Numerical method for fractional dispersive partial differential equations

In this work, fractional variational iteration method (FVIM) has been applied successfully to find the solution of fractional dispersive partial differential equations of third-order in multi-dimensional spaces. The contemplated graphs explain the character of the solution for different values of fractional order α. The mightiness and accurateness of the proposed techniques are studied with the help of two test examples.


Introduction
Now days, the superb improvements have been visualized in the area of fractional differential equations and fractional calculus.Differential equations containing fractional order derivatives are applied for modeling a numerous of schemes, of which the vital uses are in areas of diffusion equation, polarization, electrode-electrolyte heat conduction, electro-magnetic waves, viscoelasticity and so on [1][2].Because of its fabulous opportunity and uses in numerous fields, an extensive consideration has been given to exact and numerical solutions of fractional differential equations.An abundant deal of researchers has exposed the beneficial use of the fractional calculus in the forming and mechanism of numerous dynamical structures [3][4][5][6][7][8][9][10][11][12][13][14].In addition to forming of different modeling features of these fractional ordered differential equations, the solution procedures and their consistency are rather more vital characteristics.It is also equally significant to knob critical points which becomes reason for unexpected divergence convergence and bifurcation of the solutions of the model.In order to attain the aim of extremely accuracy and consistent solutions, numerous approaches have been suggested to crack the fractional order differential equations.Some of the current analytical/numerical methods are Adomian decomposition method (ADM) [15][16][17][18][19][20], finite difference method [21], Operational matrix method [22], Homotopy analysis method [23,-24], generalized differential transform method [25,26], finite element method [27], fractional differential transform method [28][29]  The key objective of present paper is to use FVIM for solving time-fractional dispersive partial differential equations of third-order [30][31][32][33][34].The FVIM gets an analytical solution in the form of sequence and series.We can get highly accurate results or exact solutions for differential equations with the help of given method FVIM.J. H.He [35][36][37][38] developed a new technique, namely, variational iteration method (VIM) to solve linear and nonlinear differential equations.Odibat et.al [39] and Molliq et al. [40] applied VIM to solve fractional Zakharov-Kuznetsov equations.J. Lu [41] and Sakar et al. [42][43] applied VIM and AVIM to Fornberg-Whitham equation and by others for different models [44][45][46][47].The proposed FVIM do not need perturbation, discretization or linearization unlike the technique argued in the literature.The key disadvantage of the ADM is to compute Adomian polynomials for a nonlinear operator where the procedure is very complex.The disadvantage of the Homotopy perturbation method is to crack functional equation in each iteration, which is sometimes difficult and unachievable.Consequently, the proposed FVIM is considerably easier when compared with ADM and HPM.The summary of this paper is as follows.In section 2 the elementary definitions of fractional calculus and fractional trigonometric function are discussed.In section 3 the solution process of FVIM method is discussed.Two test examples of fractional third-order dispersive partial differential equations are given to elucidate the proposed methods in section 4. At the end, we write the conclusions of the work.

The Proposed FVIM method for the Fractional third-order dispersive partial differential equation
To define solution process of third-order fractional dispersive partial differential equation by using fractional variational iteration method, we study the ensuing fractional differential equation According to the FVIM, a correction functional ( [3]) can be built for above equation as Now by the variational theory  must satisfy      = 0 and 1 + | = = 0. From these equations, we obtain  = −1 and a new correction functional We can build consecutive iterations   ,  ≥ 0 after by using , a common Lagrange's multiplier, that can be obtained by variational theory.The functions  ̃ is restricted variation that means  ̃ = 0.
Consequently, first we elect the Lagrange multiplier , which can be obtained using integration by parts.In this way we can obtain sequences  +1 (, ),  ≥ 0 of the solution and finally the exact solution can be obtained as (, ) = lim →∞   (, ).

Numerical Experiments
In this section, we apply proposed method to some test examples.
By given initial condition, we can take initial solutions as http://www.ispacs.com/journals/cna/2017/cna-00266/International Scientific Publications and Consulting Services Continuing in this way the remaining components of the iteration formulae can be found with the help of Mathematica package.Finally, we can obtain solution as (, ) =  →∞   (, ) =       −       .We can observe from table1 that the above series solutions converges very fast as the absolute error between exact solution and numerical solution is very small.Abbaoui and Cherruault [51] have proved the convergence of this type of series.We compare the tenth-order approximation  10 (, ) with the exact solution for different values of fractional order  to illustrate the efficiency of the FVIM.It can be observed from the numerical results that FVIM works very well for this problem, though low-order approximate solution  10 (, ) is used.We can improve the exactness by using higher-order approximate solutions.We represent numerical results for different particular cases of  in fig.1-4.It can be detected from fig. 1-4 that the solution obtained by proposed method are almost identical with the exact solution when  = 0.5.

Conclusion
In this paper, Fractional variational iteration method (FVIM) has been applied successfully for solving time-fractional third-order dispersive partial differential equations.It is apparently seen that FVIM is a very efficient and powerful numerical method to obtain the approximate solution.The method is used in a direct way without using adomain polynomial, linearization, perturbation or restrictive assumptions.Therefore, FVIM is easier and more convenient than other methods.

Table 1 :
Absolute error for different values of fractional order .
−5http://www.ispacs.com/journals/cna/2017/cna-00266/International Scientific Publications and Consulting Services [51] =  ( + )    − ( + )    .We can observe from table 2 that above series solutions converges very fast as the absolute error between exact solution and numerical solution is very small.Abbaoui and Cherruault[51]have proved the convergence of this type of series.We compare the tenth-order approximation  10 (, ) with the exact solution for different values of fractional order  to illustrate the efficiency of the FVIM.The numerical consequences display that FVIM works very well for this problem, even if low-order approximate solution  10 (, ) is used.The exactness can be enhanced by means of higher-order approximate solutions.Numerical results for different particular cases of  are presented in fig.5-7.It can be detected from fig.5-7that the solution obtained by proposed method are almost identical with the exact solution when  = 0.5.http://www.ispacs.com/journals/cna/2017/cna-00266/ →∞  International Scientific Publications and Consulting Services

Table 2 :
Absolute error for different values of fractional order .