Ain Shams Engineering Journal (2013) xxx, xxx-xxx

Ain Shams University Ain Shams Engineering Journal

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ENGINEERING PHYSICS AND MATHEMATICS

Application of the simplest equation method to some time-fractional partial differential equations

N. Taghizadeh a, M. Mirzazadeh a'*, M. Rahimian b, M. Akbari a

a Department of Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, P.O. Box 1914, Rasht, Iran b Department of Mathematics, Islamic Azad University, Masjed Soleiman Branch, Masjed Soleiman,Iran

Received 18 September 2012; revised 30 December 2012; accepted 22 January 2013

KEYWORDS

The simplest equation method;

Nonlinear fractional KleinGordon equation; Generalized Hirota-Satsuma coupled KdV system of time-fractional order; Nonlinear fractional Shar-ma-Tasso-Olever equation

Abstract In this paper, we establish exact solutions for some time fractional differential equations. The simplest equation method is used to construct the exact solutions of nonlinear fractional KleinGordon equation, Generalized Hirota-Satsuma coupled KdV system of time fractional order and nonlinear fractional Sharma-Tasso-Olever equation. The simplest equation method presents a wide applicability to handling nonlinear wave equations.

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1. Introduction

In the past four decades, the study of nonlinear evolution equations (NLEEs) modelling various physical phenomena, has played a significant role in many scientific applications such as water waves, nonlinear optics, plasma physics and solid state physics. Searching for exact solutions to these NLEEs has been one of the central issues in the field of mathematical physics and applied mathematics. Many powerful methods for finding exact solutions of NLEEs have been proposed, such as ansatz method and topological solitons [1-4], tanh method

* Corresponding author. Tel.: +98 09113445754. E-mail addresses: taghizadeh@guilan.ac.ir (N. Taghizadeh), mirzazadehs2@guilan.ac.ir (M. Mirzazadeh). q Peer review under responsibility of Ain Shams University.

[5,6], multiple exp-function method [7], simplest equation method [8-11], Hirotas direct method [12,13], transformed rational function method [14] and so on.

Khan et al. [15,16] proposed travelling waves methodology for solving Fluid Mechanics model.

The simplest equation method is a very powerful mathematical technique for finding exact solutions of nonlinear ordinary differential equations. It has been developed by Kudryashov [8,9] and used successfully by many authors for finding exact solutions of ODEs in mathematical physics [10,11]. Using simplest equation method (see Ref. [17]) exact solutions of the perturbed nonlinear Schrodinger's equation with Kerr law nonlinearity, the nonlinear Schrodinger's equation were obtained.

The aim of this paper is to find exact solutions of nonlinear fractional Klein-Gordon equation, Generalized Hirota-Satsuma coupled KdV system of time fractional order and nonlinear fractional Sharma-Tasso-Olever equation by using the simplest equation method.

The paper is arranged as follows. In Section 2, we describe briefly the Modified Riemann-Liouville derivative with

2090-4479 © 2013 Ain Shams University. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asej.2013.01.006

properties and simplest equation method. In Section 3, we apply this method to nonlinear fractional Klein-Gordon equation, Generalized Hirota-Satsuma coupled KdV system of time fractional order and nonlinear fractional Sharma-Tasso-Olever equation.

2. Modified Riemann-Liouville derivative and the simplest equation method

The Jumaries modified Riemann-Liouville derivative of order a is defined by the expression

Dfx) = -— (x - n)-a-1 m-A0№,

C (-a) Jo

if a <0,

1 d tx

dfx Jo(x - n)-a/(n)-/(0M, (2)

if 0 < a <1,

D/(x) = (/n)(x))<a-n), (3)

if n 6 a 6 n + 1,n P 1, where f.R fi R is a continuous function.

Some properties of the fractional modified Riemann-Liouville derivative were summarized in, three useful formulas of them are

c > 0, C(1 + c - a)

DX(u(x)(x)) = (x)DXu(x) + u(x)DX(x), D"x\f(u(x))} = fU(u)DXu(x) = DUf(u)(uX)a

which are direct consequences of the equality dax(t) = C (1 + a)d x(t).

The main steps of the simplest equation method are summarized as follows:

Step 1. We first consider a general form of the time fractional differential equation

P(u, Datu, ux, D2tau, uxx,...) = 0.

