COMPARISON AND COUPLING OF POLYNOMIALS FOR FLIERL-PETVIASHIVILI EQUATION

This paper outlines a comparison of the couplings of He’s and Adomian’s polynomials with correction functional of variational iteration method (VIM) to investigate a solution of Flierl-Petviashivili (FP) equation which plays a very important role in mathematical physics, engineering and applied sciences. These elegant couplings give rise to two modified versions of VIM which are very efficient in solving initial and boundary value problems of diversified nature. Moreover, we also introduces a new transformation which is required for the conversion of the Flierl-Petviashivili equation to a first order initial value problem and a reliable framework designed to overcome the difficulty of the singular point at . 0 = x The proposed modified versions are applied to the reformulated first order initial value problem which gives the solution in terms of transformed variable. The desired series of solution is obtained by making use of the inverse transformation. It is observed that the modification based on He’s polynomials is much easier to implement and is more user friendly. Key wordsFlierl-Petviashivili equation, variational iteration method, He’s polynomials, Adomian’s polynomials, Pade ́ approximants.


INTRODUCTION
The Flierl-Petviashivili (FP) equation is used to model several phenomena in mathematical physics, astrophysics, theory of stellar structure, thermal behavior of a spherical cloud of gas, isothermal gas spheres and theory of thermionic currents, (see Adomian [5], Russell and Shampine [41], Shawagfeh [44] Wazwaz [48]).Several techniques including decomposition and homotopy perturbation have been applied for solving FP equation, (see Adomian [5], Russell and Shampine [41,Shawagfeh [44] Wazwaz [48]).Most of the developed techniques have their limitations like limited convergence, divergent results, linearization, discretization unrealistic assumptions and non-compatibility with the physical problems.He foresaw the potential and compatibility of variational iteration and homotopy perturbation methods and exploited this reliable technique for solving physical problems of diversified nature, (see He [17][18][19][20][21][22][23][24][25][26][27][28][29]).These methods are fully synchronized with the versatile nature of the problems and have been applied to solve a wide class of initial and boundary value problems, (see Abbasbandy [1,2] Abdou and Soliman [6,7], Abassy et.al. [8], Baitha et.al. [10], Bizar and Ghazvini [11], Wakil et.al. [12], Ganji et.al. [13], Ghorbani and Nadifi [14,15], Golbabi and Javidi [16], He [17][18][19][20][21][22][23][24][25][26][27][28][29], Inokuti et.al. [30], Lu [31], Ma and You [32], Momani and Odibat [33], Noor and Mohyud-Din [34][35][36][37][38][39][40], Russel and Shampine [41], Rafi and Danili [42], Sweilman [43]).Abbasbandy introduced the coupling of Adomian's polynomials with the correction functional of the VIM and applied this reliable version for solving Riccati differential and Klein Gordon equations (see Abbasbandy [1,2]).In a later work, Noor and Mohyud-Din exploited this concept for solving various singular and non singular boundary and initial value problems (see Noor and Mohyud-Din [35,40]).Recently, Ghorbani et.al. introduced He's polynomials by splitting the nonlinear term and also proved that He's polynomials are fully compatible with Adomian's polynomials but are easier to calculate and are more user friendly (see Ghorbani et.al. [14,15]).More recently, Noor and Mohyud-Din combined He's polynomials and correction functional of the variational iteration method (VIM) and applied this reliable version to a number of physical problems; (see Noor and Mohyud-Din [37][38][39]).The basic motivation of the present study is the implementation and comparison of these two modified versions of VIM for solving FP equation.The singularity behavior at x = 0 is a difficult element in this type of equations which has been tackled by transforming the Flierl-Petviashivili (FP) equation to a first order initial value problem.The proposed modified versions are applied to the reformulated first order initial value problem which leads the solution in terms of transformed variable.The desired series of solutions is obtained by implementing the inverse transformation.To make the work more concise and for the better understanding of the solution behavior the diagonal Pade´ approximants are applied.It is observed that the modification based on He's polynomials (VIMHP) is much easier to implement as compare to the one (VIMAP) where the so-called Adomian's polynomials along with their complexities are used.It is to be highlighted that the variational iteration method using He's polynomials (VIMHP) has certain advantages as compare to the decomposition method.Firstly, the use of Lagrange multiplier reduces the successive applications of the integral operator and hence minimizes the computational work to a tangible level while still maintaining a very high level of accuracy.Moreover, He's polynomials are easier to calculate as compare to Adomian's polynomials and this gives it a clear edge over the traditional decomposition method.The VIMHP is also independent of the small parameter assumption (which is either not there in the physical problems or difficult to locate) and hence is more convenient to apply as compare to the traditional perturbation method.It is worth mentioning that the VIMHP is applied without any discretization, restrictive assumption or transformation and is free from round off errors.We apply the proposed VIMHP for all the nonlinear terms in the problem without discretizing either by finite difference or spline techniques at the nodes, involves laborious calculations coupled with a strong possibility of the illconditioned resultant equations which is a complicated problem to solve.Moreover, unlike the method of separation of variables that requires initial and boundary conditions, the VIMHP provides the solution by using the initial conditions only.Finally, the variational iteration method using Adomian's polynomials (VIMAP) is also easier to implement as compare to the traditional decomposition method due to the fact that it involves Lagrange multiplier which reduces the successive application of integral operator and hence minimizes the computational work.Moreover, the VIMAP is also independent of the small parameter assumption, discretization, linearization or transformation and so may be considered as a more efficient and convenient algorithm as compare to the traditional techniques which involve these deficiencies.Moreover, the use of Lagrange multiplier in VIMHP gives it a clear advantage over the traditional homotopy perturbation method (HPM) since it avoids the successive application of the integral operator.The proposed modified versions (VIMHP and VIMAP) can be applied to a number of physical problems related to fluid mechanics including Blasius' viscous flow, boundary layer flow with exponential or algebraic properties, Von Karman swirling viscous flow, nonlinear progressive waves in deep water, porous medium, financial mathematics, deep shallow water waves, electrical signals along a telegraph line, digital image processing, telecommunication, signals and systems, beam deflection theory, quantum field theory, relativistic physics, dispersive wave-phenomena, plasma physics, astrophysics, nonlinear optics, engineering and applied sciences, (see Noor and Mohyud-Din [37][38][39]).

