Couple of the Variational Iteration Method and Legendre Wavelets for Nonlinear Partial Differential Equations

This paper develops amodified variational iterationmethod coupled with the Legendre wavelets, which can be used for the efficient numerical solution of nonlinear partial differential equations (PDEs).The approximate solutions of PDEs are calculated in the form of a serieswhose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficientmatrices of the nonlinear terms.Themain advantage of the newmethod is that it can avoid solving the nonlinear algebraic system and symbolic computation. Furthermore, the developed vector-matrix form makes it computationally efficient. The results show that the proposed method is very effective and easy to implement.

In recent years, wavelets have found their way into many different fields of science and engineering.Various wavelets [24][25][26][27][28][29] have been used for studying problems with greater computational complexity and proved to be powerful tools to explore a new direction in solving differential equations.Unlike the variational iteration method that requires symbolic computations, the wavelets method converts the PDE into algebraic equations by the operational matrices, which can be solved by an iterative procedure.It is worthy to mention here that the method based on operational matrices of an orthogonal function for solving differential equations is computer oriented.The problem with this approach is that the algebraic equations may be singular and nonlinear.
Recently, some efficient modifications of ADM (using [55,56]) and VIM or HAM [57] (using Legendre polynomials) are presented to approximate nonhomogeneous terms in nonlinear differential equations.Motivated and inspired by the ongoing research in these areas, we implement Legendre wavelets within the framework of VIM to facilitate the computational work of the method while still keeping the accuracy.The remainder of the paper is organized as follows.Section 2 introduces the VIM.In Section 3, we describe the basic formulation of Legendre wavelets and the operational matrix required for our subsequent development.In Section 4, we propose a new variational iteration method using Legendre wavelets (VIMLW).In order to demonstrate the validity and applicability of VIMLW, four examples are given in Section 5. Finally, a brief summary is presented.

Variational Iteration Method
This section introduces the basic ideas of variational iteration method (VIM).Here a description of the method [15][16][17][18][19][20][21][22][23] is given to handle the general nonlinear problem: where  is a linear operator,  is a nonlinear operator, and () is a known analytic function.According to He's VIM, we can construct a correction functional as follows: where  is a general Lagrange multiplier which can be optimally identified via variational theory and ũ is a restricted variation which means ũ  = 0. Therefore, the Lagrange multiplier  should be first determined via integration by parts.The successive approximation   () ( ≥ 0) of the solution () will be readily obtained by using the obtained Lagrange multiplier and any selective function  0 .
The zeroth approximation  0 may select any function that just meets, at least, the initial and boundary conditions.With  determined, several approximations   (),  ≥ 0, follow immediately.Consequently, the exact solution may be obtained as The VIM depends on the proper selection of the initial approximation  0 ().Finally, we approximate the solution of the initial value problem (1) by the th-order term   ().It has been validated that VIM is capable of effectively, easily, and accurately solving a large class of nonlinear problems.
The integration of the vector Ψ() defined in ( 9) can be obtained as where  is a 2 −1  × 2 −1  matrix given by [58].The derivative of the vector Ψ() can be expressed by where  is the 2 −1  × 2 −1  operational matrix of derivative given by [59].
The integration of (, ) = Ψ  ()Ψ() with respect to variable  can be expressed as Similarly, the integration of (, ) = Ψ  ()Ψ() with respect to variable  can be expressed as The derivative of (, ) = Ψ  ()Ψ() with respect to variable  can be expressed as Similarly, the derivative of (, ) = Ψ  ()Ψ() with respect to variable  can be expressed as

Block Pulse Functions.
The block pulse functions (BPFs) form a complete set of orthogonal functions that are defined on the interval [0, ) by for  = 1, 2, . . ., .It is also known that for arbitrary absolutely integrable function () on [0, ) can be expanded in block pulse functions: in which where   are the coefficients of the block pulse function given by The elementary properties of BPFs are as follows.
Proof.According to the disjointness property of BPFS in ( 16), we have where  =  ⊗ .

Nonlinear Term Approximation.
The Legendre wavelets can be expanded into m-set of block pulse functions as Taking the collocation points as follow, The m-square Legendre matrix Φ × is defined as The operational matrix of product of Legendre wavelets can be obtained by using the properties of BPFs.Let (, ) and (, ) be two absolutely integrable functions, which can be expanded in Legendre wavelets as (, ) = Ψ  ()Ψ() and (, ) = Ψ  ()Ψ(), respectively.

