Opt Quant Electron (2017) 49:64 DOI 10.1007/s11082-017-0895-9

Exact solutions for nonlinear fractional differential equations using exponential rational function method Syed Tauseef Mohyud-Din1 • Sadaf Bibi1

Received: 3 October 2016 / Accepted: 9 January 2017  Springer Science+Business Media New York 2017

Abstract Exponential rational function method is a relatively new mechanism to get exact solutions of nonlinear fractional differential equations. In this paper, fractional derivatives in the sense of Jumarie’s modified Riemann–Liouville are defined. Fractional complex transform is used to convert fractional differential equations into ordinary differential equations. This method is more appropriate for solving different kind of nonlinear fractional differential equations emerging in mathematical physics. Keywords Modified Riemann–Liouville derivative  Exponential rational function method  Exact solutions  Space–time fractional order Boussinesq equation  Space–time fractional order SRLW equation  (2 ? 1)-dimensional breaking soliton equations

1 Introduction Fractional differential equations (FDEs) of fractional order have acquired a lot of interest and attracted attention of many studies due to their frequent occurrence in biochemical, mathematics, viscoelasticity, economics and other areas of science. The fractional calculus (Miller and Ross 1993; Podlubny 1999) is a unique topic and attained importance during the modern past. A variety of many authentic methods have been suggested to get the exact solutions of fractional differential equations (FDEs), i.e. homotopy analysis method (Mohyud-Din et al. 2011; Yildirim and Mohyud-Din 2010; Mohyud-Din et al. 2012), differential transformation method (Merdan et al. 2012), homotopy perturbation method (Anil Sezer et al. 2011; Gepreel 2011), reduced differential transform method (Ahmad and Mohyud-Din 2014), variational iteration method (Merdan and Mohyud-Din 2011; Ghaneai et al. 2012), Adomian’s decomposition method (Ahmad and Mohyud-Din 2013), Exp-

& Syed Tauseef Mohyud-Din [email protected] 1

Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan

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function method (Ul Hassan and Mohyud-Din 2013, 2015; Zhang et al. 2010; Bekir et al. 2013), (G0 /G)-expansion method (Shakeel and Mohyud-Din 2015; Zheng 2012; Biswas et al. 2014; Bekir and Gu¨ner 2013), finite difference method (Cui 2009), fractional sub equation method (Bekir and Aksoy 2015; Bekir et al. 2014), modified simple equation method (Zayed 2011), improved tan (/2 )-expansion method (Aghdaei and Manafian 2016), Hirota method (Liu and Tian 2012), extended fan sub-equation method (Younis et al. 2016) and ansatz method (Gu¨ner et al. 2016; Bekir 2013). There are several approaches to the generalization of the concept of differentiation to fractional orders, e.g., Riemann–Liouville and Caputo but the first major contribution to give proper definition is due to Jumarie’s modified Riemann–Liouville derivative (Jumarie 2006). In this paper, we consider the space–time fractional order Boussinesq equation (Jafari et al. 2013) 2a 2a 2 4a D2a t u þ bDt u þ bDx ðu Þ þ cDx u ¼ 0; t [ 0; 0\a  1

ð1:1Þ

In the above equation the dispersion parameter c ¼ constant [ 0 is depending on the rigidity characteristics of the material and compression,uðx; tÞ is the vertical deflection,b ¼ constant is the coefficient controlling nonlinearity and the quadratic nonlinearity ðu2 Þxx accounts for the curvature of the bending beam. a is a parameter describing the order of the fractional time and space derivative. We next consider the (2 ? 1)-dimensional breaking soliton equations as follows (Wen and Zheng 2013) oa u o3a u oa v oa u þ a 2a a þ 4au a þ 4a a v ¼ 0; a ot ox y ox ox oa u oa v  ¼ 0: oya oxa

ð1:2Þ

Where 0\a  1: When a ¼ 1, Eq. (1.2) are called the (2 ? 1)-dimensional breaking soliton equations. We have also considered the space–time fractional order SRLW equation (Xu 2008; Alzaidy 2013) of the form:  a   2a  2a a a a 2a ð1:3Þ D2a t u þ Dx u þ uDt Dx u þ Dt uDx u þ Dt Dx u ¼ 0: Where t [ ; 0\a  1; Eq. (1.3) arises in several physical applications including ion sound waves in plasma. Exponential rational function method (Bekir and Kaplan 2016) is a new technique to construct the exact solutions of nonlinear fractional differential equation. Fractional complex transformation can be used to convert FDEs into integer-order differential equations (Ibrahim 2012). The rest of the paper is organized as follows. In Sect. 2, some properties and definition of modified Riemann–Liouville derivative are laid down. In Sect. 3, exponential rational function method is explained. In Sect. 4, we give implementation of the exponential rational function method. Some conclusions are given in last section.

