Approximate Solutions for Nonlinear Initial Value Problems Using the Modified Variational Iteration Method

Abstract

We have used the modified variational iteration method (MVIM) to find the approximate solutions for some nonlinear initial value problems in the mathematical physics, via the Burgers-Fisher equation, the Kuramoto-Sivashinsky equation, the coupled Schrodinger-KdV equations, and the long-short wave resonance equations together with initial conditions. The results of these problems reveal that the modified variational iteration method is very powerful, effective, convenient, and quite accurate to systems of nonlinear equations. It is predicted that this method can be found widely applicable in engineering and physics.

1. Introduction

Nonlinear partial differential equations are known to describe a wide variety of phenomena not only in physics, where applications extend over magnetofluid dynamics, water surface gravity waves, electromagnetic radiation reactions, and ion acoustic waves in plasma, but also in biology, chemistry, and several other fields. It is one of the important tasks in the study of the nonlinear partial differential equations to seek exact and explicit solutions. In the past several decades both mathematicians and physicists have made many attempts in this direction. Various methods for obtaining exact solutions to nonlinear partial differential equations have been proposed. Among these methods are the Bäcklund transformation method [1, 2], the Hirota’s bilinear method [3], the inverse scattering transform method [4], extended tanh method [5–7], the Adomian-Pade approximation [8–10], the variational method [11–14], the variational iteration method [15, 16], the various Lindstedt-Poincare methods [17–20], the Adomian decomposition method [8, 21, 22], the 𝐹-expansion method [23, 24], the Exp-function method [25–27] and others [28–35]. Zayed et al. [36] investigated the travelling wave solutions for nonlinear initial value problems using the homotopy perturbation method. The modified variational iteration method is the couples of the variational iteration method with the homotopy pertirbation method. Recently Akbarzade and Langari [37] and Mei and Zhang [38] had used the modified variational iteration method for some nonlinear partial differential equations.

The main objective of the present paper is to use the modified variational iteration method (MVIM) for constructing the traveling wave solutions of the following nonlinear partial differential equations in mathematical physics:(i)the nonlinear Burgers-Fisher equation [39]: 𝑢_𝑡−𝑢_𝑥𝑥−𝑢𝑢_𝑥−𝑢(1−𝑢)=0, (1.1)(ii)the nonlinear Kuramoto-Sivashinsky equation [40]: 𝑢_𝑡+𝑎𝑢𝑢_𝑥+𝑏𝑢_𝑥𝑥+𝑐𝑢_{𝑥𝑥𝑥𝑥}=0, (1.2)(iii)the nonlinear coupled Schrodinger KdV equations [41]: 𝑖𝑢_𝑡−𝑢_𝑥𝑥𝑣−𝑢𝑣=0,_𝑡+𝑣_𝑥𝑥𝑥+6𝑣𝑣_𝑥−|𝑢|²_𝑥=0, (1.3)(iv)the nonlinear long-short wave resonance equations [42]: 𝑖𝑢_𝑡+𝑢_𝑥𝑥𝑣−𝑢𝑣=0,_𝑡+𝛽|𝑢|²_𝑥=0, (1.4)

together with initial conditions, where 𝑎, 𝑏, 𝑐, and 𝛽 are arbitrary constants while 𝑖=√−1. It is interesting to point out that (1.1) includes the convection term 𝑢𝑢_𝑥 and the dissipation term 𝑢_𝑥𝑥. Equation (1.2) describes the fluctuations of the position of a flame front, the motion of a fluid going down a vertical wall, or a spatially uniform oscillating chemical reaction in a homogeneous medium. Equation (1.3) describe various processes in dusty plasma such as Langmuri, dust-acoustic wave and electromagnetic waves, while in (1.4) 𝑢 is the envelope of the short wave and is a complex function, and 𝑣 is the amplitude of the long wave which is a real function.

