International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01
23
Using a complex transformation to get an exact solution for fractional generalized coupled MKDV and KDV equations Mohamed Saad 1, S. K. Elagan 2, 3, Y. S. Hamed 3, 4 and M. Sayed 3, 4
Abstract— In this paper, we use the fractional derivatives in the sense of Caputo to construct exact solution for fractal derivative generalized coupled mKdV and KdV equation Many 0 , 1 . D u a bu cu 2 x e D 3 u 0 ,
t
x
authors got to an approximate solution for fractal derivative generalized coupled mKdV and KdV equation. We apply a generalized fraction complex transformation [12, 13, 15] to convert this equation to ordinary differential equation. Finally, we obtain to a new exact solution for it by using a novel technique.
Index Term— Generalized Coupled mKdV and KdV equation, Caputo Derivative, Fractal Index. I. INT RODUCT ION The class of fraction calculus is one of the most convenient classes of fraction differential equation which viewed as generalized differential equation [10, 14]. In the sense that, much of the theory and, hence, applications of differential equation can be extended smoothly to fraction differential equation with the same flavor and spirit of the realm of differential equation fraction differential equation. The seeds of fractional calculus (that is, the theory of integrals and derivatives of any arbitrary real or complex order) were planted over 300 years ago. Since then, many researchers have contributed to this field. Recently, it has turned out that differential equations involving derivatives of non-integer
order can be adequate models for various physical phenomena
[1]. For example, the nonlinear oscillation of earthquakes can be modeled with fractional derivatives [2]. There has been some attempt to solve linear problems with multiple fractional derivatives (the so-called multi-term equations) [1, 3]. Not much work has been done on nonlinear problems and only a few numerical schemes have been proposed for solving nonlinear fractional differential equations. More recently, applications have included classes of nonlinear equation with multi-order fractional derivatives and this motivates us to develop a numerical scheme for their solution [4]. Numerical and analytical methods have included the Adomian decomposition method (ADM) [5, 6], the variational iteration method (VIM) [7], and the homotopy perturbation method [8, 9, 11]. In this project, we will use the fractional derivatives in the sense of Caputo to construct exact solution for fractal derivative generalized coupled mKdV and KdV equation Dt u a bu cu 2 x e Dx3 u 0 , 0 , 1 . Many
authors got to an approximate solutions for fractal derivative generalized coupled mKdV and KdV equation. We apply a generalized fraction complex transformation [12, 13, 15, 16, 17, 18, 19] to convert this equation to ordinary differential equation. Finally, we obtain to exact solution for it by using a novel technique. II. PRELIMINARIES AND NOT AT ION In this section, we give some basic definitions and properties of the fractional calculus theory which will be used further in this work. For more details see [1]. For the finite derivative in [a, b] , we define the following fractional integral and derivatives.
f ( x), x > 0 , is said to be in the space C , R , if there exists a real number ( p > ) p such that f ( x) = x f1 ( x) , where f1 ( x) C (0, ) , and Definition 1 A real function
1
2
3
4
Mohamed Saad is with Faculty of Applied Medical Science, Medical Records, T aif University, T aif, Kingdom of Saudi Arabia. (e-mail:
[email protected]). S. K. Elagan is with Mathematics Department, College of Science, Menofiya University, Shebin Elkom, Egypt. (e-mail:
[email protected]). S. K. Elagan, Y. S. Hamed and M. Sayed are with Department of Mathematics and Statistics, Faculty of Science, T aif University, Kingdom of Saudi Arabia. Y. S. Hamed and M. Sayed are with Department of Engineering Mathematics, Faculty of Electronic Engineering, Menofiya University, Menouf 32952, Egypt . (e-mail:
[email protected],
[email protected]).
it is said to be in the space
Cm if f m C , m N .
Definition 2 The Riemann-Liouville fractional integral operator of order 0 of a function f C , 1 , is defined as
J x =
x
1 1 x t f t d t , > 0, x > 0, J 0 x = f x . 0
135501-2828-IJBAS-IJENS @ February 2013 IJENS
IJENS
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01
J can be found in [1]; we mention
Properties of the operator only the following: For (1)
u u s . In [16] the authors shows t s t
u t
u s ' , where denotes the sigma s t
J J f ( x) = J
that
f ( x),
J J f ( x) = J J f ( x) , ( 1) (3) J x = x . ( 1) (2)
This shows that,
f C , 1, , 0 and > 1 :
x 1 m 1 f t dt , form 1 0 is defined as
chain rule
index, for more details, see [16].
The Riemann--Liouville derivative has certain disadvantages when trying to model real-world phenomena with fractional differential equations. Therefore, we shall introduce a modified fractional differential operator D proposed by M. Caputo in his work on the theory of viscoelasticity [1].
III.
