CC BY-NC-ND
0Journal of the Association of Arab Universities for Basic and Applied Sciences (2017) xxx, xxx-xxx
University of Bahrain
Journal of the Association of Arab Universities for Basic and Applied Sciences
www.elsevier.com/locate/jaaubas www.sciencedirect.com
The Exp-function method for solving nonlinear space-time fractional differential equations in mathematical physics
Ozkan Guner a'*, Ahmet Bekirb
a Cankiri Karatekin University, Faculty of Economics and Administrative Sciences, Department of International Trade, Cankiri, Turkey
b Eskisehir Osmangazi University, Art-Science Faculty, Department of Mathematics-Computer, Eskisehir, Turkey Received 3 October 2016; revised 21 November 2016; accepted 9 December 2016
KEYWORDS
Exact solution; Space-time fractional Telegraph equation; Space-time fractional KPP equation;
Exp-function method
Abstract Using the Exp-function method, we derive exact solutions of the nonlinear space-time fractional Telegraph equation and space-time fractional KPP equation. As a result, we obtain many exact analytical solutions including hyperbolic function. The fractional derivative is described in Jumarie's modified Riemann-Liouville sense. This method is very effective and convenient for solving nonlinear fractional differential equations.
© 2016 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Fractional calculus is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders. Fractional differential equations (FDEs) are used in various fields of physics, mathematics and engineering. So, they have gained much attention many mathematicians. Applications of FDEs for instance; signal processing, control theory, viscoelastic materials, systems identification, biomedical sciences, biology, finance, fractional dynamics as well as other disciplines (Miller and Ross, 1993;Podlubny, 1999; Kilbas et al., 2006).
In the last two decades, a lot of attention is paid to finding appropriate solutions of FDEs. There are many effective methods to obtain numerical and analytical solutions of these kinds
* Corresponding author.
E-mail addresses: ozkanguner@karatekin.edu.tr (O. Guner), abekir@ogu.edu.tr (A. Bekir).
Peer review under responsibility of University of Bahrain.
of FDEs, such as homotopy analysis method, adomian decomposition method, variational iteration method, homotopy perturbation method (Daftardar-Gejji and Bhalekar, 2008; Erturk et al., 2008; Gepreel and Mohamed, 2013; Sweilam et al., 2007; Gepreel, 2011; Mohamed et al., 2012) and the tanh method, the Exp-function method, the first integral method, the functional variable method, the (G//G)-expansion method, the sub-equation method, the modified trial equation method, the simplest equation method, the generalized Kudryashov method, the ansatz method and so on (Naher et al., 2013; Zheng, 2012, 2013; Bekir and Guner, 2013, 2014; Bekir et al., 2015a,b,c; Bekir et al., 2016; Bulut et al., 2013; Yan and Xu, 2015; Zhang et al., 2010; Guner, 2015; Guner and Bekir, 2016; Guner et al., 2015a,b; Taghizadeh et al., 2013; Khan and Akbar, 2013, 2014; Lu, 2012; Eslami et al., 2014; Zhang and Zhang, 2011; Liu and Chen, 2013).
The Exp-function method was first proposed by He and Wu in 2006 (He, 2006) and systematically studied in He and Abdou (2007), Noor et al. (2008), Ebaid (2012) and
http://dx.doi.org/10.1016/jjaubas.2016.12.002
1815-3852 © 2016 University of Bahrain. Publishing services by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Navickas et al. (2010). The Exp-function method was originally proposed to solve PDEs. However, its application to fractional calculus is rare and primary. In 2013, this method was successfully extended to fractional calculus, and it becomes an effective tool for fractional differential equations (see He, 2013; Guner and Bekir, 2017).
There are different kinds of fractional derivative operators. The most famous one is the Caputo definition that the function should be differentiable (Caputo, 1967). Recently, Jumarie derived definitions for the fractional derivative called modified Riemann-Liouville, which are suitable for continuous and non-differentiable functions. This paper adopts the Jumarieis derivative, which has been disadvantages, and has already been updated by some new ones, for example Atangana-Baleanu derivative and He derivative (Atangana and Koca, 2016;Atangana and Baleanu, 2016; Sayevand and Pichaghchi, 2016; He, 2011). The order a of Jumarie's derivative is defined by Jumarie (2006, 2009):
ifnb dWJW (w - nr/(n)-/(0M, 0 < a < 1
.(/V))
e 6 a < e+1, e p 1.
where f: R ! R, w ! f(w) denotes a continuous (but not necessarily differentiable) function and satisfying above properties
DWwe ==f!±C) w>
w f(1 + y - a) DW(C/(w)) = CD/w),
y > 0,
Df/w + bg(w)} - aD/w) + bDWg(w),
Da, C = 0,
where a, b and C are constant.
