Application of Optimal Homotopy Asymptotic Method to Doubly Wave Solutions of the Coupled Drinfel’d-Sokolov-Wilson Equations

Abstract

The approximate solution of the doubly periodic wave solutions of the coupled Drinfel’d-Sokolov-Wilson equations has been considered by using the optimal homotopy asymptotic method (OHAM). We obtained the numerical solution of the problem and compared that with the OHAM solution. The obtained solutions show that OHAM is effective, simpler, easier, and explicit and gives a suitable way to control the convergence of the approximate solution.

1. Introduction

The coupled nonlinear partial differential equations (NPDEs) are widely used in applied mathematics, physics, and engineering sciences to offer the description of complex phenomena. Here we consider doubly periodic wave solutions of the coupled Drinfel’d-Sokolov-Wilson equation of the form [1] 𝜕𝑢(𝑥,𝑡)()()𝜕𝑡+3𝑣𝑥,𝑡𝜕𝑣𝑥,𝑡()𝜕𝑥=0,𝜕𝑣𝑥,𝑡𝜕𝜕𝑡+2³𝑣(𝑥,𝑡)𝜕𝑥³()()+2𝑢𝑥,𝑡𝜕𝑣𝑥,𝑡+()𝜕𝑥𝜕𝑢𝑥,𝑡𝑣()𝜕𝑥𝑥,𝑡=0, (1) with 𝑢(𝑥,0)=𝑥,𝑣(𝑥,0)=−𝑥. (2) The exact and explicit solution of the NPDEs in mathematical physics, engineering, and science plays an important role. The exact solution of NPDEs cannot be found easily as all NPDEs have infinitely many solutions. The analytical and exact solution of such problems is either not available in the literature or may be found by using transformation based on the invariance group analysis method [2], the Lie infinitesimal criterion [3], the symbolic computation [4], and the Backlund transformation [5]. All these methods reduced the complex equations into simple equations by using the transformation. In the literature most of the methods like the variational iterative method (VIM) [6], Adomian decomposition method (ADM) [7], differential transform method (DTM) [8], and homotopy perturbation method (HPM) [9] have been used for the solution of weakly NPDEs and few for strongly NPDEs. To tackle the strongly NPDEs the perturbation methods were introduced [10, 11]. These methods contain a small parameter which cannot be found easily. New analytic methods such as the artificial parameters method [12], homotopy analysis method (HAM) [13], and homotopy perturbation method (HPM) [9] were introduced. These methods combined the homotopy with the perturbation techniques. Recently, Vasile Marinca et al. introduced OHAM [14–18] for the solution of nonlinear problems which made the perturbation methods independent of the assumption of small parameters and huge computational work.

The motivation of this paper is to boost OHAM for the solution of coupled NPDEs. In [19–24] OHAM has been proved to be valuable for obtaining an approximate solution of the single partial differential equation (PDE). Before these coupled NPDEs were not solved by OHAM. We have proved that OHAM is useful and reliable for NPDEs, showing its validity and great potential for the solution of transient physical phenomena in science and engineering.

In the succeeding section, the basic idea of OHAM is formulated for the solution of NPDEs. The effectiveness and efficiency of OHAM are shown in Section 3.

2. Fundamental Mathematical Theory of OHAM

Let us see the partial differential equation of the following form: 𝒜(𝑢(()ℬ𝑥,𝑡))+𝑓𝑥,𝑡=0,𝑥∈Ω,𝑢,𝜕𝑢𝜕𝑥=0,𝑥∈Γ, (3) where 𝒜 is a differential operator, 𝑢(𝑥,𝑡) is an unknown function, 𝑥 and 𝑡 denote spatial and temporal independent variables, respectively, Γ is the boundary of Ω, and 𝑓(𝑥,𝑡) is a known analytic function. 𝒜 can be divided into two parts ℒ and 𝒩 such that 𝒜=ℒ+𝒩. (4)ℒ is the simpler part of the partial differential equation which is easier to solve, and 𝒩 contains the remaining part of 𝒜.

