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Approximate solution to the time–space fractional cubic nonlinear Schrodinger equation
Applied Mathematical Modelling 36 (2012) 5678–5685
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Approximate solution to the time–space fractional cubic nonlinear Schrodinger equation Mohamed A.E. Herzallah a,b,⇑, Khaled A. Gepreel a,c a
Faculty of Science, Zagazig University, Zagazig, Egypt Faculty of Science, Majmaah University, Saudi Arabia c Faculty of Science, Taif University, Saudi Arabia b
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 13 June 2011 Received in revised form 27 December 2011 Accepted 4 January 2012 Available online 21 January 2012 Keywords: Adomian decomposition method Fractional calculus Cubic nonlinear fractional Schordinger equation
By introducing the fractional derivatives in the sense of Caputo, we use the adomian decomposition method to construct the approximate solutions for the cubic nonlinear fractional Schordinger equation with time and space fractional derivatives. The exact solution of the cubic nonlinear Schrodinger equation is given as a special case of our approximate solution. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equation. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction In recent years, there has been a great deal of interest in fractional differential equations. First there were almost no practical applications of fractional calculus, and it was considered by many as an abstract area containing only mathematical manipulations of little or no use. Nearly 30 yr ago, the paradigm began to shift from pure mathematical formulations to applications in various fields. During the last decade fractional calculus has been applied to almost every field of science, engineering and mathematics. Several fields of application of fractional differentiation and fractional integration are already well established, some others have just started. Many applications of fractional calculus can be found in turbulence and fluid dynamics, stochastic dynamical system, plasma physics and controlled thermonuclear fusion, nonlinear control theory, image processing, nonlinear biological systems, astrophysics [1–11]. Historical summaries of the developments of fractional calculus can be found in [1–3]. Various methods for obtaining exact solutions to nonlinear partial differential equations have been proposed such as the Bäcklund transformation method [12], the Hirota’s bilinear method [13], the inverse scattering transform method [14], the extended tanh method [15], the adomian pade approximation [16], the variational method [17], the variational iteration method [18], the various Lindstedt–Poincare methods [19], the adomian decomposition method [20,21], the F-expansion method [22], the exp-function method [23], the homotopy perturbation method [24,25] and so on. Consider the Schrodinger equation with a cubic nonlinearity
iut þ uxx þ 2juj2 u ¼ 0;
uðx; 0Þ ¼ f ðxÞ;
i¼
pffiffiffiffiffiffiffi 1:
⇑ Corresponding author at: Faculty of Science, Zagazig University, Zagazig, Egypt. E-mail addresses: [email protected] (M.A.E. Herzallah), [email protected] (K.A. Gepreel). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2012.01.012
ð1:1Þ
M.A.E. Herzallah, K.A. Gepreel / Applied Mathematical Modelling 36 (2012) 5678–5685
5679
Which occurs in various chapters of physics, including nonlinear optics, superconductivity and plasma physics. The cubic nonlinearity is the most common nonlinearity in applications. It arises as a simplified model for studying Bose–Einstein condensates, Kerr media in nonlinear optics freak waves in the ocean (see [26–31]). In [26] Kanth and Aruna studied (1.1) with applying the differential transform method to obtain its approximate solution. In this article, we give a new model of the time–space fractional cubic nonlinear Schrodinger equation of the form
it C Da0þ u þ x C Dbþ ðx C Dbþ uÞ þ 2juj2 u ¼ 0;
t > 0; 0 < a; b 6 1:
ð1:2Þ
This equation can be represented some dynamical system in quantum mechanics especially in the exact solution for nonlinear Schrodinger equation of motion. Also in fluid dynamic and nonlinear dynamical system. We use adomian decomposition method to calculate an approximate solution of the cubic nonlinear fractional Schordinger equation (1.2) which is a generalization of the given in [26]. We use the Caputo fractional derivative on the half axis Rþ (i.e. t 2 Rþ ) t C Da0þ for time and the Caputo fractional derivative on the whole axis R (i.e. x 2 R) x C Dbþ for space. For some special values of a and b, the exact solution of the nonlinear Schordinger with a cubic nonlinearity will obtained. 2. Preliminaries and notations We give some basic definitions and properties of the fractional calculus theory which are used further in this paper [2,3]. For the finite derivative [a, b], we define the following fractional integral and derivatives. Definition 2.1. If f ðtÞ 2 L1 ða; bÞ, the set of all integrable functions, and a > 0 then the Riemann–Liouville fractional integral of order a, denoted by Iaaþ is defined by
Iaaþ f ðtÞ ¼
1 CðaÞ
Z
t
ðt sÞa1 f ðsÞds:
a
Definition 2.2. For a > 0, the Caputo fractional derivative of order a, denoted by C Daaþ , is defined by C
1 Cðn aÞ
Daaþ f ðtÞ ¼
Z
t
ðt sÞna1 Dn f ðsÞds;
a
where n is such that n 1 < a < n and D ¼ dds. If a is an integer, then this derivative takes the ordinary derivative C
Daaþ ¼ Da ;
a ¼ 1; 2; 3; . . .
