C om mun.Fac.Sci.U niv.A nk.Series A 1 Volum e 67, N umb er 1, Pages 60–67 (2018) D O I: 10.1501/C om mua1_ 0000000830 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
APPLICATION OF THE (G0=G)-EXPANSION METHOD FOR
SOME SPACE-TIME FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS
MINE AYLIN BAYRAK
Abstract. In this paper, the (G0=G)-expansion method is presented for
…nd-ing the exact solutions of the space-time fractional travel…nd-ing wave solutions for the Joseph-Egri (TRLW) equation and Gardner equation. The fractional derivatives are described by modi…ed Riemann-Liouville sense. Many exact so-lutions are obtained by the hyperbolic functions, the trigonometric functions and the rational functions. This method is e¢ cient and powerful in perform-ing a solution to the fractional partial di¤erential equations. Also, the method reduces the large amount of calculations.
1. Introduction
In recent years, fractional partial di¤erential equations which are generalizations of classical partial di¤erential equations of integer order have been the focus of many studies [1, 2, 3]. Many powerful methods for obtaining the exact solutions of fractional partial di¤erential equations, such as the fractional the (G0=G)-expansion method [4, 5, 6, 7], the fractional …rst integral method [8, 9], the fractional exp-function method [10, 11, 12], the fractional exp-functional variable method [13] and the fractional sub-equation method [14, 15] have been developed to …nd exact analytic solutions.
In this paper, the (G0=G)-expansion method [16, 17] to solve nonlinear fractional di¤erential equations in the sense of modi…ed Riemann-Liouville derivative by Ju-marie is used [18]. The JuJu-marie’s modi…ed Riemann-Liouville derivative of order is de…ned by
Received by the editors: September 03, 2016; Accepted: April 12, 2017. 2010 Mathematics Subject Classi…cation. 35Q53; 35Q51.
Key words and phrases. Exact traveling wave solutions, (G0=G)-expansion method, space-time
fractional partial di¤erential equations,modi…ed Riemann-Liouville derivative.
c 2 0 1 8 A n ka ra U n ive rsity C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra . S é rie s A 1 . M a th e m a t ic s a n d S t a tis tic s .
A PPLIC AT IO N O F T H E (G=G)-EX PA N SIO N M ET H O D 61 Dtf (t) = 8 > < > : 1 (1 ) d dt t R 0 (t ) (f ( ) f (0))d ; 0 < < 1 (f(n)(t)) n ; n < n + 1; n 1 (1)
Some important properties of the fractional modi…ed Riemann-Liouville derivative were given [19] as
Dtx = (1 + )
(1 + )x ; > 0 (2)
Dx(u(x)v(x)) = v(x)Dxu(x) + u(x)Dxv(x) (3)
Dx[f (u(x))] = fu0(u)Dxu(x) (4)
Dx[f (u(x))] = Duf (u)(u0 (5)
Consider the following general fractional partial di¤erential equations P (u; Dtu; Dxu; Dt2 u; DtDxu; D2x u; :::) = 0
0 < ; < 1 (6)
where u = u(x; t) is an unknown function, and P is a polynomial of u = u(x; t) and its partial fractional derivatives, in which the highest order derivatives and the nonlinear terms are involved.
Li and He [20, 21] proposed a fractional complex transform to convert fractional di¤erential equations into ordinary di¤erential equations, so all analytical methods which are devoted to the advanced calculus can be easily applied to the fractional calculus. By using traveling wave variable
u(x; t) = U ( ) (7)
= cx
(1 + )
kx
(1 + ) (8)
where k and c are nonzero arbitrary constants, and Eq. (6) can be written as follows:
Q(U; U0; U00; U000; :::) = 0: (9)
where the prime denotes the derivation with respect to . If the possibility has, then Eq.(9) can be integrated term by term one or more times.
Suppose that the solution of Eq.(9) can be expressed by a polynomial in (G0=G) in
the form: U ( ) = m X i=0 ai G0 G i ; am6= 0 (10)
where ai(i = 0; 1; 2; :::; m) are constants, while G( ) satis…es the following
second-order linear ordinary di¤erential equation
G00( ) + G0( ) + G( ) = 0 (11)
The positive integer m can be found by balancing the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in Eq.(9). Substituting Eq.(10) into Eq.(9) and using Eq.(11) and equating each coe¢ cient of the resulting polynomial to zero, a set of algebraic equations for ai(i = 0; 1; 2; :::; m);
; ; k and c is obtained.