2. Basic Idea of He’s Homotopy Perturbation Method

We illustrate the following nonlinear differential equation [43–54]: 𝐴(𝑢)−𝑓(𝑟)=0,𝑟∈Ω, (2.1) with the boundary conditions: 𝐵𝑢,𝜕𝑢𝜕𝑛=0,𝑟∈Γ, (2.2) where 𝐴 is a general differential operator, 𝐵 is a boundary operator, 𝑓(𝑟) is an analytic function, and Γ is the boundary of the domain Ω. Generally speaking, the operator 𝐴 can be divided into two parts 𝐿 and 𝑁, where 𝐿 is linear but 𝑁 is nonlinear. Therefore, (2.1) can be rewritten in the following form: 𝐿(𝑢)+𝑁(𝑢)−𝑓(𝑟)=0. (2.3) By the homotopy technique, we construct a homotopy 𝑉(𝑟,𝑝)∶Ω×[0,1]→𝑅which satisfies 𝐻𝑉,𝑝=1−𝑝𝐿(𝑉)−𝐿𝑢₀+𝑝𝐴(𝑉)−𝑓(𝑟)=0,𝑟∈Ω, (2.4) or 𝐻𝑉,𝑝=𝐿(𝑉)−𝐿𝑢₀+𝑝𝐿𝑢₀+𝑝𝑁(𝑉)−𝑓(𝑟)=0,𝑟∈Ω, (2.5) where 𝑝∈[0,1] is an embedding parameter and 𝑢₀ is an initial approximation of (2.1) which satisfies the boundary conditions (2.2). Obviously, from (2.4) and (2.5), we have 𝐻(𝑉,0)=𝐿(𝑉)−𝐿𝑢₀𝐻()(𝑉)(𝑟)=0,𝑉,1=𝐴−𝑓=0. (2.6) The changing process of 𝑝 from zero to unity is just that of 𝑉(𝑟,𝑝) from 𝑢₀(𝑟)to 𝑢(𝑟). In topology, this is called the deformation but 𝐿(𝑉)−𝐿(𝑢₀)and 𝐴(𝑉)−𝑓(𝑟)are called the homotopies. According to the homotopy perturbation method, we can first use the embedding parameter “𝑝” as a small parameter and assume that (2.4) or (2.5) can be written as a power series in “𝑝” as follows: 𝑉=𝑉₀+𝑝𝑉₁+𝑝²𝑉₂+⋯. (2.7) Letting 𝑝→1 in (2.7), the approximate solution of (2.3) takes the following form: 𝑢=lim_𝑝→1𝑉=𝑉₀+𝑉₁+𝑉₂+⋯. (2.8) The combination of the perturbation method and the homotopy method is called the homotopy perturbation method which has eliminated the limitations of the traditional perturbation methods. On the other hand, this technique can have full advantage of the traditional perturbation techniques.

3. Variational Iteration Method

Consider the following nonhomogeneous, nonlinear partial differential equation: 𝐿𝑢(𝑥,𝑡)+𝑁(𝑢(𝑥,𝑡))=𝑓(𝑥,𝑡), (3.1) where 𝐿 is a linear differential operator with respect to time, 𝑁 is a nonlinear operator and 𝑓(𝑥,𝑡) is a given function.

According to the variational iteration method, we can construct correct functionals as follows:𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡₀𝜆(𝜏)𝐿𝑢_𝑛(𝑥,𝜏)+𝑁𝑢_𝑛(𝑥,𝜏)−𝑓(𝑥,𝜏)𝑑𝜏 (3.2) which is variational iteration algorithm I, and 𝜆 is a general Lagrange multipliers. The variational iteration method can be identified optimally via variational theory [6, 7]. The second term on the right-hand side in (3.2) is called the corrections, the subscript 𝑛 denotes the 𝑛th order approximation, and 𝑢_𝑛 is restricted variations. We can assume that the aforementioned correctional functionals are stationary (i.e., 𝛿𝑢_𝑛+1=0), and then the Lagrange multipliers can be identified.

Now we can start with the given initial approximation and by the previous iteration formulas we can obtain the approximate solutions. He [55] has used the fractional iteration method to obtain the approximate solutions for nonlinear fractional differential equations.

4. The Modified Variational Iteration Method

To convey the basic idea of the variational homotopy perturbation method [2, 3], we consider the following general differential equation: 𝐿𝑢+𝑁(𝑢)=𝑓(𝑥), (4.1) where 𝐿 is a linear differential operator, 𝑁 is a nonlinear operator and 𝑓(𝑥,𝑡) is an inhomogeneous term. According to the variational iteration method [4–13], we can construct a correct functional as follows: 𝑢_𝑛+1=𝑢_𝑛+^𝑥₀𝜆(𝜏)𝐿𝑢_𝑛+𝑁𝑢_𝑛−𝑓(𝜏)𝑑𝜏, (4.2) where 𝜆 is a Lagrange multipliers, which can be identified optimally via variational theory [6, 7]. The subscripts 𝑛denote the 𝑛th approximation, and 𝑢_𝑛 is considered as a restricted variation. That is, 𝛿𝑢_𝑛=0 is called a correct functional. Now, we apply the homotopy perturbation method to (4.2): ^∞_𝑖=0𝑝^𝑖𝑢_𝑖=𝑢₀+^𝑥₀𝜆(𝜏)𝑁^∞_𝑖=0𝑝^𝑖𝑢_𝑖𝑑𝜏−^𝑥₀𝜆(𝜏)𝑓(𝜏)𝑑𝜏, (4.3) which is the variational iteration algarithm II and is formulated by the modified variational iteration method. The embedding parameter 𝑝∈[0,1] can be considered as an expanding parameter [14–19].

