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Modified fractional variational iteration method for solving the generalized time-space fractional Schrodinger equation.

1. Introduction

In the past decades, due to the numerous applications of fractional differential equations (FDEs) in the areas of nonlinear science [1], many important phenomena can be described successfully using the FDEs models such as materials and processes [2], engineering and physics [3], dielectric polarization [4], and quantitative finance [5]. Searching for solutions of these FDEs plays an important and significant role in all aspects of this subject. But because of the complexity of nonlinear terms and fractional derivative, it is very difficult for us to obtain the exact analytic solutions of most FDEs, so approximate and numerical methods must be considered. A great deal of efforts have been proposed for these problems, like the homotopy analysis method (HAM)

[6], the homotopy perturbation method (HPM) [7], the adomian decomposition method (ADM) [8], the generalized differential transform method [9], and so forth [10].

The variational iteration method (VIM) established in 1999 by He in [11] is thoroughly used by many researchers to construct the approximate solutions of a wide variety of scientific and engineering models [12, 13]. After some modifications, the fractional variational iteration method (FVIM) was applied to fractional differential equations by He and many authors [14-19]. The motivation of this paper is to construct some analytical approximate solutions for the GFNLS powerfully. Firstly, we give some modifications for the FVIM and extend the application of the FVIM. Secondly, we use the modified fractional variational iteration method (MFVIM) to the GFNLS and compare the efficiency of MFVIM with some other traditional perturbation methods. The results show that MFVIM gives rapid and standard convergence to the exact solution if such a solution exists.

We give some basic definitions and properties of the fractional calculus theory which are used further in this paper; we define the following fractional integral and derivatives [20, 21].

Definition 1. A real function f(x) is said to be in the space [C.sub.[mu]], where [mu] [member of] R, x > 0, if there exists a real number p(> [mu]) such that f(x) = [x.sup.p] [f.sub.1](x), where [f.sub.1](x) [member of]C[0, [infinity]) and it is said to be in the space [C.sup.m.sub.[mu]] if and only if [f.sup.(m)] [member of] [C.sub.[mu]], m [member of] N.

Definition 2. The Riemann-Liouville fractional integral operator of order [alpha] > 0 for a function f(x) [member of] [C.sub.[mu]], [mu] [greater than or equal to] -1, is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

Also one has the following properties:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Definition 3. For [alpha] > 0, x > 0, and f(%) [member of] [C.sup.n.sub.-1] the Caputo fractional derivative operator of order a on the whole space is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Also one has the following properties:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

2. Analysis of the MFVIM and the FGNLS

Consider the following generalized time and space fractional nonlinear Schrodinger equation with variable coefficients [22, 23]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where u = u(x, t), [[partial derivative].sup.[alpha]]u/[partial derivative][t.sup.[alpha]] = [D.sup.[alpha].sub.t]u, [[partial derivative].sup.2[beta]]u/[partial derivative][x.sup.2[beta]] = [D.sup.[beta].sub.x] x ([D.sup.[beta].sub.x]u), v(x) is the trapping potential, and a, [gamma] are the slowly increasing dispersion coefficient and nonlinear coefficient, respectively. If we select [alpha] = [beta] = 1, v(x) = 0, this equation turns to the famous nonlinear Schroodinger equations in optical fiber [24-26].

According to the FVIM [14-19], we can build a correction functional for (5) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

with the initial condition [u.sub.0] = u(x, 0) = f(x), where [lambda](t, x) is a general Lagrange's multiplier which can be identified optimally with the variational theory. The function [[??].sub.n] is a restricted variation which means Sun = 0. Therefore, we first determine the Lagrange multiplier [lambda](t, x) that will be identified optimally via integration by parts [27]. The successive approximations [u.sub.n+1], n [greater than or equal to] 0, of the solution u(x, t) will be readily obtained through [lambda](t, x) and any selective function [u.sub.0]. The initial values are usually used for choosing the zeroth approximation [u.sub.0]. With [lambda](t, x) determined, then several approximations [u.sub.k], k = 1,2, ... follow immediately. Consequently, the exact solution may be procured by using u = [lim.sub.n[right arrow][infinity]][u.sub.n]. The convergence of FVIM has been proved in [28]. In this paper, notice that (5) is a complex differential equation with complex modulus term [[absolute value of u].sup.2], as we all know, a complex function u([xi]) can be written as c ([xi])[e.sup.i[theta]([xi])], where c([xi]) and d([theta]) are real functions, noticed that [absolute value of u([xi])] = [[absolute value of c([xi])].sup.2], we can give some modification for the iteration formulation (6), assume that [lim.sub.n[right arrow][infinity] [[absolute value of [[??].sub.n].sup.2] = [[absolute value of u].sup.2] = [[absolute value of [u.sub.0].sup.2], we get the MFVIM for (5). This modification should enhance rapidly the efficiency of our iteration.