24
u s mt m (1 ) m (1 )t m . s t
q U U U U = 0. Let r be an antiderivative of q . 1 Then r U U 2 = 0. 2 dU We can separate variables = d . Since r is a 2r U polynomial of degree 4 , we obtain x as an elliptic integral of U . The inverse function of an elliptic integral is an elliptic function. So U ( x) would become an elliptic function of x . In special cases the analysis can simplify, of course. For example, 2 if p U = 2 6U
135501-2828-IJBAS-IJENS @ February 2013 IJENS
IJENS
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01
2
we can take r U = 1 1 U 2
2
. Then
gives arctan U = C or
dU =d 1U 2
Kx Lt Mx U = tan C = tan C . 1 1 1 where
C
is a constant.
V. CONCLUSIONS In this paper we give a counter example to show that the chain rule in [12-15] is not correct. The fractional derivatives in the sense of Caputo, is used to construct a new exact solutions for fractional generalized coupled MKDV and KDV, we apply complex transformation using a modefied chain rule to convert fractional generalized coupled MKDV and KDV an ordinary differential equation and then we obtain to a new exact solution. In the future, One can study the fractional generalized coupled MKDV and KDV equation 3 2 Dt u a t b t u c t u x e t Dx u 0,0 , 1 as a
generalization
to
our
problem, i.e.
we
a t , b t , c t and e t all are functions of t .
can
25
[12] Z. B. Li, J. H. He, Fractional complex transformation for fractional differential equations, Math. Comput. Appl., 2010, vol. 15, pp. 970-973. [13] Z. B. Li, J. H. He, Application of the fractional complex transform to fractional differential equations, Nonlinear Sci. Lett. A, 2011, vol. 2, no. 3, pp. 121-126. [14] A. Rafiq, M. Ahmed, S. Hussain, A general approach to specific second order ordinary differential equations using homotopy perturbation method, Phys. Lett. A, 2008, vol. 372, pp. 49734976. [15] Z. B. Li, J. H. He, Fractional complex transform for fractional differential equations, Math. and Comput. Appl., 2010, vol. 15, no. 5, pp. 970-973. [16] J. H. He, S. K. Elagan, Z. B. Li, Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A, 2012, vol. 376, pp. 257–259. [17] R. W. Ibrahim, Complex transforms for systems of fractional differential equations, Abstra. and Appl. Analys., 2012, vol. 2012, Article ID 814759, 15 pages. [18] R. W. Ibrahim, Fractional complex transforms for fractional differential equations, Advan. in Diff. Equat., 2012, vol. 2012:192 doi:10.1186/1687-1847-2012-192. [19] R. W. Ibrahim, Numerical solution for complex systems of fractional order, J. of Appl. Math., 2012, vol. 2012, Article ID 678174, 11 pages.
take
REFERENCES [1] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [2] J. H. He, Some applications of nonlinear fractional differential equations and their applications, Bull. Sci. Technol., 1999, vol. 15, no. 2, pp. 86-90. [3] K. Diethelm, Y. Luchko, Numerical solution of linear multiterm differential equations of fractional order, J. Comput. Anal. Appl., 2004, vol. 6, pp. 243-263. [4] V. S. Erturk, Sh. Momani, Z. Odibat, Application of generalized differential transform method to multi-order fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 2008, vol. 13, pp. 1642-1654. [5] V. Daftardar-Gejji, S. Bhalekar, Solving multi-term linear and non-linear diffusion wave equations of fractional order by Adomian decomposition method, Appl. Math. Comput., 2008, vol. 202, pp. 113-120. [6] V. Daftardar-Gejji, H. Jafari, Solving a multi-order fractional differential equation using Adomian decomposition, Appl. Math. Comput., 2007, vol. 189, pp. 541-548. [7] N. H. Sweilam, M. M. Khader, R. F. Al-Bar, Numerical studies for a multi-order fractional differential equation, Phys. Lett. A, 2007, vol. 371, pp. 26-33. [8] A. Golbabai, K. Sayevand, Fractional calculus- A new approach to the analysis of generalized-fourth-order-diffusion wave equations, Comput. Math. Appl., 2011, vol. 61, no. 8, pp. 2227-2231. [9] A. Golbabai, K. Sayevand, T he homotopy perturbation method for multi-order time fractional differential equations, Nonlinear Sci. Lett. A, 2010, vol. 1, pp. 147-154. [10] A. G. Nikitin, T . A. Barannyk, Solitary waves and other solutions for nonlinear heat equations, Cent. Eur. J. Math., 2005, vol. 2, pp. 840-858. [11] J. H. He, New interpretation of homotopy perturbation method, Int. J. Modern Phys. B, 2006, vol. 20, pp. 1-7.
135501-2828-IJBAS-IJENS @ February 2013 IJENS
IJENS