The homotopy perturbation method uses the homotopy parameter 𝑝 as an expanding parameter [14–19] to obtain 𝑢=^∞_𝑖=0𝑝^𝑖𝑢_𝑖=𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯. (4.4) If 𝑝→1, then (4.4) becomes the approximate solution of the following form: 𝑢=𝑢₀+𝑢₁+𝑢₂+⋯. (4.5)

A comparison of like powers of 𝑝 gives solutions of various orders.

The application of the Adomain polynomial is too complex so that we consider the variational iteration method and He’s polynomial to calculate the approximate solutions (see, e.g., [56–60]).

5. Applications

In this section, we construct the approximate solutions for some nonlinear evolution equations in the mathematical physics, namely, the Burgers-Fisher equation (1.1), the Kuramoto-Sivashinsky equation (1.2), the coupled Schrodinger-KdV equations (1.3), and the long-short wave resonance equations (1.4) together with initial conditions by using the the modified variational iteration method. Applications of this method to similar equations can be found in [61–67].

5.1. Approximate Solution of Burgers-Fisher Equation with Initial Conditions Using Modified Variational Iteration Method

In this subsection, we use the MVIM to find the solution 𝑢(𝑥,𝑡) of an initial value problem consisting of the nonlinear Burgers-Fisher equation (1.1) and the following initial condition [39]: 𝑉₀()=1=𝑢𝑥,02𝑥1+tanh4. (5.1) This initial condition follows by setting 𝑡=0 in the following exact solution of (1.1): 𝑢()=1𝑥,𝑡211+tanh45𝑥+2𝑡. (5.2) This exact solution has been derived by Wazwaz [39] using the tanh-coth method. To this end, we construct the modified variational iteration method for the nonlinear Burgers- Fisher equation (1.1) which satisfies^∞_𝑖=0𝑝^𝑖𝑢_𝑖=𝑢(𝑥,0)+𝑝^𝑡₀^∞_𝑖=0𝑝^𝑖𝑢_{𝑖,𝑥𝑥}+^∞_𝑖=0𝑝^𝑖𝑢_𝑖^∞_𝑖=0𝑝^𝑖𝑢_𝑖,𝑥+^∞_𝑖=0𝑝^𝑖𝑢_𝑖−^∞_𝑖=0𝑝^𝑖𝑢_𝑖²𝑑𝜏. (5.3) Comparing the different coefficient of like power of 𝑝, we have 𝑝⁰()=1∶𝑢𝑥,0211+tanh45𝑥+2𝑡,𝑝¹∶𝑢₁(𝑥,𝑡)=^𝑡₀𝑢_0,𝑥𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₀(𝑥,𝜏)𝑢_0,𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₀(𝑥,𝜏)𝑑𝜏−^𝑡₀𝑢²₀()𝑝𝑥,𝜏𝑑𝜏,₂∶𝑢₂(𝑥,𝑡)=^𝑡₀𝑢_1,𝑥𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₀(𝑥,𝜏)𝑢_1,𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₁(𝑥,𝜏)𝑢_0,𝑥()+𝑥,𝜏𝑑𝜏_𝑡0𝑢₁(𝑥,𝜏)𝑑𝜏−2^𝑡₀𝑢₀(𝑥,𝜏)𝑢₁()𝑝𝑥,𝜏𝑑𝜏,₃∶𝑢₃(𝑥,𝑡)=^𝑡₀𝑢_2,𝑥𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₀(𝑥,𝜏)𝑢_2,𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₁(𝑥,𝜏)𝑢_1,𝑥()+𝑥,𝜏𝑑𝜏_𝑡0𝑢₂(𝑥,𝜏)𝑢_0,𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₂(𝑥,𝜏)𝑑𝜏+2^𝑡₀𝑢₀(𝑥,𝜏)𝑢₂()+𝑥,𝜏𝑑𝜏_𝑡0𝑢²₁(𝑥,𝜏)𝑑𝜏, (5.4) and so on. Consequently after some reduction with help of Maple or Mathematica, we get: 𝑉₀()=1𝑥,𝑡2𝑥1+tanh4,𝑉₁(𝑥,𝑡)=5𝑡16sech²𝑥4,𝑉₂(𝑥,𝑡)=−25𝑡²128sech²𝑥4𝑥tanh4,𝑉₃(𝑥,𝑡)=−375𝑡³3072sech⁴𝑥4+125𝑡³1536sech²𝑥4. (5.5) In this manner the other components can be obtained.

Substituting from (5.5) into (4.5), we obtain the following approximate solution of the initial value problem (1.1) and (5.1): 𝑢()=1𝑥,𝑡2𝑥1+tanh4+5𝑡16sech²𝑥4−25𝑡²128sech²𝑥4𝑥tanh4−375𝑡³3072sech⁴𝑥4+125𝑡³1536sech²𝑥4+⋯. (5.6) Note that if we expand the exact solution (5.2) in Taylor series near 𝑡=0, we obtain the approximate solution (5.6). To demonstrate the convergence of the variational homotopy perturbation method, the results of the numerical example are presented and only few terms are required to obtain accurate solutions. The accuracy of the modified variational iteration method for the nonlinear Burgers-Fisher equation is controllable and absolute errors are very small with the present choice of 𝑥 and 𝑡. These results are listed in Table 1. Both the exact solution (5.2) and the approximate solution (5.6) obtained for the first three approximations are plotted in Figure 1. There are no visible differences in diagrams. It is also evident that when more terms for the modified variational iteration method are computed, the numerical results get much more closer to the corresponding exact solution with the initial condition (5.1).