The outline of the present paper is as follows. In Section 2, we have a brief description of the Exp-function method for solving fractional partial differential equations. In Section 3, we apply the Exp-function method on the nonlinear space-time fractional Telegraph equation and the nonlinear fractional Kolmogorov Petrovskii-Piskunov (KPP) equation respectively. Finally, some conclusions are given in Section 4.
2. Description of the Exp-function method for solving fractional partial differential equations
We take into consideration the following general nonlinear FDE of the type
F(u, Dltu, Dßu, DjlM, DtDxu, Dfu,...
ß 6 1
where F is a polynomial of u, u is an unknown function and Da partial fractional derivative of u, in which the highest order derivatives and the nonlinear terms are involved.
The fractional complex transform was proposed by He and Li to convert a fractal space (time) to its continuous partner, the physical basis was illustrated by He in his recent review articles (Li and He, 2010;He et al., 2012; Hu and He, 2016). Wu and Liang gave a very clear mathematic analysis of relationship between fractal dimensions and fractional calculus in Wu and Liang (2017).
Using the fractional complex transform
u(x, y) - U(n),
f(1 + ß) f(1 + a) '
where k — 0 and c — 0 are constants. The fractional chain rule is defined as
Dlu - rtf Dln DXu - rxfDXn
where and are named as sigma indexes (Aksoy et al., 2016). We can choose = = L, where L is a constant.
When we substitute, (2.2) with (1.2) and (2.4) into (2.1), we can rewrite Eq. (2.1) in the following NODE;
Q(U, U, U", U",...)- 0.
where U(n) is the nth derivative of U with respect to
Suppose that the solution of NODE (2.5) can be expressed in the following form
U(n) -
Hn—can exp Kl ELb» exP
where p, q, c and d are positive integers which are known to be further determined, an and bm are unknown constants. To determine the value of c and p, we balance the linear term of the lowest order of equation Eq. (2.5) with the lowest order nonlinear term. Similarly, the value of d and q can be determined.
3. Applications of the proposed method Example 1:
We consider the nonlinear space-time fractional Telegraph equation (Jafari et al., 2014)
Djtlu - DXXXu + Dy + yu + ßu3 - 0,
a is a parameter describing the order of the fractional space and time derivative. When a — 1, Eq. (3.1) is called the nonlinear Telegraph equation. Wang and Li, presented to construct the traveling wave solutions of this equation (Wang and Li, 2008). Also, Jafari et al. applied the sub equation method Eq. (3.1) and obtained the many exact solutions. Zayed and Al-Nowehy solved the equation by the generalized Kudrya-shov method and obtained the exact solutions of nonlinear PDEs in mathematical physics (Zayed and Al-Nowehy, 2016).
Using the fractional complex transform,
u(x, t) - U(n),
f(1 + a) f(1 + a) '
where k and c are constants. When we substitute (3.3) with (1.2) and (2.4) into (3.1), this Eq. (3.1) can reduced to an ODE
(c2 - k2)U" - cU + yU + ßU3 - 0, where U -
- 0, 0 < a,
Balancing the order of U" and U3 in Eq. (3.4), we obtain
U = ci exp[-(c + 3p)n] + ---C2exp[-4pn]H---- '
_ C3 exp[-3сП] +----
C4exp[-3pn] + •
where ct is determined coefficient only for simplicity. Balancing the lowest order of Exp-function in Eqs. (3.5) and (3.6) we obtain
-(c + 3p) —-(3c + p), (3.7)
which leads to the result
p — c. (3.8)
In the same way also we have 3q + d — 4d, (3.9)
and this gives
q — d. (3.10)
If we set p — c — 1 and q — d — 1, Eq. (2.5) reduces to
_ ai exp(n)+ao+a-i exp(-£)
= bi exp(n)+bo+b-i exp(-n) '
(3.11)
When we substitute Eq. (3.11) into Eq. (3.1), by the help of Maple, we get the following results, Case 1:
ai = ±bi ^J-b, ao = 0 , a-i = 0 ,
bi = bi, bo = 0 , b-i = b-i,
(3.i2)
c = 3c
where b1 is free parameter. When substitute these results into (3.11), we get the following exact solution
ui ,2 (x ,t) = -
±bH -bexp
±V 9y2-2yx"-3yt" 4Г(1+")
I I ±V9y2-2yx"-3yt" \ , , II ±л/9y2-2yx"-3yt"
biexp| 4r(i+a) I + b-1exp( V
4Г(1+")
(3.i3)
If we take b_ — 1 and b — 1 Eq. (3.13) becomes
i^/y cosh
ui ,2 (x ,t)=±-
where i2 = -1. Case 2:
±V 9y2-2 yx"-3y t" 4Г(1+о)
+ г^/y sinh
±v 9y2-2yx"-3yt" 4Г(1+")
2^/b cosh
±V 9 y2-2 yx"-3yt" 4Г(1+")
(3.14)
ai = 0 , ao = ±bo\J-b a-i = 0 ,
bi = 0 , bo = bo , b-i = b-i,
c = Щ, k= i^P.