Step 2. To find the exact solution of Eq. (7) we introduce the variable transformation

u(x, t) =y(n), n = Ix -

C(1 + a)

where I and k are constants to be determined later.Using Eq. (8) changes the Eq. (7) to an ODE

where y = y(Q is an unknown function, Q is a polynomial in the variable and its derivatives.

Step 3. The basic idea of the simplest equation method consists in expanding the solutions y(£) of Eq. (9) in a finite series

y(n) = °iZ', aN-o,

where the coefficients at are independent of n and z = z(n) are the functions that satisfy some ordinary differential equations.In this paper, we use the Bernoulli equation as simplest equation

- = az(n) + bz2(n),

Eq. (11) admits the following exact solutions

z(n)=-

a exp\a(n + no)]

1 - b exp\a(n + no)] for the case a >0, b <0 and

z(n) = -

a exp\a(n + n0)] 1 + b exp\a(n + n0)];

for the case a <0, b >0, where n° is a constant of integration.

Step 4. Substituting Eq. (10) into Eq. (9) with Eq. (11), then the left hand side of Eq. (9) is converted into a polynomial in z(n), equating each coefficient of the polynomial to zero yields a set of algebraic equations for a, a, b, I, k. Step 5. Solving the algebraic equations obtained in step 4, and substituting the results into (10), then we obtain the exact solutions for Eq. (7).

Remark 1. N is a positive integer, in most cases, that will be determined. To determine the parameter N, we usually balance the linear terms of highest order in the resulting equation with the highest order nonlinear terms.

Remark 2. In Eq. (11), when a = A and b = -Bernoulli equation dZ

- Az(n)-z2(n).

Eq. (14) admits the following exact solutions

z(n)=A

1 we obtain the

1 + tanh(-(n + n0)

when A >0, and

1 - tanh(- (n + n0)

when A <0.

3. Application

In this section, we present three examples to illustrate the applicability of the simplest equation method to solve nonlinear time-fractional partial differential equations.

3.1. Nonlinear fractional Klein-Gordon equation

We first consider the nonlinear fractional Klein-Gordon equation [18]

= + h1u(x, t) + 02u3(x, t), t > 0, 0 < a 6 1, (17)

where 0! and h2 are arbitrary constants. By make the transformation

u(x, t)=y(n), n = Ix -

Eq. (17) becomes

C(1 + a)

(k2 - l2)ytt - hiy - 02У3 = 0. (19)

For the solutions of Eq. (19), we make the following ansatz

У(П) = ^a;z\ ai - 0,

where at are all real constants to be determined, N is a positive integer which can be determined by balancing the highest order derivative term with the highest order nonlinear term after substituting ansatz (20) into Eq. (19), where z satisfies Eq. (11).

When balancing ynn with y3 then gives N + 2 = 3N ) N — 1. Therefore, we may choose

y(n) — ao + a1z(n). (21)

Substituting Eq. (21) along with Eq. (11) in Eq. (19) and then setting the coefficients of Z(j = 3,2,1,0) to zero in the resultant expression, we obtain a set of algebraic equations involving a0, ai, a, b, k and I as

2(k2 - l2)b2ax — 02a3 — 0, (22)

3(k2 — 12)aba1 — 302a0a2 — 0, (k2 — 12)a2a1 — 302a0a1 — 01a1 — 0, — 02a0 — 01 a0 — 0.

With the aid of Maple, we shall find the special solution of the above system

V—0102

ao = ±—5-,

b = ±

2(k2 - l2

Г20Т

Vl2- k2

where l, k and are arbitrary constants.

Assuming a >0 and choosing b <0. Therefore, using solutions (12) and (13) of Eq. (11), ansatz (21) , we obtain the following travelling-wave solution of Eq. (19)

' rlh_ ffiffi a1exP [y^n + gp)] x "02

у(П) = ±

^k21 - ai VSK exp [^(П + no)] J

Then the exact solution to Eq. (17) can be written as

U(X; t) = ± J- — -

ai exp

Vl2 - k2 1 - a

(k2-2 )

Substituting Eq. (21) along with Eq. (14) in Eq. (19) and setting all the coefficients of powers z to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain

a1 = ±

, , 01 , ao = ±l/-^", A = 02

П20Т

Vl2- k2,

where k and I are arbitrary constants.

Therefore, using solution (16) of Eq. (14), ansatz (21), we obtain the following exact solution of Eq. (19)

у(П) = ±А/-h^tanh 02

dn + По) .

