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: 𝐿(𝑢)+𝑁(𝑢)=𝑔(𝑡), (1) where 𝐿 is a linear operator, 𝑁 is a nonlinear operator, and 𝑔(𝑡) is a known analytic function. According to He’s VIM, we can construct a correction functional as follows: 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+^𝑡₀𝜆(𝜏)𝐿𝑢_𝑛(𝜏)+𝑁𝑢_𝑛(𝜏)−𝑔(𝜏),𝑛≥0, (2) where 𝜆 is a general Lagrange multiplier which can be optimally identified via variational theory and 𝑢_𝑛 is a restricted variation which means 𝛿𝑢_𝑛=0. Therefore, the Lagrange multiplier 𝜆 should be first determined via integration by parts. The successive approximation 𝑢_𝑛(𝑡) (𝑛≥0) of the solution 𝑢(𝑡) will be readily obtained by using the obtained Lagrange multiplier and any selective function 𝑢₀. The zeroth approximation 𝑢₀ may select any function that just meets, at least, the initial and boundary conditions. With 𝜆 determined, several approximations 𝑢_𝑛(𝑡), 𝑛≥0, follow immediately. Consequently, the exact solution may be obtained as 𝑢(𝑡)=lim_𝑛→∞𝑢_𝑛(𝑡). (3) The VIM depends on the proper selection of the initial approximation 𝑢₀(𝑡). Finally, we approximate the solution of the initial value problem (1) by the 𝑛th-order term 𝑢_𝑛(𝑡). 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 𝜓_𝑛𝑚(𝑡)=𝜓(𝑘,𝑛,𝑚,𝑡) have four arguments: 𝑘 is any positive integer, 𝑛=2𝑛−1 (𝑛=1,2,3,…,2^𝑘−1), 𝑚 is the order for Legendre polynomials, and 𝑡 is the normalized time. They are defined on the interval [0, 1) as follows: 𝜓_𝑛𝑚(𝑡)=𝑚+1/22^(𝑘/2)𝐿_𝑚2^𝑘𝑡−𝑛,for𝑛−12^𝑘≤𝑡≤𝑛+12^𝑘,0,otherwise, (4) where 𝑚=0,1,2,…,𝑀−1, 𝑛=1,2,…,2^𝑘−1. The coefficient 𝑚+1/2 is for orthonormality, the dilation parameter is 𝑎=2^−𝑘, and the translation parameter 𝑏=𝑛2^−𝑘. Here, 𝐿_𝑚(𝑡) are the well-known Legendre polynomials of order 𝑚 defined on the interval [−1,1].

A function 𝑓(𝑡) defined over [0,1) may be expanded by Legendre wavelet series as 𝑓(𝑡)=^+∞_𝑛=1^+∞_𝑚=0𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑡), (5) with 𝑐_𝑛𝑚=𝑓(𝑡),𝜓_𝑛𝑚(𝑡), (6) in (6); ⟨⋅,⋅⟩ denotes the inner product.

If the infinite series in (5) is truncated, then it can be written as 𝑓(𝑡)=^2𝑘−1_𝑛=1^𝑀−1_𝑚=0𝑐_𝑛𝑚𝜓_𝑛𝑚(𝑡)=𝐶^𝑇Ψ(𝑡), (7) where 𝐶 and Ψ(𝑡) are 2^𝑘−1𝑀×1 matrices given by 𝐶(𝑡)=𝑐₁₀,𝑐₁₁,…,𝑐_1𝑀−1,𝑐₂₀,…,𝑐_2𝑀−1𝑐,…,_2𝑘−10,…,𝑐₂^𝑘−1_𝑀−1^𝑇,Ψ(𝑡)=𝜓₁₀(𝑡),𝜓₁₁(𝑡),…,𝜓_1𝑀−1(𝑡)𝜓,…,_2𝑘−10(𝑡),…,𝜓_{2𝑘−1𝑀−1}(𝑡)^𝑇. (8)(9)

