An Extension of the Optimal Homotopy Asymptotic Method to Coupled Schrödinger-KdV Equation

Abstract

We consider the approximate solution of the coupled Schrödinger-KdV equation by using the extended optimal homotopy asymptotic method (OHAM). We obtained the extended OHAM solution of the problem and compared with the exact, variational iteration method (VIM) and homotopy perturbation method (HPM) solutions. The obtained solution shows that extended OHAM is effective, simpler, easier, and explicit and gives a suitable way to control the convergence of the approximate solution.

1. Introduction

The nonlinear Schrödinger equations are of great interest due to their numerous applications in physical phenomena. The coupled Schrödinger-KdV equations are extensively used to model nonlinear dynamics of one-dimensional Langmuir and ion acoustic waves in the system of coordinates moving at the speed of ion acoustic. This problem remains under consideration from many years and has been investigated by many researchers. Many authors have investigated the nonlinear Schrödinger-KdV equation by various techniques such as the following: Wang [1] used finite difference method, Küçükarslan [2] used HPM, Bai and Zhang [3] used quadratic B-Spline finite element method, Fan and Hon [4] used extended tanh method, Kaya and El-Sayed [5] used adomian decomposition method (ADM), Doosthoseini and Shahmohamadi [6] used VIM, Alomari et al. [7] used homotopy analysis method (HAM), Qing et al. [8] used element free Galerkin method (EFG), and Golbabai and Safdari-Vaighani [9] used meshless method using RBF collocation scheme. The perturbation methods like HPM required a small parameter and are difficult to determine.

Recently, Marinca et al. introduced OHAM [10–14] 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 extend the OHAM formulation for a system of three partial differential equations and to apply the extended OHAM formulation to coupled nonlinear Schrödinger-KdV equation. In [15–17] OHAM has been proved to be valuable for obtaining an approximate solution of ordinary/partial differential equations (O/PDEs). Before, this system of nonlinear partial differential equations (NPDEs) was not solved by OHAM. We have proved that extended OHAM is useful and reliable for NPDEs, showing its validity and great potential for the solution of transient physical phenomenon in science and engineering.

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

2. Extended Mathematical Formulation of OHAM

Consider a system of three partial differential equations: 𝒜₁𝑓(𝑥,𝑡)+𝑠₁()𝒜𝑥,𝑡=0,₂𝑔(𝑥,𝑡)+𝑠₂()𝒜𝑥,𝑡=0,₃(ℎ(𝑥,𝑡))+𝑠₂()ℬ𝑥,𝑡=0,𝑥∈Ω₁𝑓,𝜕𝑓ℬ𝜕𝑥=0,₂𝑔,𝜕𝑔ℬ𝜕𝑥=0,₃ℎ,𝜕ℎ𝜕𝑥=0,𝑥∈Γ, (1) where 𝒜₁, 𝒜₂, 𝒜₃ are differential operators, 𝑓(𝑥,𝑡), 𝑔(𝑥,𝑡), ℎ(𝑥,𝑡) are unknown functions, 𝑥 and 𝑡 denote spatial and temporal independent variables, respectively, Γ is the boundary of Ω, and 𝑠₁(𝑥,𝑡), 𝑠₂(𝑥,𝑡), 𝑠₃(𝑥,𝑡) are known analytic functions. 𝒜₁, 𝒜₂, 𝒜₃ can be divided into two parts: 𝒜₁=ℒ₁+𝒩₁,𝒜₂=ℒ₂+𝒩₂,𝒜₃=ℒ₃+𝒩₃. (2)ℒ₁, ℒ₂, ℒ₃ contain the linear parts while 𝒩₁, 𝒩₂, 𝒩₃ contain the nonlinear parts of the system of partial differential equations.