According to OHAM, one can construct an optimal homotopy 𝜙(𝑥,𝑡;𝑝)∶Ω×[0,1]→ℜ which satisfies 𝐻𝜙=ℒ𝜙(𝑝𝒜𝜙(𝑥,𝑡;𝑝,𝑝1−𝑝𝑥,𝑡;𝑝+𝑓𝑥,𝑡)−𝐻𝑥,𝑡;𝑝+𝑓𝑥,𝑡)=0, (5) where the auxiliary function 𝐻(𝑝) is nonzero for 𝑝≠0 and 𝐻(0)=0. Equation (5) is called an optimal homotopy equation. Clearly, we have 𝜙()𝜙()()𝜙()(1𝒜𝜙(𝑝=0⟹𝐻𝑥,𝑡;0,0=ℒ𝑥,𝑡;0+𝑓𝑥,𝑡=0,𝑝=1⟹𝐻𝑥,𝑡;1,1=𝐻)𝑥,𝑡;𝑝+𝑓𝑥,𝑡)=0. (6) Obviously, when 𝑝=0 and 𝑝=1 we obtain 𝜙(𝑥,𝑡;0)=𝑢₀(𝑥,𝑡),𝜙(𝑥,𝑡;1)=𝑢(𝑥,𝑡), (7) respectively. Thus, as 𝑝 varies from 0 to 1, the solution 𝜙(𝑥,𝑡;𝑝) approaches from 𝑢₀(𝑥,𝑡) to 𝑢(𝑥,𝑡), where 𝑢₀(𝑥,𝑡) is obtained from (5) for 𝑝=0: ℒ𝑢₀(𝑥,𝑡)+𝑓(𝑥,𝑡)=0,ℬ𝑢₀,𝜕𝑢₀𝜕𝑥=0. (8) Next, we choose the auxiliary function 𝐻(𝑝) in the form 𝐻𝑝=𝑝𝐶₁+𝑝²𝐶₂+⋯. (9)

To get an approximate solution, we expand 𝜙(𝑥,𝑡;𝑝,𝐶_𝑖) by Taylor’s series about 𝑝 in the following manner: 𝜙𝑥,𝑡;𝑝,𝐶_𝑖=𝑢₀(𝑥,𝑡)+^∞_𝑘=1𝑢_𝑘𝑥,𝑡;𝐶_𝑖𝑝^𝑘,𝑖=1,2,…. (10) Substituting (10) into (5) and equating the coefficient of the like powers of 𝑝, we obtain the zeroth order problem, given by (8), the first and second order problems are given by (11)-(12), respectively, and the general governing equations for 𝑢_𝑘(𝑥,𝑡) are given by (13): ℒ𝑢₁(𝑥,𝑡)=𝐶₁𝒩₀𝑢₀(𝑥,𝑡),ℬ𝑢₁,𝜕𝑢₁ℒ𝑢𝜕𝑥=0,₂(𝑥,𝑡)−ℒ𝑢₁(𝑥,𝑡)=𝐶₂𝒩₀𝑢₀(𝑥,𝑡)+𝐶₁ℒ𝑢₁(𝑥,𝑡)+𝒩₁𝑢₀(𝑥,𝑡),𝑢₁(),ℬ𝑢𝑥,𝑡₂,𝜕𝑢₂ℒ𝑢𝜕𝑥=0,_𝑘(𝑥,𝑡)−ℒ𝑢_𝑘−1(𝑥,𝑡)=𝐶_𝑘𝒩₀𝑢₀(+𝑥,𝑡)_{𝑘−1𝑖=1}𝐶_𝑖ℒ𝑢_𝑘−𝑖(𝑥,𝑡)+𝒩_𝑘−𝑖𝑢₀(𝑥,𝑡),𝑢₁(𝑥,𝑡),…,𝑢_𝑘−𝑖(),𝑢𝑥,𝑡𝑘=2,3,…,ℬ_𝑘,𝜕𝑢_𝑘𝜕𝑥=0, (11)(12)(13) where 𝒩_𝑘−𝑖(𝑢₀(𝑥,𝑡),𝑢₁(𝑥,𝑡),…,𝑢_𝑘−𝑖(𝑥,𝑡)) are the coefficients of 𝑝^𝑘−𝑖 in the expansion of 𝒩(𝜙(𝑥,𝑡;𝑝)) about the embedding parameter 𝑝: 𝒩𝜙𝑥,𝑡;𝑝,𝐶_𝑖=𝒩₀𝑢₀(𝑥,𝑡)+_𝑘≥1𝒩_𝑘𝑢₀,𝑢₁,𝑢₂,…,𝑢_𝑘𝑝^𝑘. (14) It should be underscored that the 𝑢_𝑘 for 𝑘≥0 are governed by the linear equations with linear boundary conditions that come from the original problem, which can be easily solved.