A two-dimensional function 𝑓(𝑥,𝑡) defined over [0,1)×[0,1) may be expanded by Legendre wavelet series as 𝑓(𝑥,𝑡)=^2𝑘𝑀2_𝑖=1^𝑘𝑀_𝑗=1𝑐_𝑖𝑗𝜓_𝑖(𝑥)𝜓_𝑗(𝑡)=Ψ^𝑇(𝑥)𝐶Ψ(𝑡), (10) with 𝑐_𝑖𝑗=¹₀𝑓(𝑥,𝑡)𝜓_𝑖(𝑥)𝑑𝑥¹₀𝑓(𝑥,𝑡)𝜓_𝑗(𝑡)𝑑𝑡. (11) Equation (10) can be written into the discrete form (in matrix form) by 𝑓(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐶Ψ(𝑡), (12) where 𝐶 is a 2^𝑘−1𝑀×2^𝑘−1𝑀 matrix given by 𝑐𝐶=^0,0𝑐_0,1⋯𝑐_0,2^𝑘−1_𝑀𝑐_1,0𝑐_1,1⋯𝑐_1,2^𝑘−1_𝑀𝑐⋮⋮⋱⋮_{2𝑘−1𝑀,0}𝑐₂^𝑘−1_𝑀,1⋯𝑐₂^𝑘−1_𝑀,2^𝑘−1_𝑀. (13)

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

The integration of the vector Ψ(𝑡) defined in (9) can be obtained as ^𝑡₀Ψ(𝑠)𝑑𝑠≅𝑃Ψ(𝑡), (14) where 𝑃 is a 2^𝑘−1𝑀×2^𝑘−1𝑀 matrix given by [58].

The derivative of the vector Ψ(𝑡) can be expressed by 𝑑Ψ(𝑡)(𝑡),𝑑𝑡=𝐷Ψ (15) where 𝐷 is the 2^𝑘−1𝑀×2^𝑘−1𝑀 operational matrix of derivative given by [59].

The integration of 𝑢(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐶Ψ(𝑡) with respect to variable 𝑡 can be expressed as ^𝑡₀𝑢(𝑥,𝜏)𝑑𝜏=^𝑡₀Ψ^𝑇(𝑥)𝐶Ψ(𝜏)𝑑𝜏=Ψ^𝑇(𝑥)𝐶^𝑡₀Ψ(𝜏)𝑑𝜏=Ψ^𝑇(𝑥)𝐶𝑃Ψ(𝑡). (16)

Similarly, the integration of 𝑢(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐶Ψ(𝑡) with respect to variable 𝑥 can be expressed as ^𝑥₀𝑢(𝜏,𝑡)𝑑𝜏=^𝑥₀Ψ^𝑇(𝜏)(𝑡)=𝐶Ψ𝑑𝜏_𝑥0Ψ^𝑇(𝜏)𝑑𝜏𝐶Ψ(𝑡)=Ψ^𝑇(𝑥)𝑃^𝑇𝐶Ψ(𝑡). (17)

The derivative of 𝑢(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐶Ψ(𝑡) with respect to variable 𝑡 can be expressed as 𝜕𝑢(𝑥,𝑡)=𝜕Ψ𝜕𝑡^𝑇(𝑥)𝐶Ψ(𝑡)𝜕𝑡=Ψ^𝑇(𝑥)𝐶𝜕Ψ(𝑡)𝜕𝑡=Ψ^𝑇(𝑥)𝐶𝐷Ψ(𝑡). (18)

Similarly, the derivative of 𝑢(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐶Ψ(𝑡) with respect to variable 𝑥 can be expressed as 𝜕𝑢(𝑥,𝑡)=𝜕Ψ𝜕𝑥^𝑇(𝑥)𝐶Ψ(𝑡)=𝜕𝑥𝜕Ψ^𝑇(𝑥)(𝑡)𝜕𝑥𝐶Ψ=Ψ^𝑇(𝑥)𝐷^𝑇𝐶Ψ(𝑡). (19)