Variational Iteration Method Using Legendre Wavelets
In this section, we present a new modification of variational iteration method using Legendre wavelets (called VIMLW).This algorithm can be implemented for solving nonlinear PDEs effectively.
To deduce the basic relations of our proposed algorithm, consider the following forms of initial value problems: where  and  are linear operator and nonlinear operator, respectively, and (, ) is a known analytic function, subject to the initial condition (, 0).It should be noted here that [(, )] contains the term   /  , where  is a positive integer.
According to the traditional VIM, we can construct the correction functional for (36) as The Lagrange multiplier of ( 37) is In order to improve the performance of VIM, we introduce Legendre wavelets to approximate   (, ) and the nonhomogeneous term (, ) as Now for the nonlinear part, by nonlinear term approximation described in Section 3.3, we have where  is matrix of order 2 −1  × 2   −1   .For the linear part, we have where  is a matrix of order 2 −1  × 2   −1   .Then the iteration formula (37) can be constructed as If  is constant, we have Substituting ( 44) into (42), we have Since we get According to the property of block pulse functions, we obtain where Substituting ( 48) into (45), we have where Finally, we get the iteration formula as follows:

Numerical Examples
To demonstrate the effectiveness and good accuracy of the VIMLW, four different examples will be examined.
We utilize the methods presented in this paper to solve (52) with  = 16 and  = 1.Table 1 shows the approximate solutions for (52) obtained for different points using the variational iteration and VIMLW method.with VIM.It indicates that the approximate solution is quite close to the exact one.
By assuming   (, ) = Ψ  ()  Ψ() and from (54), we have where We employ the methods presented in this paper to solve (54) with  = 16 and  = 1.The numerical results are presented in Table 2 and shown in Figure 2. It is to be noted that only the fifth-order terms are used in evaluating the approximate solutions.The results obtained using the VIMLW are in good agreement with the results of VIM.Example 6.We consider the following equation [40]: where   =    − ,   =   ⊗ (    ).
Table 3 shows the approximate solutions for (56) with  = 16 and  = 1 using the VIM and the VIMLW methods and the results are plotted in Figure 3.It is to be noted that only the fourth-order terms of VIM and VIMLW are used in evaluating the approximate solutions in Table 3.We observe that the approximate solution of (56) with VIMLW gives analogous results to that obtained by VIM, which shows that the approximate solution remains closed form to the exact one.
Example 7. Consider the following Burgers-Poisson (BP) equation of the form [41]: with the initial conditions (, 0) = , and the exact solution is (, ) = (1 + )/(1 + ) − 1. Table 4 shows the approximate solutions to (58) with  = 16 and  = 1 with VIM and VIMLW, and Figure 4 presents the Exact solution and VIMLW approximate solution of Example 7.Only the fourth-order terms are used in evaluating the approximate solutions in Table 4. From Table 4 and Figure 4 the approximate solution of the given Example 7 by using VIMLW is in good agreement with the results of VIM and it clearly appears that the approximate solution remains closed form to exact solution.

Conclusion
A new modification of variational iteration method using Legendre wavelets is proposed and employed to solve a number of nonlinear partial differential equations.The proposed method can give approximations of higher accuracy and closed form solutions if existed.There are four important points to make here.First, unlike the VIM, the VIMLW can easily overcome the difficulty arising in the evaluation integration and the derivative of nonlinear terms and does not need symbolic computation.Second, by using the properties of BPFs, operational matrices of product of Legendre wavelets are derived and utilized to deal with nonlinear terms.Third, compared with Legendre wavelets method, the VIMLW only needs a few iterations instead of solving a system of nonlinear algebraic equations.Fourth and most important, VIMLW is computer oriented and can use existing fast algorithms to reduce the computation cost.

Figure 1 :
Figure 1: Exact solution and VIMLW approximate solution of Example 4.

Figure 2 :
Figure 2: Exact solution and VIMLW approximate solution of Example 5.

Figure 3 :
Figure 3: Exact solution and VIMLW approximate solution of Example 6.

Figure 4 :
Figure 4: Exact solution and VIMLW approximate solution of Example 7.