Finally the Caputo fractional derivative on the whole space R is defined by, Definition 2.3. For a > 0 the Caputo fractional derivative of order a on the whole space, denoted by C Daþ , is defined by C
Daþ f ðxÞ ¼
1 Cðn aÞ
Z
x
ðx nÞna1 Dn f ðnÞdn:
1
3. Main results Consider the nonlinear fractional Schrodinger equation
it C Da0þ u þ x C Dbþ ðx C Dbþ uÞ þ 2juj2 u ¼ 0;
t > 0;
0 < a;
b 6 1;
uðx; 0Þ ¼ f ðxÞ ¼ eix ;
ð3:1Þ
pffiffiffiffiffiffiffi where i ¼ 1. Operating on both sides on the system (3.1) by the operator t Ia0þ (the Riemann–Liouville fractional integral with parameter t) we get:
uðx; tÞ ¼ uðx; 0Þ þ it Ia0þ ½x C Dbþ ðx C Dbþ uÞ þ 2juj2 u; ¼ f ðxÞ þ it Ia0þ ½x C Dbþ ðx C Dbþ uÞ þ 2GðuÞ; where GðuÞ ¼ juj2 u and uðx; 0Þ ¼ f ðxÞ. According to the Adomain decomposition method [20,21], we assume that a series solution of the function uðx; tÞ is given by
uðx; tÞ ¼
1 X
un ðx; tÞ:
ð3:2Þ
n¼0
The nonlinear term GðuÞ can be decomposed into an infinite series of polynomials given by
Gðx; tÞ ¼
1 X n¼0
An ;
ð3:3Þ
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M.A.E. Herzallah, K.A. Gepreel / Applied Mathematical Modelling 36 (2012) 5678–5685
where the components un ðx; tÞ will be determined recursively while An ’s are the so called adomian polynomials of un ’s. Specific algorithms have been set in [20,21] for calculating adomian’s polynomials for nonlinear terms
( n " ! ! !#) n n n X X X 1 d k k k An ðu0 ; u1 ; u2 ; . . . ; un Þ ¼ k uk k uk k uk n! dkn k¼0 k¼0 k¼0
ð3:4Þ
:
k¼0
Consequently, we have
unþ1 ðx; tÞ ¼ it Ia0þ ½x C Dbþ ðx C Dbþ un Þ þ 2An ;
n P 0:
ð3:5Þ
Eq. (3.4) leads to get:
A0 ¼ u0 ju0 j2 ; A1 ¼ u20 u1 þ 2u1 ju0 j2 ; A2 ¼ u20 u2 þ u21 u0 þ 2u0 ju1 j2 þ 2u2 ju0 j2 ; A3 ¼ u20 u3 þ 2u0 u1 u2 þ 2u0 u2 u1 þ 2u1 u2 u0 þ 2u3 ju0 j2 þ u1 ju1 j2 ; A4 ¼ u20 u4 þ 2u0 u1 u3 þ 2u0 u3 u1 þ 2u1 u3 u0 þ u21 u2 þ u22 u0 þ 2u0 ju2 j2 þ 2u2 ju1 j2 þ 2u4 ju0 j2 ; and so on. Using the above recursive relationships in (3.5), with using (see [1,2]) a
t I 0þ
tb t bþa ¼ ; Cðb þ 1Þ Cðb þ a þ 1Þ
x
C
pb
Dbþ eix ¼ eiðxþ 2 Þ :
We construct the approximate solutions of uðx; tÞ as follows:
ta ; Cða þ 1Þ t 2a ; u2 ¼ c2 eix Cð2a þ 1Þ t 3a ; u3 ¼ c3 eix Cð3a þ 1Þ t 4a ; u4 ¼ c4 eix Cð4a þ 1Þ t 5a ; u5 ¼ c5 eix Cð5a þ 1Þ 6a t ; u6 ¼ c6 eix Cð6a þ 1Þ ...
u1 ¼ c1 eix
ð3:6Þ ð3:7Þ ð3:8Þ ð3:9Þ ð3:10Þ ð3:11Þ
and so on. After some calculation, we get:
c1 ¼ iðeipb þ 2Þ;
ð3:12Þ
c2 ¼ iðeipb c1 þ 2c1 þ 4c1 Þ; "
ð3:13Þ
c3 ¼ i eipb c2 þ 2 c2 þ 2c2 þ " c4 ¼ i eipb c3 þ 2 c3 þ 2c3 þ (
"
c5 ¼ i eipb c4 þ 2 c4 þ 2c4 þ
Cð2a þ 1Þ ð2jc1 j2 þ c21 Þ ðCða þ 1ÞÞ2
#!