Table 1 

The approximate solution (5.6) in comparison with the exact solution (5.2) if 𝑡=0.5.

(a)

(b)

Figure 1 

The approximate solution (5.6) shown in (b) in comparison with the exact solution (5.2) shown in (a) if 𝑡=0.5.

5.2. Approximate Solution of the Nonlinear Kuramoto-Sivashinsky Equation with Initial Conditions Using MVIM

In this subsection, we use the MVIM to find the solution 𝑢(𝑥,𝑡) of an initial-value problem consisting of the nonlinear Kuramoto-Sivashinsky equation (1.2) with the following initial condition [40]: 𝑉₀=𝑢(𝑥,0)=15𝑏𝜆19𝑎2+3tanh𝜆𝑥2−tanh³𝜆𝑥2, (5.7) where 𝜆=−𝑏/19𝑐,𝑏/𝑐<0, and 𝑎,𝑏,and𝑐 are constants. This initial condition follows by setting 𝑡=0 in the following exact solution of (1.2): 𝑢(𝑥,𝑡)=15𝑏𝜆𝜆19𝑎2+3tanh2𝑥−30𝑏𝜆𝑡19−tanh³𝜆2𝑥−30𝑏𝜆𝑡19. (5.8) This exact solution has been derived by Wazwaz [40] using the tanh method and the extended tanh method. Let us now apply the MVIM to the initial value problem (1.2) and (5.7). To this end, we construct an MVIM for the nonlinear Kuramoto-Sivashinsky equation (1.2) which satisfies ^∞_𝑖=0𝑝^𝑖𝑢_𝑖=𝑢(𝑥,0)−𝑝^𝑡₀𝑎^∞_𝑖=0𝑝^𝑖𝑢_𝑖^∞_𝑖=0𝑝^𝑖𝑢_𝑖,𝑥+𝑏^∞_𝑖=0𝑝^𝑖𝑢_{𝑖,𝑥𝑥}+𝑐^∞_𝑖=0𝑝^𝑖𝑢_{𝑖,𝑥𝑥𝑥𝑥}𝑑𝜏. (5.9) Comparing the different coefficient of like power of 𝑝, we have𝑝⁰∶𝑢(𝑥,0)=15𝑏𝜆19𝑎2+3tanh𝜆𝑥2−tanh³𝜆𝑥2,𝑝¹∶𝑢₁(𝑥,𝑡)=−𝑎^𝑡₀𝑢₀(𝑥,𝜏)𝑢_0,𝑥(𝑥,𝜏)𝑑𝜏−𝑏^𝑡₀𝑢_0,𝑥𝑥(𝑥,𝜏)𝑑𝜏−𝑐^𝑡₀𝑢_{0,𝑥𝑥𝑥𝑥}()𝑝𝑥,𝜏𝑑𝜏,₂∶𝑢₂(𝑥,𝑡)=−𝑎^𝑡₀𝑢₀(𝑥,𝜏)𝑢_1,𝑥(𝑥,𝜏)𝑑𝜏−𝑎^𝑡₀𝑢₁(𝑥,𝜏)𝑢_0,𝑥(𝑥,𝜏)𝑑𝜏−𝑏^𝑡₀𝑢_1,𝑥𝑥(𝑥,𝜏)𝑑𝜏−𝑐^𝑡₀𝑢_{1,𝑥𝑥𝑥𝑥}()𝑝𝑥,𝜏𝑑𝜏,₃∶𝑢₃(𝑥,𝑡)=−𝑎^𝑡₀𝑢₀(𝑥,𝜏)𝑢_2,𝑥(𝑥,𝜏)𝑑𝜏−𝑎^𝑡₀𝑢₁(𝑥,𝜏)𝑢_1,𝑥(𝑥,𝜏)𝑑𝜏−𝑎^𝑡₀𝑢₂(𝑥,𝜏)𝑢_0,𝑥(𝑥,𝜏)𝑑𝜏−𝑏^𝑡₀𝑢_2,𝑥𝑥(𝑥,𝜏)𝑑𝜏−𝑐^𝑡₀𝑢_{2,𝑥𝑥𝑥𝑥}(𝑥,𝜏)𝑑𝜏, (5.10) and so on. Consequently after some reduction with help of Maple or Mathematica, we get 𝑉₀(𝑥,𝑡)=15𝑏𝜆19𝑎2+3tanh𝜆𝑥2−tanh³𝜆𝑥2,𝑉₁(𝑥,𝑡)=−3𝜆𝑡𝑎15𝑏𝜆19𝑎²sech⁴𝜆𝑥2,𝑉₂(𝑥,𝑡)=−6𝜆⁵𝑡²𝑎²15𝑏𝜆19𝑎³sech⁴𝜆𝑥2tanh𝜆𝑥2. (5.11) and so on. Substituting from (5.11)-(5.12) into (4.5), we obtain the approximate solution of the initial value problem (1.2): 𝑢(𝑥,𝑡)=15𝑏𝜆19𝑎2+3tanh𝜆𝑥2−tanh³𝜆𝑥2−3𝜆𝑡𝑎15𝑏𝜆19𝑎²sech⁴𝜆𝑥2−6𝜆⁵𝑡²𝑎²15𝑏𝜆19𝑎³sech⁴𝜆𝑥2tanh𝜆𝑥2+⋯, (5.12) which is in agreement with the exact solution (5.8) using Taylor series expansion near 𝑡=0.The comparison between the exact solution (5.8) and the approximate solution (5.12) is shown in Table 2 and Figure 2. It seems that the errors are very small if 𝑏=−1,𝑎=1,𝜆=0.1,and𝑡=0.1.