3.2. Block Pulse Functions

The block pulse functions (BPFs) form a complete set of orthogonal functions that are defined on the interval [0,𝑏) by 𝑏_𝑖(𝑡)=1,𝑖−1𝑚𝑖𝑏≤𝑡<𝑚𝑏0,elsewhere (20) for 𝑖=1,2,…,𝑚. It is also known that for arbitrary absolutely integrable function 𝑓(𝑡) on [0,𝑏) can be expanded in block pulse functions: 𝑓(𝑡)≃𝜉^𝑇𝐵_𝑚(𝑡), (21) in which 𝜉^𝑇=𝑓₁,𝑓₂,…,𝑓_𝑚,𝐵_𝑚(𝑡)=𝑏₁(𝑡),𝑏₂(𝑡),…,𝑏_𝑚(𝑡), (22) where 𝑓_𝑖 are the coefficients of the block pulse function given by 𝑓_𝑖=𝑚𝑏^𝑏₀𝑓(𝑡)𝑏_𝑖(𝑡)𝑚𝑑𝑡=𝑏_))𝑏^{(𝑖/𝑚)𝑏}_{((𝑖−1/𝑚}𝑓(𝑡)𝑏_𝑖(𝑡)𝑑𝑡. (23) The elementary properties of BPFs are as follows.(1) Disjointness: the BPFs are disjoined with each other in the interval 𝑡∈[0,𝑇): 𝑏_𝑖(𝑡)𝑏_𝑗(𝑡)=𝛿_𝑖𝑗𝑏_𝑖(𝑡) (24)

for 𝑖,𝑗=1,2,…,𝑚.(2) Orthogonality: the BPFs are orthogonal with each other in the interval 𝑡∈[0,𝑇): ^𝑇₀𝑏_𝑖(𝑡)𝑏_𝑗(𝑡)𝑑𝑡=ℎ𝛿_𝑖𝑗 (25)

for 𝑖,𝑗=1,2,…,𝑚.(3) Completeness: the BPFs set is complete when 𝑚 approaches infinity. This means that for every 𝑓∈𝐿²([0,𝑇)), when 𝑚 approaches to the infinity, Parseval’s identity holds: ^𝑇₀𝑓²(𝑡)𝑑𝑡=^∞_𝑖=1𝑓²_𝑖𝑏_𝑖(𝑡)², (26)where 𝑓_𝑖=1ℎ^𝑇₀𝑓(𝑡)𝑏_𝑖(𝑡)𝑑𝑡. (27)

Definition 1. Let 𝐴 and 𝐵 be two matrices of 𝑚×𝑚, then 𝐴⊗𝐵=(𝑎_𝑖𝑗×𝑏_𝑖𝑗)_𝑚𝑚.

Lemma 2. Assuming that 𝑓(𝑡) and 𝑔(𝑡) are two absolutely integrable functions, which can be expanded in block pulse function as 𝑓(𝑡)=𝐹𝐵(𝑡), and 𝑔(𝑡)=𝐺𝐵(𝑡) respectively, then one has 𝑓(𝑡)𝑔(𝑡)=𝐹𝐵(𝑡)𝐵^𝑇(𝑡)𝐺^𝑇=𝐻𝐵(𝑡), (28) where 𝐻=𝐹⊗𝐺.

Proof. According to the disjointness property of BPFS in (16), we have 𝐹𝐵(𝑡)𝐵^𝑇(𝑡)𝐺^𝑇=𝑓₁₁𝑔₁₁𝜙₁(𝑡)𝑓₁₂𝑔₁₂𝜙₂(𝑡)⋯𝑓_1𝑚𝑔_1𝑚𝜙₂(𝑡)=𝐻𝐵(𝑡). (29)

Lemma 3. Let 𝑓(𝑥,𝑡) and 𝑔(𝑥,𝑡) be two absolutely integrable functions, which can be expanded in block pulse function as 𝑓(𝑥,𝑡)=𝐵^𝑇(𝑥)𝐹𝐵(𝑡) and 𝑔(𝑥,𝑡)=𝐵^𝑇(𝑥)𝐺𝐵(𝑡), respectively, one has 𝑓(𝑥,𝑡)𝑔(𝑥,𝑡)=𝐵^𝑇(𝑥)𝐻𝐵(𝑡), (30) where 𝐻=𝐹⊗𝐺.