Then, the exact solution to Eq. (17) can be written as

u(x, t) = -1 02

ix —

k2)V C(1 + a

■f + По

Comparing our results with Lu's results [18] then it can be seen that some results are same.

3.2. Nonlinear fractional Sharma-Tasso-Olever equation

In this section, we study nonlinear fractional Sharma-Tasso-Olever equation [18]

Datu + 3kuX + 3ku2ux + 3kuuxx + kuxxx = 0, 0 < a 6 1.

We use the transformation

t > 0,

i(x, t) = у(П), П = x -

C(1 + a)

Substituting Eq. (30) into Eq. (29), we obtain ordinary differential equation:

kyn + 3k(yn) + 3ky yn + 3kyynn + kynnn = 0.

Integrating Eq. (31) with respect to g yields

R - ky + 3kyyn + ky3 + kytl = 0. (32)

When balancing y3 with ygg then gives N =1. Therefore, we may choose

У (П) = a0 + a1z(n).

Substituting Eq. (33) along with Eq. (11) in Eq. (32) and setting all the coefficients of powers z to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain

a = -a0, b = - -1, R = 0, k = ka^,

where a0 and a1 are arbitrary constants.

Assuming a >0 and choosing b <0. Therefore, using solution (12) of Eq. (11), ansatz (33) , we obtain the following exact solution of Eq. (32)

2a 1 exp[—a0(n + &)]

у(п) = m 1

2 + a1 exp[-a0(g + &,)]

Then the exact solution to nonlinear fractional Sharma Tasso-Olever equation can be written as

^ _____ T (ka0 i £ 1 ^

i(x, t) = a0

2 - a1 exp [-a^x - ta + П0)]

2 + a1 exp a0(^x -

—ka°— ^

Г(1+а) '

Substituting Eq. (33) along with Eq. (14) in Eq. (32) and setting all the coefficients of powers z to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain

A = -a0, r = 0, k = ka0, a1 = 2, where a0 is arbitrary constant.

r — 3ka0 A + 2ka0 + a0kA2, k — 3ka0A + 3ka0 + kA2, ai — 1,

where a0 and A are arbitrary constants.

Therefore, using solution (16) of Eq. (14), ansatz (33), we obtain the following exact solutions of Eq. (32)

y(n) = -a0 tanh g (n + &)).

y (n) =a0 +

1 - tanh ( — (n + n0)

Then, the exact solutions to nonlinear fractional Sharma-Tas-so-Olever equation can be written as

u(x, t) — -a0 tanh ( y ^x- ^(1 + a)

u(x, t) = a0 + -

, , A ( 3ka0A + 3ka0 + kA2 a

1 - tanM 2 [x--"TÔTO-ta + n0

3.3. Generalized Hirota-Satsuma coupled KdV system o/ time-fractional order

Let us consider generalized Hirota-Satsuma coupled KdV system of time-fractional order [18], which reads

Dfu — 1 uxxx + 3uux + 3(-2 + w)x,

Da —-1 xxx - 3ux, 0 < a 6 1,

Daw = -2 Wxxx - 3uwx.

We use the following transformations:

u(x, t)= 1 U2(n), (x, t) = -k + U(n), w(x, t)

= 2k2 - 2kU(n),

where n = x - C1+)ta.

Substituting Eq. (44) into Eq. (43), we can know that Eq. (43) are reduced into ordinary differential equations.

kUnn + 2U3 - 2k2U — 0, (45)

k(Un)2 + kUUnn + 3U4 - 4k2U2 + 6k4 + 2k2R — 0, (46)

where R is an integration constant.

Case A: Balancing Unn with U3 in Eq. (45) gives N = 1. Therefore, we may choose

U(n) — a0 + a1z(n). (47)

Substituting Eq. (47) along with Eq. (11) in Eq. (45) and setting all the coefficients of powers z to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain

where a and b are arbitrary constants.Assuming a >0 and choosing b <0. Therefore, using solution (12) of Eq. (11), ansatz (47) , we obtain the following exact solution of Eq. (45)

TJ( A = +aV1 I 2b exp\a(n + n0)]

(n) 4 V + 1 - b exp\a(n + &)]/

Then the exact solution to Eq. (43) can be written as

u(x, t) — - — 1 +

22 a2 a2

(x, t)—T ± J

1 2bexp [a(x + ^ta + ]