Table 2 

The approximate solution (5.12) in comparison with the exact solution (5.8) if𝑏=−1,𝑎=1,𝜆=0.1,and𝑡=0.1.

(a)

(b)

Figure 2 

The approximate solution (5.12) shown in (b) in comparison with the exact solution (5.8) shown in (a) if𝑏=−1,𝑎=1,𝜆=0.1, and 𝑡=0.1.

5.3. Approximate Solutions for the Nonlinear-Coupled Schrodinger-KdV Equations with Initial Conditions Using MVIM

In this subsection, we find the solutions 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡) satisfying the nonlinear coupled Schrodinger-KdV equations (1.3) with the following initial conditions [41]: 𝑢()(),𝑣()𝑥,0=−𝑐𝛼+2𝛼𝑘tanh𝑖𝑘𝑥𝑥,0=−2𝑘²sech²(𝑖𝑘𝑥), (5.13) where 𝑘,𝛼,and𝑐 are arbitrary constants and 𝛼=2𝑘²+𝑐/2. These initial conditions follow by setting 𝑡=0 in the following exact solutions of (1.3): 𝑢()((,𝑣()𝑥,𝑡=−𝑐𝛼+2𝛼𝑘tanh𝑖𝑘𝑥−𝑐𝑡))𝑥,𝑡=−2𝑘²sech²(𝑖𝑘(𝑥−𝑐𝑡)). (5.14)(5.15) These exact solutions have been derived by Zhang [41] using a direct algebraic approach. Let us now apply the MVIM to the initial value problem (1.3) and (5.13): 𝑖^∞_𝑖=0𝑝^𝑖𝑢_𝑖=𝑖𝑢(𝑥,0)+𝑝^𝑡₀^∞_𝑖=0𝑝^𝑖𝑢_{𝑖,𝑥𝑥}+^∞_𝑖=0𝑝^𝑖𝑢_𝑖^∞_𝑖=0𝑝^𝑖𝑣_𝑖𝑑𝜏,_∞𝑖=0𝑝^𝑖𝑣_𝑖=𝑣(𝑥,0)−𝑝^𝑡₀^∞_𝑖=0𝑝^𝑖𝑣_{𝑖,𝑥𝑥𝑥}+6^∞_𝑖=0𝑝^𝑖𝑣_𝑖^∞_𝑖=0𝑝^𝑖𝑣_𝑖,𝑥−^∞_𝑖=0𝑝^𝑖𝑢_𝑖,𝑥^∞_𝑖=0𝑝ⁱ𝑢_𝑖−^∞_𝑖=0𝑝^𝑖𝑢_𝑖^∞_𝑖=0𝑝^𝑖𝑢_𝑖,𝑥𝑑𝜏. (5.16) Comparing the different coefficient of like power of 𝑝, we have 𝑝⁰()(),𝑝∶𝑢𝑥,0=−𝑐𝛼+2𝛼𝑘tanh𝑖𝑘𝑥₁∶𝑖𝑢₁(𝑥,𝑡)=^𝑡₀𝑢_0,𝑥𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₀(𝑥,𝜏)𝑣₀()𝑝𝑥,𝜏𝑑𝜏,₂∶𝑖𝑢₂(𝑥,𝑡)=^𝑡₀𝑢_1,𝑥𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₀(𝑥,𝜏)𝑣₁(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₁(𝑥,𝜏)𝑣₀()𝑝𝑥,𝜏𝑑𝜏,₃∶𝑖𝑢₃(𝑥,𝑡)=^𝑡₀𝑢_2,𝑥𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₀(𝑥,𝜏)𝑣₂(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₁(𝑥,𝜏)𝑣₁()+𝑥,𝜏𝑑𝜏_𝑡0𝑢₂(𝑥,𝜏)𝑣₀