3.3. Nonlinear Term Approximation

The Legendre wavelets can be expanded into 𝑚-set of block pulse functions as Ψ(𝑡)=Φ_𝑚×𝑚𝐵_𝑚(𝑡). (31) Taking the collocation points as follow, 𝑡_𝑖=𝑖−1/22^𝑘−1𝑀,𝑖=1,2,…,2^𝑘−1𝑀. (32) The m-square Legendre matrix Φ_𝑚×𝑚 is defined as Φ_𝑚×𝑚≜Ψ𝑡₁Ψ𝑡₂⋯Ψ𝑡₂^𝑘−1_𝑀. (33)

The operational matrix of product of Legendre wavelets can be obtained by using the properties of BPFs. Let 𝑓(𝑥,𝑡) and 𝑔(𝑥,𝑡) be two absolutely integrable functions, which can be expanded in Legendre wavelets as 𝑓(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐹Ψ(𝑡) and 𝑔(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐺Ψ(𝑡), respectively.

From (31), we have 𝑓(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐹Ψ(𝑡)=𝐵^𝑇(𝑥)Φ^𝑇_𝑚𝑚𝐹Φ_𝑚𝑚𝐵(𝑡),𝑔(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐺Ψ(𝑡)=𝐵^𝑇(𝑥)Φ^𝑇_𝑚𝑚𝐺Φ_𝑚𝑚𝐵(𝑡), (34) and let 𝐹_𝑏=Φ^𝑇_𝑚𝑚𝐹Φ_𝑚𝑚, 𝐺_𝑏=Φ^𝑇_𝑚𝑚𝐺Φ_𝑚𝑚, 𝐻_𝑏=𝐹_𝑏⊗𝐺_𝑏.

By employing Lemma 3, we get 𝑓(𝑥,𝑡)𝑔(𝑥,𝑡)=𝐵^𝑇(𝑥)𝐻_𝑏𝐵(𝑡)=𝐵^𝑇(𝑥)Φ^𝑇_𝑚𝑚invΦ^𝑇_𝑚𝑚𝐻_𝑏invΦ_𝑚𝑚×Φ_𝑚𝑚𝐵(𝑡)=Ψ^𝑇(𝑥)𝐻Ψ(𝑡), (35) where 𝐻=inv(Φ^𝑇_𝑚𝑚)𝐻_𝑏inv(Φ_𝑚𝑚).

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: 𝐿[𝑢(𝑥,𝑡)]+𝑁[𝑢(𝑥,𝑡)]=𝑔(𝑥,𝑡),𝑥∈[0,1],𝑡>0, (36) where 𝐿 and 𝑁 are linear operator and nonlinear operator, respectively, and 𝑔(𝑥,𝑡) is a known analytic function, subject to the initial condition 𝑢(𝑥,0). It should be noted here that 𝐿[𝑢(𝑥,𝑡)] contains the term 𝜕^𝑚𝑢/𝜕𝑡^𝑚, where 𝑚 is a positive integer.

According to the traditional VIM, we can construct the correction functional for (36) as 𝑢_𝑘+1(𝑥,𝑡)=𝑢_𝑘(𝑥,𝑡)+^𝑡₀𝜆𝐿𝑢_𝑘(𝑥,𝜏)+𝑁𝑢_𝑘()(𝑥,𝜏−𝑔𝑥,𝜏)𝑑𝜏. (37) The Lagrange multiplier of (37) is 𝜆()()𝑡,𝜏=−𝑡−𝜏^𝑚−1()!=()𝑚−1−1^𝑚(𝜏−𝑡)^𝑚−1()!.𝑚−1 (38)

In order to improve the performance of VIM, we introduce Legendre wavelets to approximate 𝑢_𝑘(𝑥,𝑡) and the nonhomogeneous term 𝑔(𝑥,𝑡) as 𝑢_𝑘(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐶_𝑘Ψ(𝑡),𝑔(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐺Ψ(𝑡). (39)

Now for the nonlinear part, by nonlinear term approximation described in Section 3.3, we have 𝑁𝑢_𝑘(𝑥,𝑡)=Ψ^𝑇(𝑥)𝑁_𝑘Ψ(𝑡), (40) where 𝑁 is matrix of order 2^𝑘−1𝑀×2^𝑘′−1𝑀^′.

For the linear part, we have 𝐿𝑢_𝑘(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐿_𝑘Ψ(𝑡), (41) where 𝐿 is a matrix of order 2^𝑘−1𝑀×2^𝑘′−1𝑀^′.

