μ_λ , u21(x, t)= 1

2 +1 2

A1cosh √ x^α

2(1+α) +₂₍³₁^tα_+α)

+A2sinh √ x^α

2(1+α)+ ₂₍³₁^tα_+α) +√

μ²+λ²σ λ

A1sinh √ x^α

2(1+α)+ ₂₍^3t_1+α)^α

+A2cosh √ x^α

2(1+α)+ ₂₍^3t_1+α)^α

+^μ_λ , u22(x, t)= 1

2 −1 2

A1cosh

−^√₂₍^xα_1+α) −₂₍³₁^tα_+α)

+A2sinh

−^√₂₍^xα_1+α) −₂₍³₁^tα_+α) + √

μ²+λ²σ λ

A1sinh

−^√₂₍^xα_1+α)− ₂₍³₁^tα_+α)

+A2cosh

−^√₂₍^xα_1+α) −₂₍³₁^tα_+α)

+ ^μ_λ , u31(x, t)= −1

2 +1 2

A1cosh √ x^α

2(1+α) −₂₍³₁^tα_+α)

+A2sinh √ x^α

2(1+α) −₂₍³₁^tα_+α) +√

μ²+λ²σ λ

A1sinh √ x^α

2(1+α)− ₂₍³₁^tα_+α)

+A2cosh √ x^α

2(1+α)− ₂₍³₁^tα_+α)

+^μ_λ , u32(x, t)= −1

2 −1 2

A1cosh

−^√₂₍^xα_1+α) +₂₍³₁^tα_+α)

+A2sinh

−^√₂₍^xα_1+α) +₂₍³₁^tα_+α) +√

μ²+λ²σ λ

A1sinh

−^√₂₍^xα_1+α) +₂₍³₁^tα_+α)

+A2cosh

−^√₂₍^xα_1+α) +₂₍³₁^tα_+α)

+ ^μ_λ ,

where

σ =A²1−A²2.

Case2. Trigonometric function solutions (λ >0) Ifλ > 0, substituting (17) into (16) and using (5) and (9), the left-hand side of eq. (16) becomes a polynomial inφandψ. Setting the coefficients of this polynomial to zero, yields a system of algebraic equations ina0, a1,b1,candkas follows:

φ³: −2μ²k²+μ²a₁²−3b₁²λ

−a1²λ²σ +2k²λ²σ =0, φ²ψ: −2μ²k²+3μ²a₁²−b₁²λ

+2k²λ²σ −3a²1λ²σ =0,

φ²: −6a0a₁²λ²σμ²−2b³₁λ²μ−ca1λ⁴σ² +3a0a1²μ⁴−k²b1λμ³−ca1μ⁴

+3a0a₁²λ⁴σ²+3a0b₁²λ³σ−3a0b₁²λμ² +2ca1μ²λ²σ+k²b1λ³μσ =0,

φψ: 3k²a1μ³+6μ²a0a1b1−μ²cb1

−3μk²a1λ²σ+6μa1b²1λ

−6a0a1b1λ²σ+cb1λ²σ =0, φ: −μ²+3μ²a0²−2μ²k²λ−3b²1λ²

+2k²λ³σ −3a0²λ²σ +λ²σ =0, ψ: b₁³λ⁴σ+3b³₁λ²μ²+3a0²b1μ⁴−b1λ⁴σ²

+ca1μ⁵+3a0²b1λ⁴σ²+ca1μλ⁴σ² +6a0b1²λμ³−6a0b²1λ³μσ +k²b1λμ⁴

−k²b1λ⁵σ²+2b1μ²λ²σ −b1μ⁴

−2ca1μ³λ²σ −6a²0b1λ²σμ²=0, cons.: 3a0b²1λ⁴σ −3a0b²1λ²μ²+2a0λ²σμ²

−2a0³λ²σμ²+2ca1λ³σμ²

−k²b1λ²μ³+a₀³λ⁴σ²−ca1λ⁵σ²

−ca1λμ⁴+k²b1λ⁴μσ −a0μ⁴+a³₀μ⁴

−2b₁³λ³μ−a0λ⁴σ² =0.

On solving the above algebraic equations using Maple, we get the following results:

a0=0, a1= ∓ 1

√−λ, b1= ∓ μ²−λ²σ

λ , c=0, k = ∓ 2

−λ

,

a0= 1

2, a1 = ∓ 1 2√

−λ, b1 = ∓ μ²−λ²σ

2λ , c= ∓ 3 2√

−λ, k = ∓ 1

−2λ

,

a0= −1

2, a1 = ∓ 1 2√

−λ, b1 = ∓ μ²−λ²σ

2λ , c= ± 3 2√

−λ, k = ∓ 1

−2λ

.