(3.15)
where b0 is free parameter. Substituting these results into (3.11), we get
U3,4(x , t) =
, , , , ±J9y2-2yx"-3yt" b0 + b-1exp( -| V 2Г(1+а)-
(3.16)
If we take b_1 — 1 and b0 — 1 Eq. (3.16) becomes
U3 ,4(x ,t)=±-
Vb + Vb cosh
±\ 9y2 2yx" 3y t" 2Г(1+")
- Vb sinh
±\/ 9y2 2yx" 3y t" 2Г(1+")
(3.17)
where i2 = -1. Case 3:
a-1 = ±b_h -
a1 — 0 , a0 — 0 , b1 — 0 , bo — bo , b_1 — b_1,
С — _32, k — .
where b_1 is free parameter. Similarly, we obtain
(3.18)
U5 ,6(x , t) = -
±b-H - b exp
_ / 9y2-2yx"+3yt" 2Г(1+")
bo + b-i exp
9y2-2 y x"+3 yt" 2Г(1+")
(3.19)
Eq. (3.19) becomes, if we take b_1 — 1 and b0 — 1
( ±д/ 9y2_2y xK+3y
v ccosh ^ —2r(îTa^— I _ V csinh I
«5,6 (x — ±-
2Г(1+")
Vb + Vb cosh
±V 9y2-2yx"+3yt" 2Г(1+")
- л/ß sinh
2Г(1+")
(3.20)
where i2 = -1. Case 4:
ai = ±bi ^J-b ao = 0 , a-i = 0 ,
b-1 = 0 ,
(3.21)
■ = 3i
I x/9y2-2y
where b1 is free parameter. In the same way we get
U7 ,8 (x, t) =■
±Ьи/-b exp^^9^^ bo + biexp( ffi^
(3.22)
Eq. (3.22) becomes, when we take b1 — 1 and b0 — 1
U7,8 (x,t)=±-
• ffi и l ±V9y2-2yx"-3yt"\ . _ . , / ±л/9y2-2yx"-3yt" ^ COsM V 2Г(1+")- + sinM V 2Г(1+")-
Vb ^Vb cosJ sin^ W 9y2-2yx"-3yt"
2Г(1+")
(3.23)
where i2 = -1. Case 5:
a1 = 0 , a0 = 0 , bi = bi, bo = 0 ,
a-1 = ±b_
(3.24)
, _ , x/9y2-2y ' ± 4 .
where b-1 is free parameter. These results give
U9,1o(x ,t)=-
±b-H - bexn -
^v 9y2-2yx"+3yt" 4Г(1+")
, I ±л/9y2-2yx"+3yt" 1 ,
bi exH V 4Г(1+»)-) + b-i exp
±\/9y2-2yx"+3yt" ) 4Г(1+") I
(3.25)
bi = bi,
^ 1 = b-1,
If we take bi — 1 and b_i — 1 Eq. (3.25) becomes
U9,10(x, t)— ±
• r- u I ±V9r2-2rx"+3rlM ■ ffi ■ u I ±V9y2—2yx"+3yf ^ cosM 4r(1+a)- - ^ slnM 4r(1+a)-
2ps cosj
(3.26)
where i2 — — 1. Case 6:
a1 — 0, ao — f, a—1 — 0,
b1 = bx, bo — bo, b—1 — 0,
c = k = ±pp.
(3.27)
where b0 is free parameter. Substituting these results into (3.11), we have
u11,12 0 — ■
b0 + b1 exp
:\/9y2—2yxa+3yta 2r(1+a)
(3.28)
If we take b_1 — 1 and b0 — 1 Eq. (3.28) becomes
"3,4(x, t) — ±"
Vb + cosh
2r(1+o)
+ sinh
2r(1+a)
(3.29)
where i2 — —1. Case 7:
a1 — a1 ;
a0 — a0 ;
a 1 b 1
b1 — T'aiy^, b0 — , b—1 — b—1,
k = k.