()𝑝𝑥,𝜏𝑑𝜏,₀∶𝑣(𝑥,0)=−2𝑘²sech²(),𝑝𝑖𝑘𝑥₁∶𝑣₁(𝑥,𝑡)=−^𝑡₀𝑣_{0,𝑥𝑥𝑥}(𝑥,𝜏)𝑑𝜏−6^𝑡₀𝑣₀(𝑥,𝜏)𝑣_0,𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₀(𝑥,𝜏)𝑢_0,𝑥()+𝑥,𝜏𝑑𝜏_𝑡0𝑢_0,𝑥(𝑥,𝜏)𝑢₀()𝑝𝑥,𝜏d𝜏,₂∶𝑣₂(𝑥,𝑡)=−^𝑡₀𝑣_{1,𝑥𝑥𝑥}(𝑥,𝜏)𝑑𝜏−6^𝑡₀𝑣₁(𝑥,𝜏)𝑣_0,𝑥(𝑥,𝜏)𝑑𝜏−6^𝑡₀𝑣_1,𝑥(𝑥,𝜏)𝑣₀()+𝑥,𝜏𝑑𝜏_𝑡0𝑢₀(𝑥,𝜏)𝑢_1,𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₁(𝑥,𝜏)𝑢_0,𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢_0,𝑥(𝑥,𝜏)𝑢₁(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢_1,𝑥(𝑥,𝜏)𝑢₀(𝑥,𝜏)𝑑𝜏, (5.17) and so on. Consequently after some reduction with help of Maple or Mathematica, we get 𝑢₀()(),𝑢𝑥,𝑡=−𝑐𝛼+2𝛼𝑘tanh𝑖𝑘𝑥₁(𝑥,𝑡)=−2𝑖𝑡𝛼𝑐𝑘²sech²(),𝑢𝑖𝑘𝑥₂(𝑥,𝑡)=2𝛼𝑡²𝑐²𝑘³sech²()(),𝑢𝑖𝑘𝑥tanh𝑖𝑘𝑥₃(𝑥,𝑡)=−2𝑖𝑡³𝛼𝑐³𝑘⁴3sech⁴(𝑖𝑘𝑥)−2sech²(,𝑣𝑖𝑘𝑥)₀(𝑥,𝑡)=−2𝑘²sech²(),𝑣𝑖𝑘𝑥₁(𝑥,𝑡)=−4𝑖𝑡𝑐𝑘³sech²()(),𝑣𝑖𝑘𝑥tanh𝑖𝑘𝑥₂(𝑥,𝑡)=−2𝑐²𝑡²𝑘⁴3sech⁴(𝑖𝑘𝑥)−2sech²(,𝑣𝑖𝑘𝑥)₃(𝑥,𝑡)=−2𝑖𝑐³𝑡³𝑘⁵−4sech⁴()()+4𝑖𝑘𝑥tanh𝑖𝑘𝑥3sech²(𝑖𝑘𝑥)tanh(𝑖𝑘𝑥). (5.18) In this manner the other components can be obtained. Substituting (5.18) into (4.5), we obtain the approximate solutions of the initial value problem (1.3) and (5.13): 𝑢(𝑥,𝑡)=−𝑐𝛼+2𝛼𝑘tanh(𝑖𝑘𝑥)−2𝑖𝑡𝛼𝑐𝑘²sech²(𝑖𝑘𝑥)+2𝛼𝑡²𝑐²𝑘³sech²(𝑖𝑘𝑥)tanh(𝑖𝑘𝑥)−2𝑖𝑡³𝛼𝑐³𝑘⁴3sech⁴(𝑖𝑘𝑥)−2sech²(𝑣()𝑖𝑘𝑥)+⋯,𝑥,𝑡=−2𝑘²sech²(𝑖𝑘𝑥)−4𝑖𝑡𝑐𝑘³sech²(𝑖𝑘𝑥)tanh(𝑖𝑘𝑥)−2𝑐²𝑡²𝑘⁴3sech⁴(𝑖𝑘𝑥)−2sech²(𝑖𝑘𝑥)+⋯, (5.19)(5.20) which are in the closed form of the exact solutions (5.14) and (5.15) using Taylor series expansion near 𝑡=0.

The comparison between the exact solutions (5.14), (5.15) and the approximate solutions (5.19), (5.20) respectively, are shown in Table 3 and Figures 3 and 4. It seems that the errors are very small if 𝑡=0.5, 𝑘=0.1,𝑐=0.2.

Table 3 

The approximate solutions (5.19) and (5.20) in comparison with the exact solutions (5.14) and (5.15), respectively, if 𝑡=0.5, 𝑘=0.1, 𝑐=0.2.