Solving the equation system, and substituting ai(i = 0; 1; 2; :::; m); ; ; k; c and
the general solutions of Eq.(11) into Eq.(10), a variety of exact solutions of Eq.(6) can be obtained.
2. The space-time fractional Joseph-Egri(TRLW) equation Consider the following space-time fractional Joseph-Egri (TRLW) equation [22]
Dtu + Dxu + uDxu + DxDt2 u = 0; t > 0;
0 < ; 1; x > 0 (12)
where is a constant.
Substituting Eqs.(7)-(8) into Eq.(12), the following ordinary di¤erential equation can be obtained
(c k)U0+ cU U02U000= 0 (13)
where U0 = dUd . By once integrating and setting the constants of integration to zero, (c k)U + cU 2 2 + ck 2U00= 0 (14) is obtained.
For the linear term of highest order U00 with the highest order nonlinear term
U2, balancing the two term in Eq. (14) gives
m + 2 = 2m (15)
so that
m = 2: (16)
Assuming that the solutions of Eq.(14) can be expressed by a polynomial in (G0=G) as U ( ) = a0+ a1 G0 G + a2 G0 G 2 ; a26= 0 (17)
By using Eq.(11), from Eq.(17), it is derived that
U00( ) = 2a2 2+ a1 + (6a2 + 2a1 + a1 2) G0 G +(8a2 + 3a1 + 4a2 2) G0 G 2 +(2a1+ 10a2 ) G0 G 3 + 6a2 G0 G 4 (18)
A PPLIC AT IO N O F T H E (G=G)-EX PA N SIO N M ET H O D 63 and U2( ) = a20+ 2a0a1 G0 G + (2a0a2+ a 2 1) G0 G 2 +2a1a2 G0 G 3 + a22 G 0 G 4 (19)
Substituting Eqs.(17)-(19) into Eq.(14), collecting the coe¢ cients of GG0 i (i = 0; 1; 2) and set it to zero, the following system is obtained:
(c k)a0+ 2ca 2 0+ 2ck2a2 2+ ck2a1 = 0; (c k)a1+ ca0a1+ 6ck2a2 + 2ck2a1 + ck2a1 2= 0; (c k)a2+ 2ca 2 1+ ca0a2+ 8ck2a2 +3ck2a1 + 4ck2a2 2= 0; ca1a2+ 2ck2a1+ 10ck2a2 = 0; 2ca2+ 6ck 2= 0 (20)
Solving this system gives
a1= 12 c2 p 2 c2+ 4 c2+ 1; a2= 12c2 p 2 c2+ 4 c2+ 1; a0= 2 2c2 4 c2 ; k = 2 c c2+ 4 c2+ 1; c = c (21)
where and , are arbitrary constants.
By using Eq.(21) expression Eq.(17) can be written as
U ( ) = 2 2c2 4 c2 12 c2 p 2 c2+ 4 c2+ 1 G0 G 12c2 p 2 c2+ 4 c2+ 1 G0 G 2 (22)
Substituting general solutions of Eq.(11) into Eq.(22) three types of traveling wave solutions of the space-time fractional Joseph-Egri(TRLW) equation are obtained as follows:
When 2 4 > 0 U1;2( ) = 2c2( 2+ 2 ) + 3c 2 2 q 1 c2( 2 4 ) 3c2( 2 4 ) q 1 c2( 2 4 ) K1sinh p 2 4 2 + K2cosh p 2 4 2 K1cosh p 2 4 2 + K2sinh p 2 4 2 ! (23) where = (1+ )cx 1 c2(c2 4 ) (1+ )t . When 2 4 < 0 U3;4( ) = 2c2( 2 + 2 ) + 3c 2 2 q 1 + c2(4 2) 3c2(4 2 ) q 1 + c2(4 2) K1sin p 4 2 2 + K2cos p 4 2 2 K1cos p 4 2 2 + K2sin p 4 2 2 ! (24) where = (1+ )cx 1 c2(c2 4 ) (1+ )t . When 2 4 = 0 U5;6( ) = 2c2( 2+ 2 ) 6c2 2 q 1 c2( 2 4 ) 12c2 q 1 c2( 2 4 ) K2 K1+ K2 (25) where = (1+ )cx 1 c2(c2 4 ) (1+ )t .