Then the iteration formula (37) can be constructed as Ψ^𝑇(𝑥)𝐶_𝑘+1Ψ(𝑡)=Ψ^𝑇(𝑥)𝐶_𝑘Ψ(𝑡)+^𝑡₀𝜆Ψ^𝑇(𝑥)𝐿_𝑘+𝑁_𝑘−𝐺Ψ(𝜏)𝑑𝜏. (42)

If 𝜆 is constant, we have 𝐶_𝑘+1=𝐶_𝑘+𝜆𝐿_𝑘+𝑁_𝑘−𝐺𝑃. (43)

When 𝜆 is a function of 𝜏, the Legendre wavelets are used to approximate 𝜆(𝜏) as𝜆(𝑡,𝜏)=Ψ^𝑇(𝑡)𝑆Ψ(𝜏). (44) Substituting (44) into (42), we have Ψ^𝑇(𝑥)𝐶_𝑘+1Ψ(𝑡)=Ψ^𝑇(𝑥)𝐶_𝑘Ψ(𝑡)+^𝑡₀Ψ^𝑇(𝑥)𝐿_𝑘+𝑁_𝑘−𝐺Ψ(𝜏)Ψ^𝑇(𝜏)𝑆^𝑇Ψ(𝑡)𝑑𝜏. (45) Since Ψ(𝜏)=𝜙₁₁𝜙₁₂⋯𝜙_1𝑚𝜙₂₁𝜙₂₂⋯𝜙_2𝑚𝜙⋮⋮⋱⋮_𝑚1𝜙_𝑚2⋯𝜙_𝑚𝑚𝐵₁(𝜏)𝐵₂(𝜏)⋮𝐵_𝑚(𝜏)=^𝑚_𝑖∑₌₁𝜙_1𝑖𝐵_𝑖(𝜏)_𝑚𝑖∑₌₁𝜙_2𝑖𝐵_𝑖(𝜏)_𝑚⋮_𝑖∑₌₁𝜙_𝑚𝑖𝐵_𝑖(𝜏), (46) we get Ψ(𝜏)Ψ^𝑇(𝜏)=^𝑚_𝑖∑₌₁𝜙_1𝑖𝐵_𝑖(𝑡)_𝑚𝑖∑₌₁𝜙_2𝑖𝐵_𝑖(𝑡)_𝑚⋮_𝑖∑₌₁𝜙_𝑚𝑖𝐵_𝑖(𝑡)×∑^𝑚_𝑖=1𝜙_1𝑖𝐵_𝑖(𝑡)∑^𝑚_𝑖=1𝜙_2𝑖𝐵_𝑖(𝑡)⋯∑^𝑚_𝑖=1𝜙_𝑚𝑖𝐵_𝑖(𝑡)=^𝑚_𝑗=1^𝑚_𝑖=1𝜙_𝑗𝑖𝐵_𝑖(𝑡)². (47) According to the property of block pulse functions, we obtain Ψ(𝜏)Ψ^𝑇(𝜏)=^𝑚𝑚_𝑗=1𝑖=1𝜙²_𝑗𝑖𝐵_𝑖(𝑡)=𝜙²₁₁𝜙²₁₂⋯𝜙²_1𝑚𝜙²₂₁𝜙²₂₂⋯𝜙²_2𝑚𝜙⋮⋮⋱⋮_2𝑚1𝜙²_𝑚2⋯𝜙²_𝑚𝑚𝐵₁(𝑡)𝐵₂(𝑡)⋮𝐵_𝑚(𝑡)=𝜙11⋯1₂₁₁𝜙²₁₂⋯𝜙²_1𝑚𝜙²₂₁𝜙²₂₂⋯𝜙²_2𝑚𝜙⋮⋮⋱⋮_2𝑚1𝜙²_𝑚2⋯𝜙²_𝑚𝑚invΦ_𝑚×𝑚Ψ(𝑡(𝑡,)11⋯1=𝐻Ψ)11⋯1 (48) where 𝜙𝐻=²¹¹𝜙²₁₂⋯𝜙²_1𝑚𝜙²₂₁𝜙²₂₂⋯𝜙²_2𝑚𝜙⋮⋮⋱⋮_2𝑚1𝜙²_𝑚2⋯𝜙²_𝑚𝑚invΦ_𝑚×𝑚. (49)