It has been observed that the convergence of the series equation (10) depends upon the auxiliary constants 𝐶₁,𝐶₂,…. If it is convergent at 𝑝=1, one has 𝑢𝑥,𝑡;𝐶_𝑖=𝑢₀(𝑥,𝑡)+_𝑘≥1𝑢_𝑘𝑥,𝑡;𝐶_𝑖. (15) Substituting (15) into (1), it results the following expression for the residual: 𝑅𝑥,𝑡;𝐶_𝑖=ℒ𝑢𝑥,𝑡;𝐶_𝑖+𝑓(𝑥,𝑡)+𝒩𝑢𝑥,𝑡;𝐶_𝑖. (16) In actual computation 𝑘=1,2,3,…,𝑚.

If 𝑅(𝑥,𝑡;𝐶_𝑖)=0, then 𝑢(𝑥,𝑡;𝐶_𝑖) is the exact solution of the problem. Generally it does not happen, especially in nonlinear problems.

For the determinations of auxiliary constants, 𝐶_𝑖, 𝑖=1,2,…,𝑚, there are different methods like Galerkin’s method, the Ritz method, the least squares method, and, the collocation method. One can apply the method of least squares as under 𝐽𝐶_𝑖=^𝑡_0Ω𝑅²𝑥,𝑡;𝐶_𝑖𝑑𝑥𝑑𝑡,𝜕𝐽𝜕𝐶₁=𝜕𝐽𝜕𝐶₂=⋯=𝜕𝐽𝜕𝐶_𝑚=0. (17)(18) The 𝑚th order approximation can be obtained by these constants. The constants 𝐶_𝑖 can also be determined by another method as under 𝑅ℎ₁;𝐶_𝑖=𝑅ℎ₂;𝐶_𝑖=⋯=𝑅ℎ_𝑚;𝐶_𝑖=0,𝑖=1,2,…,𝑚, (19) at any time 𝑡, where ℎ_𝑖∈Ω.

The more general auxiliary function 𝐻(𝑝) is useful for convergence, which depends upon constants 𝐶₁,𝐶₂,…, can be optimally identified by (18), and is useful in error minimization.

3. Application of OHAM to Doubly Periodic Wave Solutions of the Coupled Drinfel’d-Sokolov-Wilson Equation

To demonstrate the effectiveness of the extended formulation of OHAM for coupled nonlinear partial differential equations (NPDEs), we consider the doubly periodic wave solutions of the coupled Drinfel’d-Sokolov-Wilson equations (1) with the boundary condition (2).

Applying the method formulated in Section 2 leads to the following: ℒ(𝑢(=()𝑥,𝑡))𝜕𝑢𝑥,𝑡(𝑣(()()𝜕𝑡,𝒩𝑥,𝑡))=3𝑣𝑥,𝑡𝜕𝑣𝑥,𝑡,ℒ(𝑣(=()𝜕𝑥𝑥,𝑡))𝜕𝑣𝑥,𝑡,𝒩()𝜕𝜕𝑡𝑢,𝑣=2³𝑣(𝑥,𝑡)𝜕𝑥³()()+2𝑢𝑥,𝑡𝜕𝑣𝑥,𝑡+()𝜕𝑥𝜕𝑢𝑥,𝑡𝑣(),()𝜕𝑥𝑥,𝑡1−𝑝𝜕𝑢𝑥,𝑡𝜕𝑡−𝐻₁𝑝𝜕𝑢(𝑥,𝑡)()()𝜕𝑡+3𝑣𝑥,𝑡𝜕𝑣𝑥,𝑡()𝜕𝑡=0,1−𝑝𝜕𝑣𝑥,𝑡𝜕𝑡−𝐻₂𝑝𝜕𝑢(𝑥,𝑡)𝜕𝜕𝑡+2³𝑣(𝑥,𝑡)𝜕𝑥³()()+2𝑢𝑥,𝑡𝜕𝑣𝑥,𝑡+()𝜕𝑥𝜕𝑢𝑥,𝑡𝑣()𝜕𝑥𝑥,𝑡=0. (20)