3. The space-time fractional Gardner equation Consider the following space-time fractional Gardner equation [23, 24]
Dtu = 6uDxu + 6"2u2Dxu + Dx3 u; t > 0;
0 < ; 1; x > 0 (26)
where " is a constant.
Substituting Eqs.(7)-(8) into Eq.(26) the ordinary di¤erential equation can be obtained as follows:
kU0 6cU U02cU2U03U000= 0 (27)
where U0 = dU
d . By once integrating and setting the constants of integration to
zero,
A PPLIC AT IO N O F T H E (G=G)-EX PA N SIO N M ET H O D 65
is obtained.
For the linear term of highest order U00 with the highest order nonlinear term
U3, balancing the two term in Eq. (28) gives
m + 2 = 3m (29)
so that
m = 1: (30)
Assuming that the solutions of Eq. (28) can be expressed by a polynomial in (G0=G) as
U ( ) = a0+ a1
G0
G ; a16= 0 (31)
By using Eq.(11), from Eq.(31), it is derived that U00( ) = a1 + (2a1 + a1 2) G0 G + 3a1 G0 G 2 + 2a1 G0 G 3 (32) and U2( ) = a20+ 2a0a1 G0 G + a 2 1 G0 G 2 (33) and U3( ) = a30+ 3a20a1 G0 G + 3a0a 2 1 G0 G 2 + a31 G0 G 3 (34) Substituting Eqs.(32)-(34) into Eq.(28), collecting the coe¢ cients of GG0
i
(i = 0; 1) and set it to zero, the following system is obtained:
ka0+ 3ca20+ 2"2ca30+ c3a1 + C0= 0;
ka1+ 6ca0a1+ 6"2ca20a1+ c3a1 2+ 2c3a1 = 0;
3ca21+ 6"2ca0a21+ 3c3a1 = 0;
2"2ca31+ 2c3a1= 0: (35)
Solving this system gives a1= ci "; a0= 1 c" i 2"2 ; k = c3 4"2( 2 4 ) + c 4"4; c = c; C0= c3 2( 2 4 ) + 3c 2"2 (36)
where and , are arbitrary constants.
By using Eq.(36) expression Eq.(31) can be written as U ( ) = 1 c" i 2"2 ci " G0 G (37)
Substituting general solutions of Eq.(11) into Eq.(37) three types of traveling wave solutions of the space-time fractional Gardner equation are obtained as follows:
U1;2( ) = 1 2"2 cip 2 4 2" K1sinh p 2 4 2 + K2cosh p 2 4 2 K1cosh p 2 4 2 + K2sinh p 2 4 2 ! (38) where = (1+ )cx [4"c32( 2 4 ) +4"c4] (1+ )t . When 2 4 < 0 U3;4( ) = 1 2"2 cip4 2 2" K1sin p 4 2 2 + K2cos p 4 2 2 K1cos p 4 2 2 + K2sin p 4 2 2 ! (39) where = (1+ )cx [4"c32( 2 4 ) +4"c4] (1+ )t . When 2 4 = 0 U5;6( ) = 1 c" i 2"2 ci " K2 K1+ K2 (40) where = (1+ )cx [4"c32( 2 4 ) + c 4"4] t (1+ ). 4. Conclusion
In this paper, three types of exact analytical solutions including the general-ized hyperbolic, trigonometric and rational function solutions for the space-time fractional Joseph-Egri(TRLW) and Gardner equation are presented by using the (G0=G)-expansion method. It can be concluded that this method is very simple, reliable and proposes a variety of exact solutions to space-time fractional partial di¤erential equation.
References
[1] Miller, K.S. and Ross, B., An introduction to the fractional calculus and fractional di¤erential equations, Wiley, New York,1993.
[2] Podlubny, I., Fractional Di¤erential Equations, Academic Press, California, 1999.
[3] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., Theory and Applictions of Fractional Dif-ferential Equations, Elsevier, Amsterdam, 2006.