Substituting (48) into (45), we have Ψ^𝑇(𝑥)𝐶_𝑘+1Ψ(𝑡)=Ψ^𝑇(𝑥)𝐶_𝑘Ψ(𝑡)+^𝑡₀Ψ^𝑇(𝑥)𝐿_𝑘+𝑁_𝑘(𝜏𝑆−𝐺×𝐻Ψ)11⋯1^𝑇Ψ(𝑡)𝑑𝜏=Ψ^𝑇(𝑥)𝐶_𝑘Ψ(𝑡)+Ψ^𝑇(𝑥)𝐿_𝑘+𝑁_𝑘(𝑡𝑆−𝐺×𝐻𝑃Ψ)11⋯1^𝑇Ψ(𝑡)=Ψ^𝑇(𝑥)𝐶_𝑘Ψ(𝑡)+Ψ^𝑇(𝑥)𝐿_𝑘+𝑁_𝑘(𝑡)Ψ−𝐺×𝐻𝑃Ψ^𝑇(𝑡)𝑆11⋯1^𝑇=Ψ^𝑇(𝑥)𝐶_𝑘Ψ(𝑡)+Ψ^𝑇(𝑥)𝐿_𝑘+𝑁_𝑘(𝑡)−𝐺×𝐻𝑃𝐻Ψ𝑠, (50) where 𝑠=[^11⋯1]𝑆[^11⋯1]^𝑇.

Finally, we get the iteration formula as follows: 𝐶_𝑘+1=𝐶_𝑘+𝐿_𝑘+𝑁_𝑘−𝐺𝐻𝑃𝐻11⋯1𝑆11⋯1^𝑇. (51)

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]: 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑢²2_𝑥=0,−∞<𝑥<∞,𝑡>0 (52) with the initial condition 𝑢(𝑥,0)=𝑥 and the exact solution is 𝑢(𝑥,𝑡)=(𝑥/(1+𝑡)).
By assuming 𝑢_𝑘(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐶_𝑘Ψ(𝑡) and from (52), we have 𝐿𝑢_𝑘=𝜕𝑢_𝑘−𝜕𝜕𝑡³𝑢_𝑘𝜕𝑥²𝜕𝑡=Ψ^𝑇(𝑥)𝐿_𝑘Ψ(𝑡),𝑁𝑢_𝑘=𝑢_𝑘𝜕𝑢_𝑘𝜕𝑥=Ψ^𝑇(𝑥)𝑁_𝑘Ψ(𝑡), (53) where 𝐿_𝑘=𝐶_𝑘𝐷−(𝐷^𝑇)²𝐶_𝑘𝐷, 𝑁_𝑘=(𝐷^𝑇𝐶_𝑘)⊗𝐶_𝑘.
We utilize the methods presented in this paper to solve (52) with 𝑀=16 and 𝑘=1. 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.

Table 1 

Numerical values when 𝑡=0.25, 0.50, 0.75, and 1.0 for (52).

Figure 1 

Exact solution and VIMLW approximate solution of Example 4.

Example 5. Consider the following equation [39]: 𝑢_𝑡+12𝑢𝑢_𝑥𝑥=𝑢_𝑥𝑥𝑥,−∞<𝑥<∞,𝑡>0 (54) with the initial conditions 𝑢(𝑥,0)=𝑥², and the exact solution is 𝑢(𝑥,𝑡)=(𝑥²/(1+𝑡)).
By assuming 𝑢_𝑘(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐶_𝑘Ψ(𝑡) and from (54), we have 𝐿_𝑘𝑢_𝑘=𝜕𝑢_𝑘−𝜕𝜕𝑡³𝑢_𝑘𝜕𝑥³=Ψ^𝑇(𝑥)𝐿_𝑘Ψ(𝑡),𝑁_𝑘𝑢_𝑘=𝑢_𝑘2𝜕²𝑢_𝑘𝜕𝑥²=Ψ^𝑇(𝑥)𝑁_𝑘Ψ(𝑡), (55) where 𝐿_𝑘=(𝐷^𝑇)³𝐶_𝑘𝐷, 𝑁_𝑘=((1/2)(𝐷^𝑇)²𝐶_𝑘)⊗𝐶_𝑘.
We employ the methods presented in this paper to solve (54) with 𝑀=16 and 𝑘=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.