According to OHAM, we construct 𝛼[]𝛽[]𝛾[]𝑥,𝑡;𝑝∶Ω×0,1⟶ℜ,𝑥,𝑡;𝑝∶𝜓×0,1⟶ℜ,𝑥,𝑡;𝑝∶𝜙×0,1⟶ℜ, (3) satisfying the following homotopies: 𝐻𝛼=ℒ𝑥,𝑡;𝑝,𝑝1−𝑝₁𝛼𝑥,𝑡;𝑝+𝑠₁(𝑥,𝑡)−𝐻₁𝑝𝒜₁𝛼𝑥,𝑡;𝑝+𝑠₁(𝐻𝛽=ℒ𝑥,𝑡)=0,𝑥,𝑡;𝑝,𝑝1−𝑝₂𝛽𝑥,𝑡;𝑝+𝑠₂(𝑥,𝑡)−𝐻₂𝑝𝒜₂𝛽𝑥,𝑡;𝑝+𝑠₂(𝐻𝛾=ℒ𝑥,𝑡)=0,𝑥,𝑡;𝑝,𝑝1−𝑝₃𝛾𝑥,𝑡;𝑝+𝑠₃(𝑥,𝑡)−𝐻₃𝑝𝒜₃𝛾𝑥,𝑡;𝑝+𝑠₃(𝑥,𝑡)=0, (4) where the auxiliary functions 𝐻₁(𝑝), 𝐻₂(𝑝), 𝐻₃(𝑝) are nonzero for 𝑝≠0 and 𝐻₁(0)=0, 𝐻₂(0)=0, 𝐻₃(0)=0. Equation (4) is called optimal homotopy equation. Clearly, we have 𝑝=0⟹𝐻(𝛼(𝑥,𝑡;0),0)=ℒ₁(𝛼(𝑥,𝑡;0))+𝑠₁()𝛽()𝑥,𝑡=0,𝑝=0⟹𝐻𝑥,𝑡;0,0=ℒ₂𝛽(𝑥,𝑡;0)+𝑠₂()𝛾()𝑥,𝑡=0,𝑝=0⟹𝐻𝑥,𝑡;0,0=ℒ₃𝛾(𝑥,𝑡;0)+𝑠₃()(𝛼())𝑥,𝑡=0,𝑝=1⟹𝐻𝑥,𝑡;1,1=𝐻₁(1)𝒜₁𝛼𝑥,𝑡;𝑝+𝑠₁(𝛽()𝑥,𝑡)=0,𝑝=1⟹𝐻𝑥,𝑡;1,1=𝐻₂(1)𝒜₂𝛽𝑥,𝑡;𝑝+𝑠₂(𝛾()𝑥,𝑡)=0,𝑝=1⟹𝐻𝑥,𝑡;1,1=𝐻₃(1)𝒜₃𝛾𝑥,𝑡;𝑝+𝑠₃(𝑥,𝑡)=0. (5)

Obviously, when 𝑝=0 and 𝑝=1 we obtain 𝛼(𝑥,𝑡;0)=𝑓₀(𝑥,𝑡),𝛽(𝑥,𝑡;0)=𝑔₀(𝑥,𝑡),𝛾(𝑥,𝑡;0)=ℎ₀(),𝛼()()()()()(),𝑥,𝑡𝑥,𝑡;1=𝑓𝑥,𝑡,𝛽𝑥,𝑡;1=𝑔𝑥,𝑡,𝛾𝑥,𝑡;1=ℎ𝑥,𝑡 (6) respectively. When 𝑝 varies from 0 to 1, the solution 𝛼(𝑥,𝑡;𝑝), 𝛽(𝑥,𝑡;𝑝), 𝛾(𝑥,𝑡;𝑝) approaches from 𝑓₀(𝑥,𝑡), 𝑔₀(𝑥,𝑡), ℎ₀(𝑥,𝑡) to 𝑓(𝑥,𝑡), 𝑔(𝑥,𝑡), ℎ(𝑥,𝑡), where 𝑓₀(𝑥,𝑡), 𝑔₀(𝑥,𝑡), ℎ₀(𝑥,𝑡) are obtained from (4) for 𝑝=0: ℒ₁𝑓₀(𝑥,𝑡)+𝑠₁(𝑥,𝑡)=0,ℬ₁𝑓₀,𝜕𝑓₀ℒ𝜕𝑥=0,₂𝑔₀(𝑥,𝑡)+𝑠₂(𝑥,𝑡)=0,ℬ₂𝑔₀,𝜕𝑔₀ℒ𝜕𝑥=0,₃ℎ₀(𝑥,𝑡)+𝑠₃(𝑥,𝑡)=0,ℬ₃ℎ₀,𝜕ℎ₀𝜕𝑥=0. (7)