We consider 𝑢=𝑢₀+𝑝𝑢₁+𝑝²𝑢₂,𝑣=𝑣₀+𝑝𝑣₁+𝑝²𝑣₂,𝐻₁𝑝=𝑝𝐶₁₁+𝑝²𝐶₁₂,𝐻₂𝑝=𝑝𝐶₂₁+𝑝²𝐶₂₂. (21)

3.1. Zeroth Order System

We have 𝜕𝑢₀(𝑥,𝑡)𝜕𝑡=0,𝜕𝑣₀(𝑥,𝑡)𝜕𝑡=0, (22) with initial conditions 𝑢₀(𝑥,0)=𝑥,𝑣₀(𝑥,0)=−𝑥. (23) Its solution 𝑢₀(𝑥,𝑡)=𝑥,𝑣₀(𝑥,𝑡)=−𝑥. (24)

3.2. First Order System

We have 𝜕𝑢₁(𝑥,𝑡)=𝜕𝑡1+𝐶₁₁𝜕𝑢₀(𝑥,𝑡)𝜕𝑡+3𝐶₁₁𝑣₀(𝑥,𝑡)𝜕𝑣₀(𝑥,𝑡),𝜕𝑥𝜕𝑣₁(𝑥,𝑡)=𝜕𝑡𝜕𝑣₀(𝑥,𝑡)𝜕𝑡+𝐶₂₁𝑣₀(𝑥,𝑡)1+𝜕𝑢₀(𝑥,𝑡)𝜕𝑡+2𝐶₂₁𝑢₀(𝑥,𝑡)𝜕𝑣₀(𝑥,𝑡)𝜕𝑥+2𝐶₂₁𝜕³𝑣₀(𝑥,𝑡)𝜕𝑥³,𝑢₁(𝑥,0)=0,𝑣₁(𝑥,0)=0. (25)

Its solutions 𝑢₁(𝑥,𝑡)=3𝑡𝑥𝐶₁₁,𝑣₁(𝑥,𝑡)=−𝑡𝑥+4𝑥𝐶₂₁. (26)

3.3. Second Order System

We have 𝜕𝑢₂(𝑥,𝑡)=𝜕𝑡1+𝐶₁₁𝜕𝑢₁(𝑥,𝑡)𝜕𝑡+𝐶₂₁𝜕𝑢₀(𝑥,𝑡)𝐶𝜕𝑥+3₁₂𝑣₀(𝑥,𝑡)+𝐶₁₁𝑣₁(𝑥,𝑡)𝜕𝑣₀(𝑥,𝑡)𝜕𝑥+3𝐶₁₁𝑣₀(𝑥,𝑡)𝜕𝑣₁(𝑥,𝑡),𝜕𝑥𝜕𝑣₂(𝑥,𝑡)=𝜕𝑡𝜕𝑣₁(𝑥,𝑡)𝜕𝑡+𝐶₂₁𝑣₁(𝑥,𝑡)+𝐶₂₂𝑣₀()+𝐶𝑥,𝑡₂₂𝑣₀(𝑥,𝑡)+𝐶₂₁𝑣₁()×𝑥,𝑡𝜕𝑢₀(𝑥,𝑡)𝐶𝜕𝑥+2₂₂𝑢₀(𝑥,𝑡)+𝐶₂₁𝑢₁(𝑥,𝑡)𝜕𝑣₀(𝑥,𝑡)𝜕𝑥+2𝐶₂₁𝑢₀(𝑥,𝑡)𝜕𝑣₁(𝑥,𝑡)𝜕𝑥+2𝐶₂₂𝜕³𝑣₀(𝑥,𝑡)𝜕𝑥³+2𝐶₂₁𝜕³𝑣₁(𝑥,𝑡)𝜕𝑥³, (27)