Table 2 

Numerical values when 𝑡=0.25, 0.50, 0.75, and 1.0 for (54).

Figure 2 

Exact solution and VIMLW approximate solution of Example 5.

Example 6. We consider the following equation [40]: 𝑢_𝑡(𝑥,𝑡)+𝑢(𝑥,𝑡)𝑢_𝑥(𝑥,𝑡)=𝑔(𝑥,𝑡),𝑡>0,𝑥∈𝑅,0<𝛼≤1 (56) with the initial conditions 𝑢(𝑥,0)=0 and the exact solution is 𝑢(𝑥,𝑡)=𝑥𝑡, where 𝑔(𝑥,𝑡)=𝑥+𝑥𝑡².
By assuming 𝑢_𝑘(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐶_𝑘Ψ(𝑡), 𝑔(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐺Ψ(𝑡), we have 𝐿_𝑘𝑢_𝑘=𝑢_𝑡(𝑥,𝑡)−𝑔(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐿_𝑘Ψ(𝑡),𝑁_𝑘𝑢_𝑘=𝑢_𝑘𝜕𝑢_𝑘𝜕𝑥=Ψ^𝑇(𝑥)𝑁_𝑘Ψ(𝑡), (57) where 𝐿_𝑘=𝐶_𝑘𝐷−𝐺, 𝑁_𝑘=𝐶_𝑘⊗(𝐷^𝑇𝐶_𝑘).

Table 3 shows the approximate solutions for (56) with 𝑀=16 and 𝑘=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.

Table 3 

Numerical values when 𝑡=0.25, 0.50, 0.75, and 1.0 for (56).

Figure 3 

Exact solution and VIMLW approximate solution of Example 6.

Example 7. Consider the following Burgers-Poisson (BP) equation of the form [41]: 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑢_𝑥+𝑢𝑢_𝑥=3𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥,−∞<𝑥<∞,𝑡>0 (58) with the initial conditions 𝑢(𝑥,0)=𝑥, and the exact solution is 𝑢(𝑥,𝑡)=(1+𝑥)/(1+𝑡)−1.
By assuming 𝑢_𝑘(𝑥,𝑡)=Ψ^𝑇(𝑥)𝐶_𝑘Ψ(𝑡), we have 𝐿𝑢_𝑘=𝜕𝑢_𝑘−𝜕𝜕𝑡³𝑢_𝑘𝜕𝑥²+𝜕𝑡𝜕𝑢_𝑘𝜕𝑥=Ψ^𝑇(𝑥)𝐿_𝑘Ψ(𝑡), (59) where 𝐿_𝑘=𝐶_𝑘𝐷−(𝐷^𝑇)²𝐶_𝑘𝐷+𝐷^𝑇𝐶_𝑘.
And 𝑁𝑢_𝑘=𝑢_𝑘𝜕𝑢_𝑘𝜕𝑥−3𝜕𝑢_𝑘𝜕𝜕𝑥²𝑢_𝑘𝜕𝑥²−𝑢_𝑘𝜕³𝑢_𝑘𝜕𝑥³=Ψ^𝑇(𝑥)𝑁_𝑘Ψ(𝑡), (60) where 𝑁_𝑘=𝐶_𝑘⊕(𝐷^𝑇𝐶_𝑘)−3(𝐷^𝑇𝐶_𝑘)⊕[(𝐷^𝑇)²𝐶_𝑘]−𝐶_𝑘⊕[(𝐷^𝑇)³𝐶_𝑘].
Table 4 shows the approximate solutions to (58) with 𝑀=16 and 𝑘=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.

Table 4 

Numerical values when 𝑡=0.25, 0.50, 0.75, and 1.0 for (58).

Figure 4 

Exact solution and VIMLW approximate solution of Example 7.

6. 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.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 41105063). The authors are very grateful to the reviewers for carefully reading the paper and for thier comments and suggestions which have improved the paper.