In this case, the exact solutions of (14) are u11(x, t) = A1cos

i_(1+α)^√2xα

−A2sin

i_(1+α)^√2xα +√

μ²−λ²σ λ

A1sin

i₍^√₁²_+α)^xα

+A2cos

i₍^√₁²_+α)^xα

+^μ_λ , u12(x, t) = −A1cos

−i₍^√₁^2x_+α)^α

−A2sin

−i₍^√₁^2x_+α)^α + √

μ²−λ²σ λ

A1sin

−i₍^√₁²_+α)^xα

+A2cos

−i₍^√₁²_+α)^xα

+^μ_λ , u21(x, t) = 1

2 +1 2

A1cos

√ ix^α

2(1+α) +₂₍^3it_1+α)^α

+A2sin

√ ix^α

2(1+α) +₂₍^3it_1+α)^α +√

μ²−λ²σ λ

A1sin

√ ix^α

2(1+α) +₂₍^3it_1+α)^α

+A2cos

√ ix^α

2(1+α) +₂₍^3it_1+α)^α

+^μ_λ , u22(x, t) = 1

2 −1 2

A1cos

√−ix^α

2(1+α) −₂₍^3it_1+α)^α

+A2sin

√−ix^α

2(1+α) −₂₍^3it_1+α)^α +√

μ²−λ²σ λ

A1sin

√−ix^α

2(1+α) −₂₍^3it_1+α)^α

+A2cos

√−ix^α

2(1+α) −₂₍^3it_1+α)^α

+^μ_λ , u31(x, t) = −1

2 +1 2

A1cos

√ ix^α

2(1+α) −₂₍^3it_1+α)^α

+A2sin

√ ix^α

2(1+α) −₂₍^3it_1+α)^α +√

μ²−λ²σ λ

A1sin

√ ix^α

2(1+α) −_2(1+α)^3itα

+A2cos

√ ix^α

2(1+α) −_2(1+α)^3itα

+^μ_λ , u32(x, t) = −1

2 −1 2

A1cos

−^√₂₍^ix₁^α_+α)+ ₂₍^3it_1+α)^α

+A2sin

−^√₂₍^ix₁^α_+α) +₂₍^3it_1+α)^α +√

μ²−λ²σ λ

A1sin

−^√₂₍^ix₁^α_+α) +_2(1+α)^3itα

+A2cos

−^√₂₍^ix₁^α_+α)+ _2(1+α)^3itα

+^μ_λ ,

where

σ =A²₁+A²₂.

5. Implementation of

G/G,1/G

-expansion method for the exact solutions of space–time fractional Klein–Gordon equation

We consider space–time fractional Klein–Gordon equation

D^α_t D^α_t u

−D^α_x D_x^αu

+u³−u=0, (18)

whereu(x, t)is an unknown function. We suppose the following transformation:

u(x, t)=U(ξ), xi = kx^α

(1+α)+ ct^α

(1+α), (19) wherecandkare constants. Then putting (19) in (18), we can get an ODE of the form:

(c²−k²)U+U³−U =0, (20) where

U=dU/dξ.

Balancing the order of U and U³ in (20), we get N=1. Consequently, we get

U(ξ)=a0+a1φ+b1ψ, (21)

wherea0,a1andb1are constants.

Case1. Hyperbolic function solutions (λ <0)

Similarly, solving the algebraic system withMaple, we get

a0 =0, a1 = ∓ 1

√−λ, b1 = ∓ λ²σ +μ²

λ , c= ∓

k²+ 2

λ, k=k

. In this case, the exact solutions of (18) are:

u1(x, t)=

A1cosh

kx^α+

k²+_λ²t^α _√

(1−λ+α)

+A2sinh

kx^α+

k²+²_λt^α _√

(1−λ+α)

+√

μ²+λ²σ λ

A1sinh

kx^α+

k²+ ²_λt^α _√

(1−λ+α)

+A2cosh

kx^α+

k²+_λ²t^α _√

(1−λ+α)

+^μ_λ

,

u2(x, t)= −

A1cosh

kx^α−

k²+²_λt^α _√

(1−λ+α)

+A2sinh

kx^α−

k²+²_λt^α _√

(1−λ+α)

+√

μ²+λ²σ λ

A1sinh

kx^α−

k²+ ²_λt^α _√

(1−λ+α)

+A2cosh

kx^α−

k²+²_λt^α _√

(1−λ+α)

+^μ_λ

.

(22)

Case2. Trigonometric function solutions (λ >0) In this case, solving the algebraic system withMaple, we get

a0 =0, a1 = ∓ 1

√−λ, b1 = ∓ μ²−λ²σ

λ , c= ∓

k²+ 2

λ, k=k

.

The exact solutions of (18) are

u1(x, t)= i

A1cos

kx^α+

k²+_λ²t^α _√

(1+α)λ

−A2sin

kx^α+

k²+²_λt^α _√

(1+α)λ

+√

μ²−λ²σ λ

A1sin

kx^α+

k²+_λ²t^α _√

(1+α)λ

+A2cos

kx^α+

k²+ ²_λt^α _√

(1+α)λ

+^μ_λ

,

u2(x, t)= − i

A1cos

kx^α−

k²+ ²_λt^α _√

(1+α)λ

−A2sin

kx^α−

k²+²_λt^α _√

(1+α)λ

+√

μ²−λ²σ λ

A1sin

kx^α−

k²+_λ²t^α _√

(1+α)λ

+A2cos

kx^α−

k²+ ²_λt^α _√

(1+α)λ

+ ^μ_λ

.

(23)

Remark2. All solutions of this work have been checked with Maple by putting them back into the original eqs (14) and (18).

6. Conclusion

In this work, we have implemented fractional complex transform with the help of (G/G,1/G)-expansion method for obtaining several travelling wave exact solutions of nonlinear evolution equations, namely, the space–time fractional Cahn–Allen and Klein–Gordon equations. With the fractional complex transform, the equation(s) can be very easily converted to the cor- responding ODE. Then, we use the (G/G,1/G)- expansion method for getting exact travelling wave solutions of the aforementioned equations. The obtained exact solutions would be very useful in various areas of applied mathematics and they can be used to interpret some physical phenomena. From our results obtained in this paper, we conclude that the (G/G,1/G)- expansion method is powerful, effective and conve- nient for NLFEEs. This method readily does not use any linearization process, unrealistic ansatzes (see also [7]). In addition, as one can see, this method has more general applications than the other methods.

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