We choose auxiliary functions 𝐻₁(𝑝), 𝐻₂(𝑝), 𝐻₃(𝑝) in the form 𝐻₁𝑝=𝑝𝐶₁₁+𝑝²𝐶₁₂+𝑝³𝐶₁₃+⋯+𝑝^𝑚𝐶_1𝑚,𝐻₂𝑝=𝑝𝐶₂₁+𝑝²𝐶₂₂+𝑝³𝐶₂₃+⋯+𝑝^𝑚𝐶_2𝑚,𝐻₃𝑝=𝑝𝐶₃₁+𝑝²𝐶₃₂+𝑝³𝐶₃₃+⋯+𝑝^𝑚𝐶_3𝑚. (8)

To get the approximate solutions, we expand 𝛼(𝑥,𝑡;𝑝,𝐶_1𝑖), 𝛽(𝑥,𝑡;𝑝,𝐶_2𝑖), 𝛾(𝑥,𝑡;𝑝,𝐶_3𝑖) by Taylor’s series about 𝑝 in the following manner: 𝛼𝑥,𝑡;𝑝,𝐶_1𝑖=𝑓₀(𝑥,𝑡)+_𝑘≥1𝑓_𝑘𝑥,𝑡;𝐶_1𝑖𝑝^𝑘,𝛽𝑥,𝑡;𝑝,𝐶_2𝑖=𝑔₀(𝑥,𝑡)+_𝑙≥1𝑔_𝑙𝑥,𝑡;𝐶_2𝑖𝑝^𝑘,𝛾𝑥,𝑡;𝑝,𝐶_3𝑖=ℎ₀(𝑥,𝑡)+_𝑛≥1ℎ_𝑛𝑥,𝑡;𝐶_3𝑖𝑝^𝑘, (9) where 𝑘=𝑙=𝑛=𝑖=1,2,3,4,…. Now substituting (8)-(9) into (4) and equating the coefficient of like powers of 𝑝, we obtain zeroth order system, given by (7) and the first and second order systems given by (10)-(11), respectively, and the general governing equations for 𝑢_𝑘(𝑥,𝑡) are given by (12): ℒ₁𝑓₁(𝑥,𝑡)−ℒ₁𝑓₀(𝑥,𝑡)=𝐶₁₁ℒ₁𝑓₀(𝑥,𝑡)+𝒩₁𝑓₀(),ℬ𝑥,𝑡₁𝑓₁,𝜕𝑓₁ℒ𝜕𝑥=0₂𝑔₁(𝑥,𝑡)−ℒ₂𝑔₀(𝑥,𝑡)=𝐶₂₁ℒ₂𝑔₀(𝑥,𝑡)+𝒩₂𝑔₀(),ℬ𝑥,𝑡₂𝑔₁,𝜕𝑔₁ℒ𝜕𝑥=0₃ℎ₁(𝑥,𝑡)−ℒ₃ℎ₀(𝑥,𝑡)=𝐶₃₁ℒ₃ℎ₀(𝑥,𝑡)+𝒩₃ℎ₀(),ℬ𝑥,𝑡₃ℎ₁,𝜕ℎ₁𝜕𝑥=0ℒ₁𝑓₂(𝑥,𝑡)−ℒ₁𝑓₁(𝑥,𝑡)=𝐶₁₁ℒ₁𝑓₁(𝑥,𝑡)+𝒩₁𝑓₀(