In what follows, in order to illustrate the strength of this method, we will apply the MFVIM to some models about (5).

3. Approximate Solutions for the FGNLS

Example 4. We first consider the time-fractional NLS equation [29]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

The correction functional for (7) reads

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Making the above correction functional stationary,

[delta][u.sub.n+1] = [delta][u.sub.n] + A (t, x) i[delta][u.sub.n] - 1/[GAMMA](1 + [alpha]) [[integral].sup.t.sub.0] [i[D.sup.([alpha]).sub.[tau]] [lambda] ([tau], x) [delta][u.sub.n]] [(d[tau]).sup.[alpha]] (9)

After getting the coefficients of [delta][u.sub.n] to zero we can determine the Lagrange multiplier

[lambda] = i. (10)

We produce the iteration formulation as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

As stated before, we can select [u.sub.0] = w(v, 0) = Asec hv; using the iteration (11) and the mathematica software, we obtain the following successive approximations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

The exact solution of (7) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)

where [E.sub.[alpha]]([ait.sup.[alpha]]) is the Mittag-Leffler function. If we let [alpha] = 1 in (13), the exact solution of the regular NLS equation (7) can be obtained as follows:

u[|.sub.[alpha]=1] = [+ or -]i [square root of 2a/[gamma]] sec hx[e.sup.iat]. (14)

Example 5. We now consider the time-space fractional NLS equation [30, 31]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

With the similar process, we get the iteration formulation as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Using the iteration (16) and the mathematica software, we obtain the following successive approximations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

The exact solution of (15) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

If we let [alpha] = 1 and let [beta] = 1 in (18), the exact solution of the regular NLS equation (15) can be obtained as follows:

u[|.sub.([alpha]=1, [beta]=1)] = [e.sup.i(x+at). (19)

Remark 6. The solution (18) is more standard than the result (3.18) in [30]. If one selects a= 1 or a= 1/2, the solution 19) is the same as the result (49) in 29], the result (3.21) in [30], and the result (29) in [31], but one can find that this iteration is much more standard and powerful than the HAM, the ADM, and the VIM mentioned in [29-31].

Example 7. Consider the following time-space fractional NLS equation [23]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

With the similar process, we get the iteration formulation as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

If we select [u.sub.0] = m(x, 0) = sin x, using the iteration 21) and the mathematica software, we obtain the following successive approximations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

where [c.sub.k](x) = [c.sub.k,0] sin x + [c.sub.k,1] sin(x + [pi][beta]) + [c.sub.k,2] sin(x + 2[pi][beta]) + ... + [c.sub.k,k] sin(x + k[pi][beta]), [c.sub.k,0] = [(-1).sup.k], [c.sub.k,1] = (1/2)[c.sub.k-1,1] - [c.sub.k-1,0], ..., [c.sub.k,k-1] = (1/2)[c.sub.k-1,k-2] - [c.sub.k-1,k-1] and [c.sub.k,k] = (1/2)[c.sub.k-1,k-1] k [greater than or equal to] 2, [c.sub.0,0] = 1; [c.sub.1,0] = -1, [c.sub.1,1] = (1/2); [c.sub.2,0] = 1, [c.sub.2,1] = -1, [c.sub.2,2] = (1/4);....

The exact solution of (20) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

If we let [alpha] = 1 and let [beta] = 1 in (23), the exact solution of the regular NLS equation (20) can be obtained as follows.

u[|.sub.([alpha]=1, [beta]=1)] = sin [xe.sup.-(3/2)it]. (24)

If we select [u.sub.0] = w(x, 0) = cos x, with the same process, we can also obtain the following exact solution of 20):

u = i cos [[x.sup.[beta]]/[GAMMA](1 + [beta])] Exp [- 1/2 [it.sup.[alpha]]/[GAMMA] (1 + [alpha])]. (25)

Remark 8. If one selects [beta] = 1, the solution (23) is more standard than the result (5.10) in [23]. The solutions (23) and (25) are new exact solutions for (20) to our knowledge.