ð3:14Þ
;
2Cð3a þ 1Þ Cð3a þ 1Þ c1 jc1 j2 ðc c þ c1 c2 þ c2 c1 Þ þ Cð2a þ 1ÞðCða þ 1ÞÞ 1 2 ðCða þ 1ÞÞ3
#! ;
ð3:15Þ
Cð4a þ 1Þ Cð4a þ 1Þ 2 ðc2 þ 2jc2 j2 Þ ð2c c þ 2c1 c3 þ 2c3 c1 Þ þ 2 Cð3a þ 1ÞðCða þ 1ÞÞ 1 3 C ð2a þ 1Þ #)
Cð4a þ 1Þ ðc21 c2 þ 2c2 jc1 j2 Þ þ 2 C ða þ 1ÞCð2a þ 1Þ Cð5a þ 1Þ ð2c c þ 2c1 c4 þ 2c4 c1 Þ c6 ¼ i eipb c5 þ 2 c5 þ 2c5 þ Cð4a þ 1ÞðCða þ 1ÞÞ 1 4 Cð5a þ 1Þ Cð5a þ 1Þ ðc21 c3 þ 2jc1 j2 c3 Þ þ ð2c2 c3 þ 2c2 c3 þ 2c3 c2 Þ þ ðCð3a þ 1ÞÞCð2a þ 1Þ Cð3a þ 1ÞðCða þ 1ÞÞ2 #) Cð5a þ 1Þ 2 2 c þ 2c jc j Þ ðc þ 1 2 2 1 ðCð2a þ 1ÞÞ2 Cða þ 1Þ ... and so on.
ð3:16Þ
ð3:17Þ
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M.A.E. Herzallah, K.A. Gepreel / Applied Mathematical Modelling 36 (2012) 5678–5685
Table 1 Shown that the absolute value of approximate solution (3.20) and the absolute value of the exact solution (3.21) when t ¼ 0:1 and 0 < x < 10. Table 1 leads to get the proposed algorithm produced a rapidly convergent series. x
absðuapp Þ
absðuex Þ
0.1
0.9999999931
1.000000000
Errorðjabsðuapp Þ absðuex ÞjÞ 0:69 108
0.5
0.9999999931
1.000000000
0:69 108
1
0.9999999934
1.000000000
0:66 108
1.5
0.9999999933
1.000000000
0:67 108
3
0.9999999934
1.000000000
0:66 108
5
0.9999999934
1.000000000
0:66 108
7
0.9999999933
1.000000000
0:67 108
9
0.9999999934
1.000000000
0:66 108
10
0.9999999934
1.000000000
0:66 108
Table 2 Shown the absolute value of approximate solution (3.18) and the absolute value of the exact solution (3.21) when t ¼ 0:1, a ¼ 0:9, b ¼ 0:9 and 0 < x < 10. Table 2 leads to get the proposed algorithm produced a rapidly convergent series. x
absðuapp Þ
absðuex Þ
Error
0.1
0.9967252929
1.000000000
3:274 103
0.5
0.9967252929
1.000000000
3:274 103
1
0.9967252931
1.000000000
3:274 103
1.5
0.9967252932
1.000000000
3:274 103
3
0.9967252927
1.000000000
3:274 103
5
0.9967252931
1.000000000
3:274 103
7
0.9967252927
1.000000000
3:274 103
9
0.9967252927
1.000000000
3:274 103
10
0.9967252929
1.000000000
3:2 103
Fig. 1. The the real part of the approximate solution (3.20) shown in the figure (a) in comparison with the real part of the exact solution (3.21) shown in (b) when 10 < x < 10 and 0 < t < 1.
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M.A.E. Herzallah, K.A. Gepreel / Applied Mathematical Modelling 36 (2012) 5678–5685
Fig. 2. The the imaginary part of the approximate solution (3.20) shown in the figure (a) in comparison with the imaginary part of the exact solution (3.21) shown in (b) when 10 < x < 10 and 0 < t < 1.