with 𝑢₂(𝑥,0)=0,𝑣₂(𝑥,0)=0. (28) Its solutions 𝑢₂()=3𝑥,𝑡2𝐶₁₁(1+2𝑥)𝑡²+2𝐶²₁₁𝑥𝑡(1+4𝑡)+𝐶₁₂,𝑣𝑥𝑡₂()=1𝑥,𝑡2−2𝑥𝑡−4𝐶₂₁𝑥𝑡(2+𝑡)−9𝐶₁₁𝐶₂₁𝑥𝑡²−16𝐶²₂₁𝑥𝑡²−8𝐶₂₂𝑥𝑡. (29) Adding (26), (29), and (28), we obtain 𝑢(𝑥,𝑡)=𝑥+3𝑡𝑥𝐶₁₁+32𝐶₁₁(1+2𝑥)𝑡²+2𝐶²₁₁𝑥𝑡(1+4𝑡)+𝐶₁₂,𝑣()𝑥𝑡𝑥,𝑡=−𝑥−𝑡𝑥+4𝑥𝐶₂₁+12−2𝑥𝑡−4𝐶₂₁𝑥𝑡(2+𝑡)−9𝐶₁₁𝐶₂₁𝑥𝑡²−16𝐶²₂₁𝑥𝑡²−8𝐶₂₂𝑥𝑡. (30) For the calculations of the constants 𝐶₁₁, 𝐶₁₂, 𝐶₂₁, and 𝐶₂₂ using (30) in (1) and applying the procedure mentioned in (16)–(19), we get 𝐶₁₁=0,𝐶₁₂𝐶=−0.6282197208603675,₂₁𝐶=0.3177122963169145,₂₂𝑢()𝑣()1=−1.689990604397596,𝑥,𝑡=𝑥−1.88466𝑥𝑡,𝑥,𝑡=−𝑥−2.27085𝑥𝑡+28.97823𝑥𝑡−2.88591𝑥𝑡². (31)

4. Results and Discussions

The formulation presented in Section 2 provides highly accurate solutions for the problems demonstrated in Section 3. We have used Mathematica 7 for most of our computational work. In Tables 1 and 3, we have presented absolute errors for 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡) at a spatial domain [0, 0.4] for 𝑡=0.01, 𝑡=0.015, 𝑡=0.1, 𝑡=0.2, 𝑡=0.3, and 𝑡=0.4, while in Tables 2 and 4 the convergence of the OHAM solution is given, through first and second order absolute errors at time 𝑡=0.1 and 0≤𝑥≤1. Here we observe that the OHAM solution converges rapidly with increasing order of approximation. From Tables 1–4 and Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 it is evident that the OHAM results are nearly identical to the numerical results. Here the results are very consistent with the increasing time.

Table 1 

Absolute error of OHAM solution of 𝑢(𝑥,𝑡) corresponding to the numerical solution.

Table 2 

Comparison of first order and second order errors of 𝑢(𝑥,𝑡) corresponding to the numerical solution at time 𝑡=0.1, and 0≤𝑥≤1.

Table 3 

Absolute error of OHAM solution of 𝑣(𝑥,𝑡) corresponding to the numerical solution.

Table 4 

Comparison of first order and second order errors of 𝑣(𝑥,𝑡) corresponding to the numerical solution at time 𝑡=0.1, and 0≤𝑥≤1.

Figure 1 

3D approximate solution of 𝑢(𝑥,𝑡) for 𝑡=0.1.

Figure 2 

3D numerical solution of 𝑢(𝑥,𝑡) for 𝑡=0.1.

Figure 3 

2D approximate solution of 𝑢(𝑥,𝑡) for 𝑡=0.1.

Figure 4 

2D numerical solution for 𝑡=0.1.

Figure 5 

2D numerical, zeroth, first, and second order solutions of 𝑢(𝑥,𝑡) for 𝑡=0.5.

Figure 6 

3D approximate solution of 𝑣(𝑥,𝑡) for 𝑡=0.1.

Figure 7 

3D exact solution of 𝑣(𝑥,𝑡) for 𝑡=0.1.

Figure 8 

2D approximate solution of 𝑣(𝑥,𝑡)for 𝑡=0.1.

Figure 9 

2D numerical solution of 𝑣(𝑥,𝑡) for 𝑡=0.1.

Figure 10 

2D numerical, zeroth, first, and second order solution of 𝑣(𝑥,𝑡) for 𝑡=0.5.

5. Conclusion

In this paper, we have seen the effectiveness of OHAM [16–20] in doubly periodic wave solutions of the coupled Drinfel’d-Sokolov-Wilson equation. By applying the basic idea of OHAM to doubly periodic wave solutions of the coupled Drinfel’d-Sokolov-Wilson equation, we found it simpler in applicability, more convenient to control convergence, and involving less computational overhead. Therefore, OHAM shows its validity and great potential for the solution of time dependant problems in science and engineering.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.