Comparisons between the real part of some numerical results and the exact solution (23) are summarized in Tables 1 and 2, and the simulations for [u.sub.4], [u.sub.abs], and u are plotted in Figures 1 and 2, which shows that the MFVIM produced a rapidly convergent series.

4. Summary

In this paper, the MFVIM is used for finding approximate and exact solutions of the GFNLS equation with Caputo derivative. The obtained results indicate that the MFVIM is effective, convenient, and powerful method for solving nonlinear fractional complex differential equations when comparing it with some other traditional asymptotic decomposition methods such as HAM, VIM, and ADM. We believe that these methods should play an important role for finding exact and approximate solutions in the mathematical physics.

http://dx.doi.org/10.1155/2014/964643

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is supported by the National Nature Science Foundation of China (no. 61070231); the Outstanding Personal Program in Six Fields of Jiangsu (2009188); and the General Program of Innovation Foundation of NanJing Institute of Technology (Grant no. CKJB201218).

References

[1] M. Dalir and M. Bashour, "Applications of fractional calculus," Applied Mathematical Sciences, vol. 4, pp. 1021-1032, 2010.

[2] I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.

[3] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.

[4] H. H. Sun, A. A. Abdelwahab, and B. Onaral, "Linear approximation of transfer function with a pole of fractional order," IEEE Transactions on Automatic Control, vol. 29, no. 5, pp. 441-444, 1984.

[5] N. Laskin, "Fractional market dynamics," Physica A, vol. 287, no. 3-4, pp. 482-492, 2000.

[6] S. Panda, A. Bhowmik, R. Das, R. Repaka, and S. C. Martha, "Application of homotopy analysis method and inverse solution of a rectangular wet fin," Energy Conversion and Management, vol. 80, pp. 305-318, 2014.

[7] K. A. Gepreel, "The homotopy perturbation method applied to the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations," Applied Mathematics Letters, vol. 24, no. 8, pp. 1428-1434, 2011.

[8] L. N. Song and W. G. Wang, "A new improved Adomian decomposition method and its application to fractional differential equations," Applied Mathematical Modelling, vol. 37, no. 3, pp. 1590-1598, 2013.

[9] Z. Odibat, S. Momani, and V. S. Erturk, "Generalized differential transform method: application to differential equations of fractional order," Applied Mathematics and Computation, vol. 197, no. 2, pp. 467-477, 2008.

[10] M. Inc and A. Akgul, "Numerical solution of seventh-order boundary value problems by a novel method," Abstract and Applied Analysis, vol. 2014, Article ID 745287, 9 pages, 2014.

[11] J. H. He, "Variational iteration method--a kind of non-linear analytical technique: some examples," International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699-708, 1999.

[12] M. A. Noor and S. T. Mohyud-Din, "Variational iteration method for solving higher-order nonlinear boundary value problems using He's polynomials," International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, pp. 141-156, 2008.

[13] S. Momani and S. Abuasad, "Application of He's variational iteration method to Helmholtz equation," Chaos, Solitons & Fractals, vol. 27, no. 5, pp. 1119-1123, 2006.

[14] J. H. He, "Some applications of nonlinear fractional differential equations and their approximations," Bulletin of Science, Technology & Society, vol. 15, no. 2, pp. 86-90, 1999.

[15] J. He, "A short remark on fractional variational iteration method," Physics Letters A, vol. 375, no. 38, pp. 3362-3364, 2011.

[16] M. Merdan, "A numeric-analytic method for time-fractional Swift-Hohenberg (S-H) equation with modified RiemannLiouville derivative," Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 37, no. 6, pp. 4224-4231, 2013.

[17] G. C. Wu, "A fractional variational iteration method for solving fractional nonlinear differential equations," Computers & Mathematics with Applications, vol. 61, no. 8, pp. 2186-2190, 2011.

[18] G. C. Wu and D. Baleanu, "Discrete fractional logistic map and its chaos," Nonlinear Dynamics, vol. 75, no. 1-2, pp. 283-287, 2014.

[19] G. C. Wu and D. Baleanu, "Variational iteration method for the Burgers' flow with fractional derivatives--new Lagrange multipliers," Applied Mathematical Modelling, vol. 37, no. 9, pp. 6183-6190, 2013.

[20] M. Inc, "The approximate and exact solutions of the space and time-fractional Burgers equations with initial conditions by variational iteration method," Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 476-484, 2008.

[21] M. G. Sakar, F. Erdogan, and A. Yildirim, "Variational iteration method for the time-fractional Fornberg-Whitham equation," Computers & Mathematics with Applications, vol. 63, no. 9, pp. 1382-1388, 2012.