Fig. 3. The the real part of the approximate solution (3.18) shown in the figure (a) in comparison with the real part of the exact solution (3.21) shown in (b) when a ¼ 0:9, b ¼ 0:9. 10 < x < 10 and 0 < t < 1.
M.A.E. Herzallah, K.A. Gepreel / Applied Mathematical Modelling 36 (2012) 5678–5685
5683
Fig. 4. The the imaginary part of the approximate solution (3.18) shown in the figure (a) in comparison with the imaginary part of the exact solution (3.21) shown in (b) when a ¼ 0:9, b ¼ 0:9. 10 < x < 10 and 0 < t < 1.
Fig. 5. The the real part of the approximate solution (3.18) shown in the figure (a) and the imaginary part of the approximate solution (3.18) shown in (b) when a ¼ 0:0001, b ¼ 0:0001. 10 < x < 10 and 0 < t < 1.
In this case the approximate solution for the nonlinear fractional Schrodinger Eq. (1.2) takes the form
uapp ¼ u0 þ u1 þ u2 þ u3 þ u4 þ u5 þ u6 þ c1 ta c2 t 2a c 3 t 3a c 4 t 4a c 5 t 5a c 6 t 6a þ þ þ þ þ þ ¼ eix 1 þ Cða þ 1Þ Cð2a þ 1Þ Cð3a þ 1Þ Cð4a þ 1Þ Cð5a þ 1Þ Cð6a þ 1Þ " # 1 X ck ðt a Þk : ¼ eix 1 þ Cðka þ 1Þ k¼1
ð3:18Þ
5684
M.A.E. Herzallah, K.A. Gepreel / Applied Mathematical Modelling 36 (2012) 5678–5685
Now, let a ! 1 and b ! 1, we get
c1 ¼ i;
c2 ¼ 1;
c3 ¼ i;
c4 ¼ 1;
c5 ¼ i; . . .
ð3:19Þ
Consequently, the approximate solution (3.18) takes the following form
it eix ðitÞ2 eix ðitÞ3 eix ðitÞ4 eix uapp ¼ u0 þ u1 þ u2 þ u3 þ u4 þ u5 þ ¼ eix þ þ þ þ þ Cð2Þ Cð3Þ Cð4Þ Cð5Þ ! it ðitÞ2 ðitÞ3 ðitÞ4 þ þ þ : ¼ eix 1 þ þ 1! 2! 3! 4!
ð3:20Þ
Using Taylor series expansion near t = 0, we get:
uex ¼ eiðxþtÞ :
ð3:21Þ
This is the exact solution of the nonlinear Schrodinger partial differential Eq. (1.1) obtained when replacing a and b by 1 in (3.18). This solution (3.21) is exactly the same solution obtained in [26]. The comparison between the exact solution (3.21) and the approximate solutions (3.20) are shown in Table 1. The comparison between the exact solution (3.21) and the approximate solutions (3.18) are shown in Table 2. when a ¼ 0:9 and b ¼ 0:9. 4. Conclusion In this paper, the adomian decomposition method has been successfully applied to obtain the numerical solutions of the time–space fractional cubic nonlinear Schrodinger fractional equation with initial conditions. The reliability of this method and reduction in computations give this method a wider applicability. From Figs. 1–5, we deduce the behavior of the approximate solutions is the same behavior of the exact solution at some different values of a and b. Tables 1 and 2 leads to see the small error between the approximate solution and exact solution for some different values of a and b. Consequently, we deduce that the approximate solution is rapidly convergent series as the exact solutions. Adomian decomposition method was clearly very efficient and powerful technique in finding the numerical solutions of the proposed equations. References [1] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, 204, Elsevier, Amsterdam, 2006. [2] I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, New York, London, 1999. [3] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Langhorne, 1993. [4] A.M.A. El-Sayed, Fractional-order diffusion-wave equation, Int. J. Theory Phys. 35 (1996) 311–322. [5] M.A.E. Herzallah, A.M.A. El-Sayed, D. 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He, Modified Lindstedt–Poincare methods for some strongly non-linear oscillations: Part I: Expansion of a constant, Int. J. Nonlinear Mech. 37 (2002) 309–314; G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135 (2,1) (1988) 501–544. [20] A.M. Wazwaz, A reliable technique for solving the wave equation in an infinite one-dimensional medium, Appl. Math. Comput. 92 (1) (1998) 1–7. [21] E.M.E. Zayed, T.A. Nofal, K.A. Gepreel, Homotopy perturbation and adomain decomposition methods for solving nonlinear Boussinesq equations, Commun. Appl. Nonlinear Anal. 15 (2008) 57–70. [22] D.S. Wang, H.Q. Zhang, Further improved F-expansion method and new exact solutions of Konopelchenko–Dubrovsky equation, Chaos Solitons Fract. 25 (3) (2005) 601–610. [23] X.H. Wu, J.H. He, Solitary solutions, Periodic solutions and compact on-like solutions using the exp-function method, Comput. Math. Appl. 54 (7–8) (2007) 966–986. [24] J.H. He, Homotopy perturbation method for solving boundary value problems, Phys. Lett. A. 350 (1-2) (2006) 87–88. [25] K.A. Gepreel, The homotopy perturbation method applied to the nonlinear fractional Kolmogorov–Petrovskii–Piskunov equations, Appl. Math. Lett. 24 (8) (2011) 1428–1434. [26] A.S. V Kanth, K. Aruna, Two-dimensional differential transform method for solving linear and non-linear Schrodinger equations, Chaos Solitons Fract. 41 (5) (2009) 2277–2281. [27] E.P. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys. 4 (1963) 195–207.
