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

Abstract

This paper develops a modified variational iteration method 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 series whose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The main advantage of the new method 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.

1. Introduction

Nonlinear phenomena are of fundamental importance in applied mathematics and physics and thus have attracted much attention. It is well known that most engineering problems are nonlinear, and it is very difficult to achieve the solution analytically or numerically. The analytical methods commonly used to solve them are very restricted, while the numerical techniques involving discretization of the variables on the other hand give rise to rounding off errors. Considerable attention has been paid to developing an efficient and fast convergent method. Recently, several approximate methods are introduced to find the numerical solutions of nonlinear PDEs, such as Adomian’s decomposition method (ADM) [1–6], homotopy perturbation method (HPM) [7–12], homotopy analysis method (HAM) [13, 14], variational iteration method (VIM) [15–23], and wavelets method [24–29].

The variational iteration method (VIM) proposed by He [15–23] has been shown to be very efficient for handling a wide class of physical problems [16–18, 30–41]. If the exact solution of the nonlinear PDEs exists, the VIM gives rapidly convergent successive approximations; otherwise, a few approximations can be used for numerical purposes. In order to improve the efficiency of these algorithms, several modifications, such as variational iteration method using He’s Polynomials [42–48] or using Adomian’s Polynomials [49–54], have been developed and successfully applied to various engineering problems. However, since the variational iteration method provides the solution as a sequence of iterates, its successive iterations may be very complex, so that the resulting integrations in its iterative relation may be impossible to perform analytically.

In recent years, wavelets have found their way into many different fields of science and engineering. Various wavelets [24–29] 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.

2. Variational Iteration Method

This section introduces the basic ideas of variational iteration method (VIM). Here a description of the method [15–23] 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 is a restricted variation which means . Therefore, the Lagrange multiplier should be first determined via integration by parts. The successive approximation () of the solution will be readily obtained by using the obtained Lagrange multiplier and any selective function . The zeroth approximation may select any function that just meets, at least, the initial and boundary conditions. With determined, several approximations , , follow immediately. Consequently, the exact solution may be obtained as The VIM depends on the proper selection of the initial approximation . 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.

3. Legendre Wavelets

3.1. Legendre Wavelets

Legendre wavelets have four arguments: is any positive integer, (), is the order for Legendre polynomials, and is the normalized time. They are defined on the interval [0, 1) as follows: where , . The coefficient is for orthonormality, the dilation parameter is , and the translation parameter . Here, are the well-known Legendre polynomials of order defined on the interval .

A function defined over may be expanded by Legendre wavelet series as with in (6); denotes the inner product.

If the infinite series in (5) is truncated, then it can be written as where and are matrices given by

A two-dimensional function defined over may be expanded by Legendre wavelet series as with Equation (10) can be written into the discrete form (in matrix form) by where is a matrix given by

The integration and derivative operation matrices of the Legendre wavelets have been derived in [58, 59].

The integration of the vector defined in (9) can be obtained as where is a matrix given by [58].

The derivative of the vector can be expressed by where is the operational matrix of derivative given by [59].

The integration of with respect to variable can be expressed as

Similarly, the integration of with respect to variable can be expressed as

The derivative of with respect to variable can be expressed as

Similarly, the derivative of with respect to variable can be expressed as

3.2. Block Pulse Functions

The block pulse functions (BPFs) form a complete set of orthogonal functions that are defined on the interval by for . It is also known that for arbitrary absolutely integrable function on 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.(1) Disjointness: the BPFs are disjoined with each other in the interval :

for .(2) Orthogonality: the BPFs are orthogonal with each other in the interval :

for .(3) Completeness: the BPFs set is complete when approaches infinity. This means that for every , when approaches to the infinity, Parseval’s identity holds: where

Definition 1. Let and be two matrices of , then .

Lemma 2. Assuming that and are two absolutely integrable functions, which can be expanded in block pulse function as , and respectively, then one has where .

Proof. According to the disjointness property of BPFS in (16), we have

Lemma 3. Let and be two absolutely integrable functions, which can be expanded in block pulse function as and , respectively, one has where .

3.3. Nonlinear Term Approximation

The Legendre wavelets can be expanded into -set of block pulse functions as Taking the collocation points as follow, The m-square Legendre matrix 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 and , respectively.

From (31), we have and let , , .

By employing Lemma 3, we get where .

4. 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 . 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 .

For the linear part, we have where is a matrix of order .

Then the iteration formula (37) can be constructed as

If is constant, we have

When is a function of , the Legendre wavelets are used to approximate as 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:

5. Numerical Examples

To demonstrate the effectiveness and good accuracy of the VIMLW, four different examples will be examined.

Example 4. Consider the regularized long-wave (RLW) equation [39]: with the initial condition and the exact solution is .
By assuming and from (52), we have where , .
We utilize the methods presented in this paper to solve (52) with and . Table 1 shows the approximate solutions for (52) obtained for different points using the variational iteration and VIMLW method. Figure 1 presents the Exact solution and VIMLW approximate solution of Example 4. Note that only the fifth-order term of their solutions is used in evaluating the approximate solutions for Example 4. We can see that the approximate solution obtained with VIMLW gives almost the same results as that with VIM. It indicates that the approximate solution is quite close to the exact one.