(a)

(b)

Figure 3 

The approximate solution (5.19) shown in the figure (a) in comparison with the exact solution (5.14) shown in figure (b) if 𝑘=0.1,∝=0.1,andc=0.2.

(a)

(b)

Figure 4 

The approximate solution (5.20) shown in (a) in comparison with the exact solution (5.15) shown in (b) if 𝑘=0.1,𝑐=0.2,∝=0.1.

5.4. Approximate Solution of the Nonlinear Long and Short Wave Resonance Equations with Initial Conditions Using MVIM

In this subsection, we find the solutions 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡) satisfying the nonlinear long–short wave resonance equations (1.4) with the following initial conditions [42]: 𝑣₀=𝑢(𝑥,0)=2𝑘𝛽1𝑥+𝜉₁𝑒^{𝑖(𝑘𝑥+𝜉0)},𝑣₀()=2=𝑣𝑥,0𝑥+𝜉₁²−𝐶, (5.21) where 𝛽,𝑘,𝐶,𝜉₀ and 𝜉₁ are arbitrary constants. These initial conditions follow by setting 𝑡=0 in the following exact solutions of (1.4): 𝑢(𝑥,𝑡)=2𝑘𝛽1𝑥−2𝑘𝑡+𝜉₁𝑒^{𝑖(𝑘𝑥+𝑤𝑡+𝜉0)}𝑣()=2,𝑥,𝑡𝑥−2𝑘𝑡+𝜉₁²−𝐶, (5.22)(5.23) where 𝑤=𝐶−𝑘²is constant. These exact solutions have been derived by Shang [42] using the extended hyperbolic function method, which describes the resonance interaction between the long wave and the short wave. Let us now apply the MVIM to the initial value problem (1.4) and (5.21): 𝑖^∞_𝑖=0𝑝^𝑖𝑢_𝑖=𝑖𝑢(𝑥,0)−𝑝^𝑡₀^∞_𝑖=0𝑝^𝑖𝑢_{𝑖,𝑥𝑥}−^∞_𝑖=0𝑝^𝑖𝑢_𝑖^∞_𝑖=0𝑝^𝑖𝑣_𝑖𝑑𝜏,_∞𝑖=0𝑝^𝑖𝑣_𝑖=𝑣(𝑥,0)−𝑝𝛽^𝑡₀^∞_𝑖=0𝑝^𝑖𝑢_𝑖,𝑥^∞_𝑖=0𝑝^𝑖𝑢_𝑖+^∞_𝑖=0𝑝^𝑖𝑢_𝑖^∞_𝑖=0𝑝^𝑖𝑢_𝑖,𝑥𝑑𝜏. (5.24) Comparing the different coefficient of like power of 𝑝, we have 𝑝⁰∶𝑢(𝑥,0)=2𝑘𝛽1𝑥+𝜉₁𝑒^{𝑖(𝑘𝑥+𝜉0)},𝑝¹∶𝑖𝑢₁(𝑥,𝑡)=−^𝑡₀𝑢_0,𝑥𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₀(𝑥,𝜏)𝑣₀()𝑝𝑥,𝜏𝑑𝜏,₂∶𝑖𝑢₂(𝑥,𝑡)=−^𝑡₀𝑢_1,𝑥𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₀(𝑥,𝜏)𝑣₁(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₁(𝑥,𝜏)𝑣₀()𝑝𝑥,𝜏𝑑𝜏,₃∶𝑖𝑢₃(𝑥,𝑡)=−^𝑡₀𝑢_2,𝑥𝑥(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₀(𝑥,𝜏)𝑣₂(𝑥,𝜏)𝑑𝜏+^𝑡₀𝑢₁(𝑥,𝜏)𝑣₁()+𝑥,𝜏𝑑𝜏_𝑡0𝑢₂(𝑥,𝜏)𝑣₀()𝑝𝑥,𝜏𝑑𝜏,₀()=2∶𝑣𝑥,0𝑥+𝜉₁²𝑝−𝐶,₁∶𝑣₁(𝑥,𝑡)=−𝛽^𝑡₀𝑢₀(𝑥,𝜏)𝑢_0,𝑥(𝑥,𝜏)𝑑𝜏−𝛽^𝑡₀𝑢_0,𝑥(𝑥,𝜏)𝑢₀()𝑝𝑥,𝜏𝑑𝜏,₂∶𝑣₂(𝑥,𝑡)=−𝛽^𝑡₀𝑢₀(𝑥,𝜏)𝑢_1,𝑥(𝑥,𝜏)𝑑𝜏−𝛽^𝑡₀𝑢₁(𝑥,𝜏)𝑢_0,𝑥(𝑥,𝜏)𝑑𝜏−𝛽^𝑡₀𝑢_0,𝑥(𝑥,𝜏)𝑢₁(𝑥,𝜏)𝑑𝜏−𝛽^𝑡₀𝑢_1,𝑥(𝑥,𝜏)𝑢₀()𝑝𝑥,𝜏𝑑𝜏,₃∶𝑣₃(𝑥,𝑡)=−𝛽^𝑡₀𝑢_0,𝑥(𝑥,𝜏)𝑢₂(𝑥,𝜏)𝑑𝜏−𝛽^𝑡₀𝑢₀(𝑥,𝜏)𝑢_2,𝑥(𝑥,𝜏)𝑑𝜏−𝛽^𝑡₀𝑢_1,𝑥(𝑥,𝜏)𝑢₁(𝑥,𝜏)𝑑𝜏−𝛽^𝑡₀𝑢₁(𝑥,𝜏)𝑢_1,𝑥(𝑥,𝜏)𝑑𝜏−𝛽^𝑡₀𝑢_2,𝑥(𝑥,𝜏)𝑢₀(𝑥,𝜏)𝑑𝜏−𝛽^𝑡₀𝑢₂(𝑥,𝜏)𝑢_0,𝑥(𝑥,𝜏)𝑑𝜏, (5.25) and so on. On substituting (5.21) into (5.25), we deduce that 𝑢₀(𝑥,𝑡)=2𝑘𝛽1𝑥+𝜉₁𝑒^{𝑖(𝑘𝑥+𝜉0)},𝑢₁(𝑥,𝑡)=2𝑡𝑘𝛽𝑖𝐶−𝑘²𝑥+𝜉₁+2𝑘𝑥+𝜉₁²𝑒^{𝑖(𝑘𝑥+𝜉0)},𝑢₂(𝑥,𝑡)=2𝑡²𝑘𝛽𝑖𝐶−𝑘²²𝑥+𝜉₁+4𝑖𝑘𝑥+𝜉₁²+4𝑘𝑥+𝜉₁³𝑒^{𝑖(𝑘𝑥+𝜉0)},𝑣₀()=2𝑥,𝑡𝑥+𝜉₁²𝑣−𝐶,₁(𝑥,𝑡)=8𝑘𝑡𝑥+𝜉₁³,𝑣₂(𝑥,𝑡)=48𝑘𝑡²𝑥+𝜉₁⁴,𝑣₃(𝑥,𝑡)=384𝑘𝑡³𝑥+𝜉₁⁵. (5.26)