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[28] K.B. Dysthe, Modelling a rogue wave – speculations or a realistic possibility?, in: Rogue Waves 2000, ´ Editions Ifremer, Plouzan´ e, France, 2001, pp 255–264. [29] K.L. Henderson, D.H. Peregrine, J.W. Dold, Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrodinger equation, Wave Motion 29 (1999) 341–361. [30] P.L. Kelley, Self-focusing of optical beams, Phys. Rev. Lett. 15 (1965) 1005–1008. [31] V.I. Talanov, Self modelling wave beams in a nonlinear dielectric, Radiophys. Quantum Electron. 9 (1967) 260–261.
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Approximate solution to the time–space fractional cubic nonlinear Schrodinger equation Mohamed A.E. Herzallah a,b,⇑, Khaled A. Gepreel a,c a
Faculty of Science, Zagazig University, Zagazig, Egypt Faculty of Science, Majmaah University, Saudi Arabia c Faculty of Science, Taif University, Saudi Arabia b
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 13 June 2011 Received in revised form 27 December 2011 Accepted 4 January 2012 Available online 21 January 2012 Keywords: Adomian decomposition method Fractional calculus Cubic nonlinear fractional Schordinger equation
By introducing the fractional derivatives in the sense of Caputo, we use the adomian decomposition method to construct the approximate solutions for the cubic nonlinear fractional Schordinger equation with time and space fractional derivatives. The exact solution of the cubic nonlinear Schrodinger equation is given as a special case of our approximate solution. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equation. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction In recent years, there has been a great deal of interest in fractional differential equations. First there were almost no practical applications of fractional calculus, and it was considered by many as an abstract area containing only mathematical manipulations of little or no use. Nearly 30 yr ago, the paradigm began to shift from pure mathematical formulations to applications in various fields. During the last decade fractional calculus has been applied to almost every field of science, engineering and mathematics. Several fields of application of fractional differentiation and fractional integration are already well established, some others have just started. Many applications of fractional calculus can be found in turbulence and fluid dynamics, stochastic dynamical system, plasma physics and controlled thermonuclear fusion, nonlinear control theory, image processing, nonlinear biological systems, astrophysics [1–11]. Historical summaries of the developments of fractional calculus can be found in [1–3]. Various methods for obtaining exact solutions to nonlinear partial differential equations have been proposed such as the Bäcklund transformation method [12], the Hirota’s bilinear method [13], the inverse scattering transform method [14], the extended tanh method [15], the adomian pade approximation [16], the variational method [17], the variational iteration method [18], the various Lindstedt–Poincare methods [19], the adomian decomposition method [20,21], the F-expansion method [22], the exp-function method [23], the homotopy perturbation method [24,25] and so on. Consider the Schrodinger equation with a cubic nonlinearity
iut þ uxx þ 2juj2 u ¼ 0;
uðx; 0Þ ¼ f ðxÞ;
i¼
pffiffiffiffiffiffiffi 1:
⇑ Corresponding author at: Faculty of Science, Zagazig University, Zagazig, Egypt. E-mail addresses: [email protected] (M.A.E. Herzallah), [email protected] (K.A. Gepreel). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2012.01.012
ð1:1Þ
M.A.E. Herzallah, K.A. Gepreel / Applied Mathematical Modelling 36 (2012) 5678–5685
5679
