# 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 , many important phenomena can be described successfully using the FDEs models such as materials and processes , engineering and physics , dielectric polarization , and quantitative finance . 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)

, the homotopy perturbation method (HPM) , the adomian decomposition method (ADM) , the generalized differential transform method , and so forth .

The variational iteration method (VIM) established in 1999 by He in  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 . 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 . 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 :

[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 . 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 , and the result (29) in , 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 :

[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 . 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).

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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

```
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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