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2 Modified Riemann–Liouville derivative and properties The Jumarie’s modified Riemann–Liouville derivative of order a is defined as (Jumarie 2006, 2009):  y 1 a Dy f ð yÞ ¼ r ðy  nÞa1 ½f ðnÞ  f ð0Þdn; a\0 Cð1  aÞ 0 1 d y ð2:1Þ r ðy  nÞa ½f ðnÞ  f ð0Þdn; 0\a\1 Cð1  aÞ dy 0 o ðf ðkÞ ð yÞÞðakÞ ; k  a\k þ 1; k  1 ; where f ð yÞ denotes a continuous f :R ! R; y ! f ð yÞ function. CðaÞ is defined as: k!ka k!a aða þ 1Þða þ 2Þ. . .:ða þ nÞ

CðaÞ ¼ lim

ð2:2Þ

Some characteristics of modified Riemann–Liouville derivative (Li and He 2010; He and Li 2012) are given below: Property 1 Day yc ¼

Cð1 þ cÞ ca y ;c[0 Cð1 þ c  aÞ

ð2:3Þ

Property 2 The Jumarie’s modified fractional differentiation is a linear operation: Day ðaf ðyÞ þ bgð yÞÞ ¼ aDay f ð yÞ þ bDay gð yÞ

ð2:4Þ

where a; b are constants. Property 3 Dat c ¼ 0; c ¼ constant

ð2:5Þ

Property 4 Modified multiplication rule of two functions is used to calculate the fractal index. h i ð2:6Þ Day ½f ð yÞgð yÞ ¼ ry gð yÞDay f ðyÞ þ f ð yÞDay gðyÞ

The left-hand side is   Day ½f ð yÞgð yÞ ¼ Day ybþc ¼

Cðb þ c þ 1Þ bþca : y Cðb þ c þ 1  aÞ

ð2:7Þ

The derivative of the right-hand side is

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h h i  i ry gð yÞDay f ðyÞ þ f ð yÞDay gðyÞ ¼ ry yb Day ðyc Þ þ yc Day yb 

 Cðc þ 1Þ Cðb þ 1Þ bþca þ y : ¼ ry Cðc þ 1  aÞ Cðb þ 1  aÞ Therefore, the fractal index ry is equal to   Cðb þ c þ 1Þ Cðb þ 1  aÞCðc þ 1  aÞ : ry ¼ Cðb þ c þ 1  aÞ Cðc þ 1  aÞCðb þ 1ÞCðb þ 1  aÞCðc þ 1Þ

ð2:8Þ

ð2:9Þ

These properties will be used in the subsequent problems.

3 Exponential rational function method Consider nonlinear FDE as   p u; Dax u; Dbx u; Dat Dat u; Dat Dbx u; Dbx Dbx u ¼ 0; 0\a; b\1:

ð3:1Þ

where p is a polynomial of u and its partial fractional derivatives and u is an unknown function. The important steps of this method for solving FDEs are explained (Yusufoglu and Bekir 2007): Step 1 Nonlinear fractional complex transformation is defined as: uðx; tÞ ¼ U ðnÞ; n¼

kxb cta þ : Cð1 þ bÞ Cð1 þ aÞ

ð3:2Þ

where c, k are nonzero arbitrary constants, using the above transformation Eq. (3.1) can be transformed to an ODE: QðU; U 0 ; U 00 ; U 000 ; . . .:Þ ¼ 0:

ð3:3Þ

In the above equation differentiation with respect to n is denoted by 0 . Step 2 The exact solution of Eq. (3.3) is defined as under: U ðnÞ ¼

N X

an n: ð 1 þ en Þ n¼0

ð3:4Þ

where constants an ðan 6¼ 0Þ to be determined later. Balancing number M can be obtained by balancing the highest order derivative term with the highest order nonlinear term in Eq. (3.3) Step 3 We substitute Eq. (3.4) in Eq. (3.3) and collect all the coefficients with the same power of ein ði ¼ 1; 2; . . .:4Þ, together, the left-hand side of ODE is converted into another polynomial in ein : A set of algebraic equations for an unknown parameters are obtained by setting the coefficients of polynomial ein ði ¼ 1; 2; . . .:4Þ to zero. Exact solution for nonlinear FDE is to be constructed in order to solve the system of equation.