Which occurs in various chapters of physics, including nonlinear optics, superconductivity and plasma physics. The cubic nonlinearity is the most common nonlinearity in applications. It arises as a simplified model for studying Bose–Einstein condensates, Kerr media in nonlinear optics freak waves in the ocean (see [26–31]). In [26] Kanth and Aruna studied (1.1) with applying the differential transform method to obtain its approximate solution. In this article, we give a new model of the time–space fractional cubic nonlinear Schrodinger equation of the form
it C Da0þ u þ x C Dbþ ðx C Dbþ uÞ þ 2juj2 u ¼ 0;
t > 0; 0 < a; b 6 1:
ð1:2Þ
This equation can be represented some dynamical system in quantum mechanics especially in the exact solution for nonlinear Schrodinger equation of motion. Also in fluid dynamic and nonlinear dynamical system. We use adomian decomposition method to calculate an approximate solution of the cubic nonlinear fractional Schordinger equation (1.2) which is a generalization of the given in [26]. We use the Caputo fractional derivative on the half axis Rþ (i.e. t 2 Rþ ) t C Da0þ for time and the Caputo fractional derivative on the whole axis R (i.e. x 2 R) x C Dbþ for space. For some special values of a and b, the exact solution of the nonlinear Schordinger with a cubic nonlinearity will obtained. 2. Preliminaries and notations We give some basic definitions and properties of the fractional calculus theory which are used further in this paper [2,3]. For the finite derivative [a, b], we define the following fractional integral and derivatives. Definition 2.1. If f ðtÞ 2 L1 ða; bÞ, the set of all integrable functions, and a > 0 then the Riemann–Liouville fractional integral of order a, denoted by Iaaþ is defined by
Iaaþ f ðtÞ ¼
1 CðaÞ
Z
t
ðt sÞa1 f ðsÞds:
a
Definition 2.2. For a > 0, the Caputo fractional derivative of order a, denoted by C Daaþ , is defined by C
1 Cðn aÞ
Daaþ f ðtÞ ¼
Z
t
ðt sÞna1 Dn f ðsÞds;
a
where n is such that n 1 < a < n and D ¼ dds. If a is an integer, then this derivative takes the ordinary derivative C
Daaþ ¼ Da ;
a ¼ 1; 2; 3; . . .
Finally the Caputo fractional derivative on the whole space R is defined by, Definition 2.3. For a > 0 the Caputo fractional derivative of order a on the whole space, denoted by C Daþ , is defined by C
Daþ f ðxÞ ¼
1 Cðn aÞ
Z
x
ðx nÞna1 Dn f ðnÞdn:
1
3. Main results Consider the nonlinear fractional Schrodinger equation
it C Da0þ u þ x C Dbþ ðx C Dbþ uÞ þ 2juj2 u ¼ 0;
t > 0;
0 < a;
b 6 1;
uðx; 0Þ ¼ f ðxÞ ¼ eix ;
ð3:1Þ
pffiffiffiffiffiffiffi where i ¼ 1. Operating on both sides on the system (3.1) by the operator t Ia0þ (the Riemann–Liouville fractional integral with parameter t) we get:
uðx; tÞ ¼ uðx; 0Þ þ it Ia0þ ½x C Dbþ ðx C Dbþ uÞ þ 2juj2 u; ¼ f ðxÞ þ it Ia0þ ½x C Dbþ ðx C Dbþ uÞ þ 2GðuÞ; where GðuÞ ¼ juj2 u and uðx; 0Þ ¼ f ðxÞ. According to the Adomain decomposition method [20,21], we assume that a series solution of the function uðx; tÞ is given by
uðx; tÞ ¼
1 X
un ðx; tÞ:
ð3:2Þ
n¼0
The nonlinear term GðuÞ can be decomposed into an infinite series of polynomials given by
Gðx; tÞ ¼
1 X n¼0
An ;
ð3:3Þ
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where the components un ðx; tÞ will be determined recursively while An ’s are the so called adomian polynomials of un ’s. Specific algorithms have been set in [20,21] for calculating adomian’s polynomials for nonlinear terms
( n " ! ! !#) n n n X X X 1 d k k k An ðu0 ; u1 ; u2 ; . . . ; un Þ ¼ k uk k uk k uk n! dkn k¼0 k¼0 k¼0
ð3:4Þ
:
k¼0
Consequently, we have
unþ1 ðx; tÞ ¼ it Ia0þ ½x C Dbþ ðx C Dbþ un Þ þ 2An ;
n P 0:
ð3:5Þ
Eq. (3.4) leads to get:
A0 ¼ u0 ju0 j2 ; A1 ¼ u20 u1 þ 2u1 ju0 j2 ; A2 ¼ u20 u2 þ u21 u0 þ 2u0 ju1 j2 þ 2u2 ju0 j2 ; A3 ¼ u20 u3 þ 2u0 u1 u2 þ 2u0 u2 u1 þ 2u1 u2 u0 þ 2u3 ju0 j2 þ u1 ju1 j2 ; A4 ¼ u20 u4 þ 2u0 u1 u3 þ 2u0 u3 u1 þ 2u1 u3 u0 þ u21 u2 þ u22 u0 þ 2u0 ju2 j2 þ 2u2 ju1 j2 þ 2u4 ju0 j2 ; and so on. Using the above recursive relationships in (3.5), with using (see [1,2]) a
t I 0þ
tb t bþa ¼ ; Cðb þ 1Þ Cðb þ a þ 1Þ
x
C
pb
Dbþ eix ¼ eiðxþ 2 Þ :
We construct the approximate solutions of uðx; tÞ as follows:
ta ; Cða þ 1Þ t 2a ; u2 ¼ c2 eix Cð2a þ 1Þ t 3a ; u3 ¼ c3 eix Cð3a þ 1Þ t 4a ; u4 ¼ c4 eix Cð4a þ 1Þ t 5a ; u5 ¼ c5 eix Cð5a þ 1Þ 6a t ; u6 ¼ c6 eix Cð6a þ 1Þ ...