In this manner the other components can be obtained. Consequently, we obtain the following approximate solutions of the initial value problem (1.4) and (5.21): 𝑢(𝑥,𝑡)=2𝑘𝛽1𝑥+𝜉₁𝑒^{𝑖(𝑘𝑥+𝜉0)}+2𝑡𝑘𝛽𝑖𝐶−𝑘²𝑥+𝜉₁+2𝑘𝑥+𝜉₁²𝑒^{𝑖(𝑘𝑥+𝜉0)}+2𝑡²𝑘𝛽𝑖𝐶−𝑘²²𝑥+𝜉₁+4𝑖𝑘𝑥+𝜉₁²+4𝑘𝑥+𝜉₁³𝑒^{𝑖(𝑘𝑥+𝜉0)}𝑣()=2+⋯,𝑥,𝑡𝑥+𝜉₁²−𝐶+8𝑘𝑡𝑥+𝜉₁³+48𝑘𝑡²𝑥+𝜉₁⁴+384𝑘𝑡³𝑥+𝜉₁⁵+⋯ (5.27)(5.28) which are in the closed forms of the exact solutions (5.22) and (5.23) using Taylor series expansion near 𝑡=0.

The comparison between the exact solutions (5.22), (5.23) and the approximate solutions (5.27), (5.28) respectively is shown in Table 4 and Figures 5 and 6. It seems that the errors are very small if𝑡=0.5, 𝑘=0.1,𝑐=0.2,𝜉₀=2,and𝜉₁=3.

Table 4 

The approximate solutions (5.27) and (5.28) in comparison with the exact solutions (5.22) and (5.23), respectively, if 𝑡=0.5, 𝑘=0.1,𝑐=0.2,𝜉₀=2, and 𝜉₁=3.

(a)

(b)

(c)

(d)

Figure 5 

The approximate solution (5.27) in comparison with the exact solution (5.22) if 𝑘=0.1,𝑐=0.2,𝜉0=2,and𝜉1=3.

(a)

(b)

Figure 6 

The approximate solution (5.28) in comparison with the exact solution (5.23) if 𝑘=0.1,𝑐=0.2,𝜉0=2,and𝜉1=3.

6. Conclusions

In this paper, the modified variational iteration method was applied for finding the approximate solutions for some nonlinear evolution equations in mathematical physics via the nonlinear Burgers-Fisher equation, nonlinear Kuramoto-Sivashinsky equation, nonlinear coupled Schrodinger KdV equations, and nonlinear long-short wave resonance equations with well-known initial conditions. It seems to us that the modified variational iteration method presents a rapid convergence solutions. It can be concluded that this method is very powerful and efficient technique in finding approximate solutions for wide classes of nonlinear problems.