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4 Applications In this section, nonlinear fractional differential equations are solved by using the above method.

4.1 The space–time fractional Boussinesq equation Consider the space–time fractional order Boussinesq equation (Jafari et al. 2013):   2a 2a 2 þ cD4a ð4:1Þ D2a t u þ bDt u þ bDx u x u ¼ 0; t [ 0; 0\a  1: Using the transformation uðx; tÞ ¼ U ðnÞ; n¼

kxa cta  : Cð1 þ aÞ Cð1 þ aÞ

ð4:2Þ

where c; k are nonzero constants, Eq. (4.1) can be transformed into an ODE  00 0000 c2 U 00 þ bk2 U 00 þ bk2 U 2 þck4 U ¼ 0;

ð4:3Þ

where U 0 ¼ dU dn . Integrating Eq. (4.3) twice with respect to n, we attain ðc2 þ bk2 ÞU þ bk2 U 2 þ ck4 U 00 ¼ 0:

ð4:4Þ

Balancing number M ¼ 2 can be obtained by balancing U 2 and U 00 in Eq. (4.4). The solution of Eq. (4.4) can be written as: a1 a2 þ : ð4:5Þ U ðnÞ ¼ a0 þ 2 n 1þe ð1 þ en Þ We substitute Eq. (4.5) in Eq. (4.4) to collect all the coefficient with the same power of ein ði ¼ 1; 2; . . .:4Þ, and then set each coefficient of ein to zero, algebraic equations are attained as follows: e4n : bk2 a20 þ bk2 a0 þ c2 a0 ¼ 0; e3n : ck4 a1 þ 4bk2 a20 þ 2bk2 a0 a1 þ 4bk2 a0 þ bk2 a1 þ 4c2 a0 þ c2 a1 ¼ 0; e2n : 4ck4 a2 þ 6bk2 a20 þ 6bk2 a0 a1 þ 2bk2 a0 a2 þ bk2 a21 þ 6bk2 a0 þ 3bk2 a1 þ bk2 a2 þ 6c2 a0 þ 3c2 a1 þ c2 a2 ¼ 0; en : ck4 a1  2ck4 a2 þ 4bk2 a20 þ 6bk2 a0 a1 þ 4bk2 a0 a2 þ 2bk2 a21 þ 2bk2 a1 a2 þ 4bk2 a0 þ 3bk2 a1 þ 2bk2 a2 þ 4c2 a0 þ 3c2 a1 þ 2c2 a2 ¼ 0; e0n : c2 a0 þ c2 a1 þ c2 a2 þ bk2 a0 þ bk2 a1 þ bk2 a2 þ bk2 a20 þ bk2 a21 þ bk2 a22 þ 2bk2 a0 a1 þ 2bk2 a0 a2 þ 2bk2 a1 a2 ¼ 0: The following results are obtained upon solving the above algebraic equations using Maple

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Case 1 a0 ¼ 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ck2 6ck2 6ck2 ; a1 ¼ ; a2 ¼  ; c ¼ ck2  bk: b b b

ð4:6Þ

Case 2 a0 ¼ 0; a1 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6ck2 6ck2 ; a2 ¼  ; c ¼ ck2  bk: b b

ð4:7Þ

Solutions of Eq. (4.1) are gained by using the above results in (4.5). U1 ðnÞ ¼  a

ck2 6ck2 6ck2  : þ b bð1 þ en Þ bð1 þ en Þ2

ð4:8Þ

a

ct where n ¼ Cðkx 1þaÞ  Cð1þaÞ :

U2 ðnÞ ¼ a

6ck2 6ck2  : bð1 þ en Þ bð1 þ en Þ2

ð4:9Þ

a

ct where n ¼ Cðkx 1þaÞ  Cð1þaÞ :Comparing our results with Ozkan Guner’s (2015) results. The solutions we obtained are new for the Eq. (4.1). The exact solutions for Eq. (4.8) with various values of a are shown in Fig. 1a–d)