[22] M. Ganjiani, "Solution of nonlinear fractional differential equations using homotopy analysis method," Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 34, no. 6, pp. 1634-1641, 2010.

[23] S. Z. Rida, H. M. El-Sherbiny, and A. A. M. Arafa, "On the solution of the fractional nonlinear Schrodinger equation," Physics Letters A, vol. 372, no. 5, pp. 553-558, 2008.

[24] R. Hao, L. Li, Z. Li, W. Xue, and G. Zhou, "A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrodinger equation with variable coefficients," Optics Communications, vol. 236, no. 1-3, pp. 79-86, 2004.

[25] Y. Chen and B. Li, "An extended subequation rational expansion method with symbolic computation and solutions of the nonlinear Schrodinger equation model," Nonlinear Analysis: Hybrid Systems, vol. 2, no. 2, pp. 242-255, 2008.

[26] B. Li and Y. Chen, "On exact solutions of the nonlinear Schroodinger equations in optical fiber," Chaos, Solitons and Fractals, vol. 21, no. 1, pp. 241-247, 2004.

[27] R. Almeida and D. F. M. Torres, "Calculus of variations with fractional derivatives and fractional integrals," Applied Mathematics Letters, vol. 22, no. 12, pp. 1816-1820, 2009.

[28] S. Yang, A. Xiao, and H. Su, "Convergence of the variational iteration method for solving multi-order fractional differential equations," Computers & Mathematics with Applications, vol. 60, no. 10, pp. 2871-2879, 2010.

[29] M. Ganjiani, "Solution of nonlinear fractional differential equations using homotopy analysis method," Applied Mathematical Modelling, vol. 34, no. 6, pp. 1634-1641, 2010.

[30] M. A. E. Herzallah and K. A. Gepreel, "Approximate solution to the time-space fractional cubic nonlinear Schrodinger equation," Applied Mathematical Modelling, vol. 36, no. 11, pp. 5678-5685, 2012.

[31] A. Wazwaz, "A study on linear and nonlinear Schrodinger equations by the variational iteration method," Chaos, Solitons and Fractals, vol. 37, no. 4, pp. 1136-1142, 2008.

Baojian Hong (1,2) and Dianchen Lu (1)

(1) Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China

(2) Department of Basic Causes, Nanjing Institute of Technology, Nanjing 211167, China

Correspondence should be addressed to Baojian Hong; hbj@njit.edu.cn

Received 8 May 2014; Revised 24 July 2014; Accepted 27 July 2014; Published 4 September 2014

Academic Editor: Mustafa Inc

TABLE 1: Comparison between the real part of [u.sub.4] and
u when [alpha] = [beta] = 1.

x     t      Approximate        Exact          Absolute error
              solution        solution
            [u.sub.4appr]

1     0.4   0.6945501509    0.6944959727        0.0000541782
5     0.4   -0.7914960963   -0.7914343559       0.0000617404
1     0.3   0.7577097797    0.7577001100    9.66968 x [10.sup.-6]
15    0.3   0.5855572741    0.5855498014    7.47272 x [10.sup.-6]
12    0.2   -0.5126082301   -0.5126076876   6.16597 x [10.sup.-6]
3     0.2   0.13481723570   0.13481709305   1.42655 x [10.sup.-7]
2     0.1   0.8990870113    0.8990869969    1.43796 x [10.sup.-8]
0.2   0.1   0.1964384915    0.1964384884    3.14175 x [10.sup.-9]

TABLE 2: Comparison between the real part of [u.sub.4]
and u when [alpha] = 0.7, [beta] = 0.9.

x      t     Approximate       Exact       Absolute error
              solution        solution
            [u.sub.4appr]

1     0.4   0.5356787528    0.5565185584   0.02083980558
2     0.4   0.5092408025    0.6018550029    0.0926142003
1     0.3   0.6273816439    0.6535277868   0.02614614289
2     0.3   0.6242882934    0.7067670289   0.08247873543
1     0.2   0.7153235950    0.7417557951   0.02643220012
0.2   0.2   0.1978265247    0.2080382267   0.01021170194
0.2   0.1   0.1989280524    0.2288404399   0.02991238755
12    0.1   0.8741657132    0.8902825407   0.01611682753
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Title Annotation:Research Article
Author:Hong, Baojian; Lu, Dianchen
Publication:The Scientific World Journal
Article Type:Report
Date:Jan 1, 2014
Words:2987
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