(3.30)
c — c,
where a1; a0 and b1 are free parameters. Substituting these results into (3.11) we get
M13,14(xi 0 —
a1 exp (t(Ηf ) + a0 ± ib—1 ^ exp (—()
y exp (w—f
± i'a^f + b—xexp (Cgf
(3.31)
where i2 — _1 and from ((3.14)), ((3.17)), ((3.24)), ((3.23)), ((3.25)), ((3.31)) it is possible to see that the solution will exist provided y — 0 and y — 9. Example 2:
Nonlinear fractional Kolmogorov-Petrovskii-Piskunov (KPP) equation has the form (Zayed and Amer, 2015):
D?m — + iM + yu2 + du3 — 0,
(3.32)
where 0 < a 6 1 and 1, y, d are non zero constants. Nonlinear fractional KPP equation is important in the physical fields, and it includes the fractional Klein-Gordon equation, fractional Fitzhugh-Nagumo equation, fractional Newell-Whitehead equation, fractional Burgers equation, fractional Fisher equation, fractional Cahn-Allen equation, fractional Chaffee-Infanfe equation and fractional Huxlay equation. Zayed and Amer applied the improved Riccati equation method and found more general exact solutions which include the solitary wave solutions, the periodic solutions and the rational function solutions. Zayed et al. applied the fractional (D|?G/G)-expansion method Eq. (3.32) and obtained three types of solutions via the solitary, trigonometric and rational solutions (Zayed et al., 2015). Song and Wang have implemented the differential transform method for the approximate solution of this equation (Song and Wang, 2012). When a — 1 Eq. (3.32) is called the nonlinear KPP equation and it has been discussed in (Feng et al., 2011) using the (GV/G)-expansion method. In Zayed and Ibrahim (2014), Zayed and Ibrahim have obtained the solitary wave solutions via the modified simple equation method. Hariharan has obtained the analytical/numerical solutions of this equation with the Homotopy analysis method (HAM) for linear and nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) and fractional KPP equations (Hariharan, 2013).
Now, we solve Eq. (3.32) using the proposed method of Section 2. To reduce this equation to the following ODE with integer order, we use the expression (2.3):
—cLU' — k2L2 U" + iM + yM2 + du3 — 0,
(3.33)
Same approach in the previous example, we have p — c and q — d. Substituting Eq. (3.11) into Eq. (3.33), and collecting all the terms with the same power of ej, j — _3,..., 3 together and then equating each coefficient equal to zero, yields a set of algebraic equations. By solving this algebraic equations with help of symbolic computation, we get.
Case 1:
ax — 0,
a0 — 0, b0 — 0,
a—1 —-
/y2—4di±c
_6id+y2TY y2—4di , _
c — 8Ld , k — T
y/d(y2—2di+yp y2 —4ffi)
(3.34)
where a_1 and b1 are free parameters. By substituting these results into (3.11), we get
b 1 b 1
— Vy2—4di±y b 2d b —1
exp — T
2di+^y2— 4di) f (6id+y2TPffi^)
4Ldr(1+a)
8Ldr(1+a)
Ml,2(x, t) —-
bi exp t
^/d(y2 — 2d1+yp2- 4dl) ta (6id+y2T^ffi-iffi)f
4Ldr(1+a)
8Ldr(1+a)
(3.35)
+ b—1 exp
y^^y2 — 2di + y pffi2 — 44 <5^ta f6id + y2 TVffiffi)^ ' 4Ldr(1 + a) 8Ldr(1 + a)
If we take b1 — 2 , b1 — 1 and \Jy2 — 4di — —2d ± y Eq. (3.35) becomes
U1,2(x,t) = 2 - 1 tanh I t
^/d(c2 - 2di + cV-2d ± y) ta
4dLr(1 + a)
(6id + c2 T V-± y)ta 8dLr(1 + a)
(3.36)
Case 2:
at = 0 , a0 = a0 ,
, dèû(2daû+cèû+è^c2 4di)
b =--V;-y-■/ , 00 = 00 ,
b-1 l^2 c2 4dlJ
A/c2 4di±c , a 1 =-2d-b 1>
b 1 = b
6id+y2T\/C2 4di
k = T-
d (c2 4di^
(3.37)
where a0 , b0 and b_1 are free parameters. Substituting these results into (3.11), we get
in (3.14), (3.17), (3.20), (3.23), (3.26), (3.31), (3.36) and (3.39)
are new exact solutions to the these equations.
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4. Conclusions
In this paper, we have used the fractional complex transform with the Exp-function method in order to reduce nonlinear FDEs to NODEs for finding exact analytic solutions for the nonlinear space-time fractional Telegraph equation and the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equation. As a result, some new exact solutions for them have been successfully found. These results are going to be very useful, very effective in different areas of applied mathematics, so this method can be applied to other nonlinear FDEs in the mathematical physics. We note that the exact solutions established
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