𝑥,𝑡),𝑓₁(𝑥,𝑡)+𝐶₁₂ℒ₁𝑓₀(𝑥,𝑡)+𝒩₁𝑓₀(,ℬ𝑥,𝑡)₁𝑓₂,𝜕𝑓₂ℒ𝜕𝑥=0,₂𝑔₂(𝑥,𝑡)−ℒ₂𝑔₁(𝑥,𝑡)=𝐶₂₁ℒ₂𝑔₁(𝑥,𝑡)+𝒩₂𝑔₀(𝑥,𝑡),𝑔₁(𝑥,𝑡)+𝐶₂₂ℒ₂𝑔₀(𝑥,𝑡)+𝒩₂𝑔₀(),ℬ𝑥,𝑡₂𝑔₂,𝜕𝑔₂ℒ𝜕𝑥=0,₃ℎ₂(𝑥,𝑡)−ℒ₃ℎ₁(𝑥,𝑡)=𝐶₃₁ℒ₃ℎ₁(𝑥,𝑡)+𝒩₃ℎ₀(𝑥,𝑡),ℎ₁(𝑥,𝑡)+𝐶₃₂ℒ₃ℎ₀(𝑥,𝑡)+𝒩₃ℎ₀(),ℬ𝑥,𝑡₃ℎ₂,𝜕ℎ₂𝜕𝑥=0,ℒ₁𝑓₃(𝑥,𝑡)−ℒ₁𝑓₂(𝑥,𝑡)=𝐶₁₁ℒ₁𝑓₂(𝑥,𝑡)+𝒩₁𝑓₀(𝑥,𝑡),𝑓₁(𝑥,𝑡),𝑓₂(𝑥,𝑡)+𝐶₁₂ℒ₁𝑓₁(𝑥,𝑡)+𝒩₁𝑓₀(𝑥,𝑡),𝑓₁(𝑥,𝑡)+𝐶₁₃ℒ₁𝑓₀(𝑥,𝑡)+𝒩₁𝑓₀()ℬ𝑥,𝑡₁𝑓₃,𝜕𝑓₃ℒ𝜕𝑥=0,₂𝑔₃(𝑥,𝑡)−ℒ₂𝑔₂(𝑥,𝑡)=𝐶₂₁ℒ₂𝑔₂(𝑥,𝑡)+𝒩₂𝑔₀(𝑥,𝑡),𝑔₁(𝑥,𝑡),𝑔₂(𝑥,𝑡)+𝐶₂₂ℒ₂𝑔₁(𝑥,𝑡)+𝒩₂𝑔₀(𝑥,𝑡),𝑔₁(𝑥,𝑡)+𝐶₂₃ℒ₂𝑔₀(𝑥,𝑡)+𝒩₂𝑔₀(),ℬ𝑥,𝑡₂𝑔₃,𝜕𝑔₃ℒ𝜕𝑥=0,₃ℎ₃(𝑥,𝑡)−ℒ₃ℎ₂(𝑥,𝑡)=𝐶₃₁ℒ₃ℎ₂(𝑥,𝑡)+𝒩₃ℎ₀(𝑥,𝑡),ℎ₁(𝑥,𝑡),ℎ₂(𝑥,𝑡)+𝐶₃₂ℒ₃ℎ₁(𝑥,𝑡)+𝒩₃ℎ₀(𝑥,𝑡),ℎ₁(𝑥,𝑡)+𝐶₃₃ℒ₃ℎ₀(𝑥,𝑡)+𝒩₃ℎ₀(),ℬ𝑥,𝑡₃ℎ₃,𝜕ℎ₃𝜕𝑥=0,ℒ₁𝑓_𝑘(𝑥,𝑡)−ℒ₁𝑓_𝑘−1()=𝑥,𝑡_𝑘𝑖=1𝐶_1𝑖ℒ₁𝑓_𝑘−𝑖(𝑥,𝑡)+𝒩₁𝑓₀(𝑥,𝑡),𝑓₁(𝑥,𝑡),…,𝑓_𝑘−𝑖(),ℬ𝑥,𝑡𝑘=2,3,…,₁𝑓_𝑘,𝜕𝑓_𝑘ℒ𝜕𝑥=0,₂𝑔_𝑘(𝑥,𝑡)−ℒ₂𝑔_𝑘−1()=𝑥,𝑡_𝑘𝑖=1𝐶_2𝑖ℒ₂𝑔_𝑘−𝑖(𝑥,𝑡)+𝒩₂𝑔₀(𝑥,𝑡),𝑔₁(𝑥,𝑡),…,𝑔_𝑘−𝑖(),ℬ𝑥,𝑡𝑘=2,3,…,₂𝑔_𝑘,𝜕𝑔_𝑘ℒ𝜕𝑥=0,₃ℎ_𝑘(𝑥,𝑡)−ℒ₃ℎ_𝑘−1()=𝑥,𝑡_𝑘𝑖=1𝐶_3𝑖ℒ₃ℎ_𝑘−𝑖(𝑥,𝑡)+𝒩₃ℎ₀(𝑥,𝑡),ℎ₁(𝑥,𝑡),…,ℎ_𝑘−𝑖(),ℬ𝑥,𝑡𝑘=2,3,…,₃ℎ_𝑘,𝜕ℎ_𝑘𝜕𝑥=0. (10)(11)(12)(13)