1 - bexp \a[x ' 2b exp

" 4C(1+a) '

4C(1+a)

4C(1+a)

a4_± a4/ 2b exp [a(* + jÇfcôta + n<>) ]

w(x, t)—- ± y | 1 +

1 - b exp ^a ^x

4C(1+a)

Substituting Eq. (47) along with Eq. (14) in Eq. (45) and setting all the coefficients of powers z to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain

A — —2a1, k — —a1, a0 — a^,

where a1 is arbitrary constant.Therefore, using solution (16) of Eq. (14), ansatz (47), we obtain the following exact solution of Eq. (45)

T(n) =-a?tanh(a1(n + n0)). (52)

Then, the exact solution to Eq. (43) can be written as

(44) u(x, t) — -a^tanh2

C(a + 1)

(x, t)= a1 ( 1 - tanh

C(a + 1)

w(x, t) — 2a{ 1 - tanh

a1 I x

C(a + 1)

Case B: Balancing UUnn with U4 in Eq. (46) gives N =1. Therefore, we may choose

U(n) — a0 + a1z(n).

Substituting Eq. (54) along with Eq. (11) in Eq. (46) and setting all the coefficients of powers z to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain

5 4 a2 ab a2

R — -12a , a0 — , a1 — ±_2~, k — ,

where a and b are arbitrary constants.

Assuming a > 0 and choosing b < 0. Therefore, using solution (12) of Eq. (11), ansatz (54) , we obtain the following exact solution of Eq. (46)

u(«) —1

2b exp[a(n + «0)] 4 V ' 1 — b exp[a(n + «0)]/

Then the exact solution to Eq. (43) can be written as

u(x, t) — — X 1 +-

2b exp[a(x + 4r(a1+a) ta + «0)]

1 — b exp [a(x + 4^ta

a?± a2/ 2b exp Kx+ird+jta+«0) ]

(x,± ^ +1 — b exp [a(.

y I__a

x + 4C(1+a) 1

w(x, t) ± y 1 +

«0)]-

2bexp [a(x + ^ f + «0)]

1 — b exp [a(x + ta + «0)],

Comparing our results with Lu's results [18] then it can be seen that some results are same.

4. Conclusion

The simplest equation method has been proposed and applied to obtain exact solutions of nonlinear fractional Klein-Gordon equation, Generalized Hirota-Satsuma coupled KdV system of time fractional order and nonlinear fractional Sharma-Tasso-Olever equation. The simplest equation method is effective in searching exact solutions of nonlinear time-fractional partial differential equations. The method proposed in this paper can also be extended to solve nonlinear time-fractional partial differential equations in mathematical physics.

Acknowledgments

The authors are highly grateful to the referees for their constructive comments.

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Nasir Taghizadeh is Professor of Pure Mathematics and Faculty of Mathematical sciences, guilan university, guilan, Iran. His research interests are theory of partial differential equations. He has published 58 papers.

[1] Biswas A. Topological 1-soliton solution of the nonlinear Schro-dinger's equation with Kerr law nonlinearity in 1 + 2 dimensions. Commun. Nonlinear Sci. Numer. Simulat. 2009;14:2845-7.

[2] Ebadi G, Biswas A. The G'/G method and topological soliton solution of the K(m, n) equation. Commun. Nonlinear Sci. Numer. Simulat. 2011;16:2377-82.

[3] Biswas A. 1-Soliton solution of the K(m, n) equation with generalized evolution. Phys. Lett. A 2008;372(25):4601-2.

[4] Biswas A. 1-Soliton solution of the K(m, n) equation with generalized evolution and time-dependent damping and dispersion. Comput. Math. Appl. 2010;59(8):2538-42.

M. Mirzazadeh is Assistant Professor of Pure Mathematics and Faculty of Mathematical sciences, guilan university, guilan, Iran. His research interests are theory of partial differential equations.

M. Rahimian is Assistant Professor of Pure Mathematics and Faculty of Mathematical sciences, Islamic Azad University, Masjed Soleiman Branch, Masjed Soleiman, Iran. His research interests include exact solution of nonlinear differential equations, Differential geometry.

M. Akbari is Assistant Professor of Pure Mathematics and Faculty of Mathematical sciences, guilan university, guilan, Iran. His research interests are theory of partial differential equations.