4.2 The space–time (2 1 1)-dimensional breaking soliton equations Consider the (2 ? 1)-dimensional breaking soliton equations (Wen and Zheng 2013): oa u o3a u oa v oa u þ a þ 4au þ 4a v ¼ 0; ota ox2a ya oxa oxa oa u oa v  ¼ 0: oya oxa

ð4:10Þ

where t [ 0; 0\a  1 By the transformations uðx; tÞ ¼ U ðnÞ; vðx; tÞ ¼ V ðnÞ , n¼

kxa xya cta þ  : Cð1 þ aÞ Cð1 þ aÞ Cð1 þ aÞ

ð4:11Þ

where c; k and x are nonzero constants, using Eq. (4.11) with Eq. (2.3) into Eq. (4.10), Eq. (4.10) can be converted into an ODE

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Fig. 1 Bright soliton solutions for Eq. (4.8) with a ¼ 0:25ðaÞ,a ¼ 0:5ðbÞ; a ¼ 0:75ðcÞ and a ¼ 1:0ðdÞ respectively, when k ¼ 1; c ¼ 2; b ¼ 2; b ¼ 1:

cU 0 þ ak2 xU 000 þ 4akUV 0 þ 4akU 0 V ¼ 0; xU 0  kV 0 ¼ 0:

ð4:12Þ

0 dV Where U 0 ¼ dU dn and V ¼ dn : Integrating Eq. (4.12) once with respect to n, we attain

xU  kV ¼ 0:

ð4:13Þ

Substituting Eq. (4.13) into first equation of the system (4.12), we obtain cU 0 þ ak2 xU 000 þ 8axUU 0 ¼ 0:

ð4:14Þ

Integrating Eq. (4.14) once with respect to n, we obtain cU þ ak2 xU 00 þ 4axU 2 ¼ 0:

ð4:15Þ

Balancing number N ¼ 2 can be obtained by balancing U 2 and U 00 in Eq. (4.15). The solution of Eq. (4.12) can be written as:

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U ðnÞ ¼ a0 þ

a1 a2 þ : 2 n 1þe ð1 þ en Þ

ð4:16Þ

We substitute Eq. (4.16) in Eq. (4.15) to collect all the coefficients with the same power of ein ði ¼ 1; 2; . . .:4Þ; and then set each coefficient of ein to zero, algebraic equations are attained as follows: e4n : 4awa20  ca0 ¼ 0; e3n : ak2 wa1 þ 16awa20 þ 8awa0 a1  4ca0  ca1 ¼ 0; e2n :¼ 4ak2 wa2 þ 24awa20 þ 24awa0 a1 þ 8awa0 a2 þ 4awa21  6ca0  3ca1  ca2 ¼ 0; en :¼ ak2 wa1  2ak2 wa2 þ 16awa20 þ 24awa0 a1 þ 16awa0a2 þ 8awa21 þ 8awa1 a2  4ca0  3ca1  2ca2 ¼ 0; e0n : 8awa0 a1 þ 8awa0 a2 þ 8awa1 a2  ca0  ca1  ca2 þ 4awa20 þ 4awa21 þ 4awa22 ¼ 0: The following results are obtained upon solving the above algebraic equations using Maple Case 1 1 3 3 a0 ¼  k2 ; a1 ¼ k2 ; a2 ¼  k2 ; c ¼ ak2 x: 4 2 2

ð4:17Þ

3 3 a0 ¼ 0; a1 ¼ k2 ; a2 ¼  k2 ; c ¼ ak2 x: 2 2

ð4:18Þ

Case 2

Solutions of Eq. (4.10) are attained by using the above results in (4.19). 1 3 k2 3 k2  : U1 ðnÞ ¼  k2 þ n 4 21 þ e 2 ð1 þ en Þ2 a

a

2

ð4:19Þ

a

xy ak xt where n ¼ Cðkx 1þaÞ þ Cð1þaÞ þ Cð1þaÞ :

U2 ðnÞ ¼ a

a

2

3 k2 3 k2  : n 21 þ e 2 ð1 þ en Þ2

ð4:20Þ

a

xy ak xt where n ¼ Cðkx 1þaÞ þ Cð1þaÞ  Cð1þaÞ : The exact solutions for Eq. (4.19) with various values of a are shown in Fig. 2a–d.