u1 ¼ c1 eix
ð3:6Þ ð3:7Þ ð3:8Þ ð3:9Þ ð3:10Þ ð3:11Þ
and so on. After some calculation, we get:
c1 ¼ iðeipb þ 2Þ;
ð3:12Þ
c2 ¼ iðeipb c1 þ 2c1 þ 4c1 Þ; "
ð3:13Þ
c3 ¼ i eipb c2 þ 2 c2 þ 2c2 þ " c4 ¼ i eipb c3 þ 2 c3 þ 2c3 þ (
"
c5 ¼ i eipb c4 þ 2 c4 þ 2c4 þ
Cð2a þ 1Þ ð2jc1 j2 þ c21 Þ ðCða þ 1ÞÞ2
#!
ð3:14Þ
;
2Cð3a þ 1Þ Cð3a þ 1Þ c1 jc1 j2 ðc c þ c1 c2 þ c2 c1 Þ þ Cð2a þ 1ÞðCða þ 1ÞÞ 1 2 ðCða þ 1ÞÞ3
#! ;
ð3:15Þ
Cð4a þ 1Þ Cð4a þ 1Þ 2 ðc2 þ 2jc2 j2 Þ ð2c c þ 2c1 c3 þ 2c3 c1 Þ þ 2 Cð3a þ 1ÞðCða þ 1ÞÞ 1 3 C ð2a þ 1Þ #)
Cð4a þ 1Þ ðc21 c2 þ 2c2 jc1 j2 Þ þ 2 C ða þ 1ÞCð2a þ 1Þ Cð5a þ 1Þ ð2c c þ 2c1 c4 þ 2c4 c1 Þ c6 ¼ i eipb c5 þ 2 c5 þ 2c5 þ Cð4a þ 1ÞðCða þ 1ÞÞ 1 4 Cð5a þ 1Þ Cð5a þ 1Þ ðc21 c3 þ 2jc1 j2 c3 Þ þ ð2c2 c3 þ 2c2 c3 þ 2c3 c2 Þ þ ðCð3a þ 1ÞÞCð2a þ 1Þ Cð3a þ 1ÞðCða þ 1ÞÞ2 #) Cð5a þ 1Þ 2 2 c þ 2c jc j Þ ðc þ 1 2 2 1 ðCð2a þ 1ÞÞ2 Cða þ 1Þ ... and so on.