[4] Wang, X.L., Li, X.Z. and Zhang, J.L., The (G0=G)-expansion method and traveling wave
solutions of nonlinear evolution equations in mathematical physics, Physics Letters A (2008),372,417-423.
[5] Zheng, B., (G0=G)-expansion method for solving fractional partial di¤erential equations in the
theory of mathematical physics, Communications in Theoretical Physics (2012),58,623-630. [6] Gepreel,K.A. and Omran,S., Exact solutions for nonlinear partial fractional di¤erential
A PPLIC AT IO N O F T H E (G=G)-EX PA N SIO N M ET H O D 67
[7] Shang, N. and Zheng, B., Exact solutions for three fractional partial di¤erential equations by
the (G0=G)method, International journal of Applied mathematics (2013),43, p114.
[8] Lu, B., The …rst integral method for some time fractional di¤erential equations, Journal of Mathematical Analysis and Applications (2012),395, 684-693.
[9] Eslami, M., Vajargah, B.F. , Mirzazadeh, M. and Biswas, A., Application of …rst integral method to fractional partial di¤erential equations, Indian Journal of Physics (2014),88, 177-184.
[10] Zhang, S., Zong, Q-A., Liu, D. and Gao, Q., A generalized exp-function method for fractional riccati di¤erential equations, Communications in Fractional Calculus (2010),1, 48-51. [11] Bekir, A., Güner, Ö. and Çevikel, A.C.,Fractional complex transform and exp-function
meth-ods for fractional di¤erential equations, Abstract and Applied Analysis (2013),2013, 426462. [12] Zhang, B., Exp-function method for solving fractional partial di¤erential equations, Scienti…c
World Journal (2013),2013, 465723.
[13] Liu, W. and Chen, K., The functional variable method for …nding exact solutions of some nonlinear time-fractional di¤erentional equations, Pramana-Journal of Physics (2013),81, 377-384.
[14] Zhang, S. and Zhang, H-Q., Fractional sub-equation method and its applications to nonlinear fractional PDEs, Physics Letters A (2011),375, 1069-1073.
[15] Alzaidy, J.F., Fractional sub-equation method and its applications to the space-time fractional di¤erential equations in mathematical physics, British Journal of Mathematics and Computer Science (2013),3, 153-163.
[16] Zhang, S, Tong, J.L. and Wang, W., A Generalized -Expansion Method for the mKdV Equa-tion with Variable Coe¢ cients, Physics Letters A (2008),372, 2254-2257.
[17] Zayed, E.M.E. and Gepreel, K.A., The (G0=G)-expansion method for …nding traveling wave
solutions of nonlinear partial di¤erential equations in mathematical physics, Journal of math-ematical Physics (2009),50, 013502.
[18] Jumarie, G., Fractional partial di¤erential equations and modi…ed Riemann-Liouville deriv-ative new methods for solution, Journal of Applied Mathematics and Computation (2007),4, 31-48.
[19] Jumarie, G., Table of some basic fractional calculus formulae derived from a modi…ed Riemann-Liouville derivative for nondi¤erentiable functions, Applied Mathematics Letters (2009),22, 378-385.
[20] Li, Z.B. and He, J., Fractional complex transform for fractional di¤erential equations, Math-ematical & Computational Applications, (2010),15, 970-973.
[21] Li, Z.B. and He, J., Application of the fractional complex transform to fractional di¤erential equations, Nonlinear Science Letter A (2011),2, 121-126.
[22] Heremant, W., Banerjeeg, P.P., Korpel, A., Assanto, G., Van Immerzeele, A. and Meerpoel, A., Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method, Journal of Physics A:Mathematical and General (1986),19,607-628.
[23] Liu, X., Tian, L. and Wu, Y., Application of (G0=G)-expansion method to two nonlinear
evolution equations, Applied Mathematics and Computation (2010),217, 1376-1384. [24] Tang, Y., Xu, W. and Shen, J., Solitary wave solutions to Gardner equation, Chinese journal
of Engineering Mathematics (2007),24, 119-127.
Current address : Mine Aylin BAYRAK: Kocaeli University, Art and Science Faculty, Depart-ment of Mathematics, 41380, Izmit,Kocaeli.