MODIFIED VARIATIONAL ITERATION METHODS
The modified variational iteration techniques are obtained by the elegant coupling of correction functional of VIM with He's and Adomian's polynomials.

Variational Iteration Method Using He's Polynomials (VIMHP)
This modified version of variational iteration method is obtained by the elegant coupling of correction functional (2) of variational iteration method (VIM) with He's polynomials and is given by Comparisons of like powers of p give solutions of various orders (see Noor and Mohyud-Din [37][38][39]).

Variational Iteration Method Using Adomian's Polynomials (VIMAP)
This modified version of VIM is obtained by the coupling of correction functional (2) of variational iteration method with Adomian's polynomials and is given by where n A are the so-called Adomian's polynomials and are calculated for various classes of nonlinearities by using the specific algorithm developed in (see Abbasbandy [1,2], Noor and Mohyud-Din [35,40]).

NUMERICAL APPLICATION
In this section, we apply and compare both the modified versions of VIM for solving Flierl-Petviashivili (FP) equation.Consider the generalized variant of the Flierl-Petviashivili equation the generalized Flierl-Petviashivili equation (10,11) can be converted to the following first order initial value problem The correction functional is given as Making the correction functional stationary, the Lagrange multiplier can easily be identified as ( ) Applying the variational iteration method using He's polynomials (VIMHP), we get Comparing the co-efficient of like powers of p, following approximants are obtained , Now, we apply the diagonal Pade´ approximants to the obtained series solution to handle the boundary conditions at infinity because power series in isolation are never useful in boundary value problems because mostly radius of convergence is not sufficiently large, (see Noor and Mohyud-Din [36], Wazwaz [48]).This makes the use of Pade´ approximants very essential in unbounded domain.(10,11) and applying the same transformation, we get the following iterative scheme where n A are the so-called Adomian's polynomials and can be generated for all types of nonlinearities according to the algorithms developed in (Wazwaz [47,48]).First few Adomian's polynomials are as follows , the solution u will be readily obtained upon using the determined Lagrange multiplier and any selective function , 0 u consequently, the solution is given by equation reduces to the standard Flierl-Petviashivili equation.The general series solution for the equation is to be constructed for all possible values of The series solution is used to obtain various Pade´ approximants [2/2],[4/4],[6/6],[8/8]. Roots of the Pade´ approximants to the Flierl-Petviashivili monopole α were obtained by using the limit of the Pade´ approximant [m/m] with other real roots are discarded since these do not meet the physical requirements.

Table 5 .
4shows that the roots of the monopole α converge to -1 as n increases.Now applying the modified version 4.2 (VIMAP) on