ð3:16Þ
ð3:17Þ
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Table 1 Shown that the absolute value of approximate solution (3.20) and the absolute value of the exact solution (3.21) when t ¼ 0:1 and 0 < x < 10. Table 1 leads to get the proposed algorithm produced a rapidly convergent series. x
absðuapp Þ
absðuex Þ
0.1
0.9999999931
1.000000000
Errorðjabsðuapp Þ absðuex ÞjÞ 0:69 108
0.5
0.9999999931
1.000000000
0:69 108
1
0.9999999934
1.000000000
0:66 108
1.5
0.9999999933
1.000000000
0:67 108
3
0.9999999934
1.000000000
0:66 108
5
0.9999999934
1.000000000
0:66 108
7
0.9999999933
1.000000000
0:67 108
9
0.9999999934
1.000000000
0:66 108
10
0.9999999934
1.000000000
0:66 108
Table 2 Shown the absolute value of approximate solution (3.18) and the absolute value of the exact solution (3.21) when t ¼ 0:1, a ¼ 0:9, b ¼ 0:9 and 0 < x < 10. Table 2 leads to get the proposed algorithm produced a rapidly convergent series. x
absðuapp Þ
absðuex Þ
Error
0.1
0.9967252929
1.000000000
3:274 103
0.5
0.9967252929
1.000000000
3:274 103
1
0.9967252931
1.000000000
3:274 103
1.5
0.9967252932
1.000000000
3:274 103
3
0.9967252927
1.000000000
3:274 103
5
0.9967252931
1.000000000
3:274 103
7
0.9967252927
1.000000000
3:274 103
9
0.9967252927
1.000000000
3:274 103
10
0.9967252929
1.000000000
3:2 103
Fig. 1. The the real part of the approximate solution (3.20) shown in the figure (a) in comparison with the real part of the exact solution (3.21) shown in (b) when 10 < x < 10 and 0 < t < 1.
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Fig. 2. The the imaginary part of the approximate solution (3.20) shown in the figure (a) in comparison with the imaginary part of the exact solution (3.21) shown in (b) when 10 < x < 10 and 0 < t < 1.
Fig. 3. The the real part of the approximate solution (3.18) shown in the figure (a) in comparison with the real part of the exact solution (3.21) shown in (b) when a ¼ 0:9, b ¼ 0:9. 10 < x < 10 and 0 < t < 1.
M.A.E. Herzallah, K.A. Gepreel / Applied Mathematical Modelling 36 (2012) 5678–5685
5683
Fig. 4. The the imaginary part of the approximate solution (3.18) shown in the figure (a) in comparison with the imaginary part of the exact solution (3.21) shown in (b) when a ¼ 0:9, b ¼ 0:9. 10 < x < 10 and 0 < t < 1.
Fig. 5. The the real part of the approximate solution (3.18) shown in the figure (a) and the imaginary part of the approximate solution (3.18) shown in (b) when a ¼ 0:0001, b ¼ 0:0001. 10 < x < 10 and 0 < t < 1.
In this case the approximate solution for the nonlinear fractional Schrodinger Eq. (1.2) takes the form
uapp ¼ u0 þ u1 þ u2 þ u3 þ u4 þ u5 þ u6 þ c1 ta c2 t 2a c 3 t 3a c 4 t 4a c 5 t 5a c 6 t 6a þ þ þ þ þ þ ¼ eix 1 þ Cða þ 1Þ Cð2a þ 1Þ Cð3a þ 1Þ Cð4a þ 1Þ Cð5a þ 1Þ Cð6a þ 1Þ " # 1 X ck ðt a Þk : ¼ eix 1 þ Cðka þ 1Þ k¼1
ð3:18Þ
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M.A.E. Herzallah, K.A. Gepreel / Applied Mathematical Modelling 36 (2012) 5678–5685
Now, let a ! 1 and b ! 1, we get
c1 ¼ i;
c2 ¼ 1;
c3 ¼ i;
c4 ¼ 1;
c5 ¼ i; . . .
ð3:19Þ
Consequently, the approximate solution (3.18) takes the following form
it eix ðitÞ2 eix ðitÞ3 eix ðitÞ4 eix uapp ¼ u0 þ u1 þ u2 þ u3 þ u4 þ u5 þ ¼ eix þ þ þ þ þ Cð2Þ Cð3Þ Cð4Þ Cð5Þ ! it ðitÞ2 ðitÞ3 ðitÞ4 þ þ þ : ¼ eix 1 þ þ 1! 2! 3! 4!
ð3:20Þ
Using Taylor series expansion near t = 0, we get:
uex ¼ eiðxþtÞ :
ð3:21Þ
This is the exact solution of the nonlinear Schrodinger partial differential Eq. (1.1) obtained when replacing a and b by 1 in (3.18). This solution (3.21) is exactly the same solution obtained in [26]. The comparison between the exact solution (3.21) and the approximate solutions (3.20) are shown in Table 1. The comparison between the exact solution (3.21) and the approximate solutions (3.18) are shown in Table 2. when a ¼ 0:9 and b ¼ 0:9. 4. Conclusion In this paper, the adomian decomposition method has been successfully applied to obtain the numerical solutions of the time–space fractional cubic nonlinear Schrodinger fractional equation with initial conditions. The reliability of this method and reduction in computations give this method a wider applicability. From Figs. 1–5, we deduce the behavior of the approximate solutions is the same behavior of the exact solution at some different values of a and b. 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