4.3 The space–time fractional SRLW equation Consider the space–time fractional order SRLW equation (Xu 2008; Alzaidy 2013):

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Fig. 2 Bright soliton solutions for Eq. (4.19) with a ¼ 0:25ðaÞ,a ¼ 0:5ðbÞ; a ¼ 0:75ðcÞ and a ¼ 1:0ðdÞ respectively, when k ¼ 1; c ¼ 1; x ¼ 1; a ¼ 2; y ¼ 1

 a   2a  2a a a a 2a D2a t u þ Dx u þ uDt Dx u þ Dt uDx u þ Dt Dx u ¼ 0; t [ ; 0\a  1

ð4:21Þ

Using the transformation uðx; tÞ ¼ U ðnÞ; n ¼

kxa cta þ : Cð1 þ aÞ Cð1 þ aÞ

where c; k are nonzero constants, Eq. (4.21) can be converted into an ODE   2k2 c2 U 00 þ 2 k2 þ c2 U þ kcU 2 ¼ 0:

ð4:22Þ

ð4:23Þ

where U 0 ¼ dU dn : Balancing number N ¼ 2 can be obtained by balancing U 2 and U 00 in Eq. (4.23). The solution of Eq. (4.23) can be written as: a1 a2 þ : ð4:24Þ U ðnÞ ¼ a0 þ 1 þ en ð1 þ en Þ2

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We substitute Eq. (4.24) in Eq. (4.23) to collect all the coefficients with the same power of ein ði ¼ 1; 2; . . .:4Þ: and then set each coefficient of ein to zero, algebraic equations are attained as follows: e4n : kca20 þ 2c2 a0 þ 2k2 a0 ¼ 0; e3n : 2k2 c2 a1 þ 4kca20 þ 2kca0 a1 þ 8c2 a0 þ 8k2 a0 þ 2c2 a1 þ 2k2 a1 ¼ 0; e2n : 8k2 c2 a2 þ 6kca20 þ 6kca0 a1 þ 2kca0 a2 þ kca21 þ 12c2 a0 þ 12k2 a0 þ 6c2 a1 þ 6k2 a1 þ 2c2 a2 þ 2k2 a2 ¼ 0; en : 2k2 c2 a1  4k2 c2 a2 þ 4kca20 þ 6kca0 a1 þ 4kca0 a2 þ 2kca21 þ 2kca1 a2 þ 8c2 a0 þ 8k2 a0 þ 6c2 a1 þ 6k2 a1 þ 4c2 a2 þ 4k2 a2 ¼ 0; e0n : 2c2 a2 þ 2k2 a2 þ 2k2 a1 þ 2c2 a0 þ 2c2 a1 þ 2k2 a0 þ 2kca1 a2 þ kca21 þ 2kca0 a2 þ 2kca0 a1 þ kca22 þ kca20 ¼ 0: The following results are obtained upon solving the above algebraic equations using Maple Case 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 2 ; a2 ¼ 12k  2 ;c ¼  2 k: a0 ¼ 0; a1 ¼ 12k  2 k þ1 k þ1 k þ1 2

ð4:25Þ

Case 2 rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 1 2 2 a0 ¼ 2k ; a1 ¼ 12k ; a2 ¼ 12k ;c ¼ k: k2  1 k2  1 k2  1 k2  1 2

Solutions of Eq. (4.21) are obtained by using the above results in (4.24). qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi 12k2  k21þ1 12k2  k21þ1 U1 ðnÞ ¼ :  2 ð 1 þ en Þ ð1 þ en Þ While n ¼

kxa Cð1þaÞ

pffiffiffiffiffiffiffiffi ffi 1 þ



k2 þ1

kta

Cð1þaÞ

ð4:26Þ

ð4:27Þ

:

qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2 1 12k 12k 2 1 k 1 k2 1 : U2 ðnÞ ¼ 2k2 þ  2 k2  1 ð1 þ en Þ ð1 þ en Þ

ð4:28Þ

pffiffiffiffiffiffi 1 kta a k2 1 While n ¼ Cðkx þ 1þaÞ Cð1þaÞ : Comparing our results with Abdullah Sonmezoglu’s results (Sonmezoglu 2015). The solutions we obtained are new for the Eq. (4.21).

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5 Conclusions Exponential rational function method is used to attain the exact solutions of space–time fractional order Boussinesq, (2 ? 1)-dimensional breaking soliton and SRLW equations. It is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical, physical and engineering problems. The acquired solutions are not quite the same as the past ones. Symbolic computational symbol Maple 16 is used to compute exact solutions. We hope that exponential rational function method can be used to generate exact solutions of others nonlinear equations.

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