It has been observed that the convergence of the series (9) depends upon the auxiliary constants 𝐶₁₁, 𝐶₁₂, 𝐶₁₃, …𝐶₂₁, 𝐶₂₂, 𝐶₂₃, …𝐶₃₁, 𝐶₃₂, 𝐶₃₃, …. If it is convergent at 𝑝=1, one has 𝛼^∗𝑥,𝑡;𝐶_1𝑖=𝑓₀(𝑥,𝑡)+_𝑘≥1𝑓_𝑘𝑥,𝑡;𝐶_1𝑖,𝛽^∗𝑥,𝑡;𝐶_2𝑖=𝑔₀(𝑥,𝑡)+_𝑙≥1𝑔_𝑙𝑥,𝑡;𝐶_2𝑖,𝛾𝑖=1,2,…𝑚_∗𝑥,𝑡;𝐶_3𝑖=ℎ₀(𝑥,𝑡)+_𝑛≥1ℎ_𝑛𝑥,𝑡;𝐶_3𝑖. (14)

Substituting (14) into (1.1), the following expression for residuals results: 𝑅₁𝑥,𝑡;𝐶_1𝑖=ℒ₁𝛼^∗𝑥,𝑡;𝐶_1𝑖+𝑠₁(𝑥,𝑡)+𝒩₁𝛼^∗𝑥,𝑡;𝐶_1𝑖,𝑅₂𝑥,𝑡;𝐶_2𝑖=ℒ₂𝛽^∗𝑥,𝑡;𝐶_2𝑖+𝑠₂(𝑥,𝑡)+𝒩₂𝛽^∗𝑥,𝑡;𝐶_2𝑖,𝑅₃𝑥,𝑡;𝐶_3𝑖=ℒ₃𝛾^∗𝑥,𝑡;𝐶_3𝑖+𝑠₃(𝑥,𝑡)+𝒩₃𝛾^∗𝑥,𝑡;𝐶_3𝑖. (15)

If 𝑅₁(𝑥,𝑡;𝐶_1𝑖)=0, 𝑅₂(𝑥,𝑡;𝐶_2𝑖)=0, 𝑅₃(𝑥,𝑡;𝐶_3𝑖)=0 then 𝛼^∗(𝑥,𝑡;𝐶_1𝑖), 𝛽^∗(𝑥,𝑡;𝐶_2𝑖), and 𝛾^∗(𝑥,𝑡;𝐶_3𝑖) will be the exact solutions of the problem. Generally it does not happen, especially in nonlinear problems.

For the computation of auxiliary constants, 𝐶_1𝑖, 𝐶_2𝑖, 𝐶_3𝑖, 𝑖=1,2,…,𝑚, there are different methods like Galerkin’s method, Ritz method, least squares method, and collocation method. One can apply the method of least squares as follows: 𝐽₁𝐶_1𝑖=^𝑡_0Ω𝑅²₁𝑥,𝑡;𝐶_1𝑖𝐽𝑑𝑥𝑑𝑡,₂𝐶_2𝑖=^𝑡_0𝜓𝑅²₂𝑥,𝑡;𝐶_2𝑖𝐽𝑑𝑥𝑑𝑡,₃𝐶_3𝑖=^𝑡_0𝜙𝑅²₃𝑥,𝑡;𝐶_3𝑖𝑑𝑥𝑑𝑡,𝜕𝐽₁𝜕𝐶₁₁=𝜕𝐽₁𝜕𝐶₁₂=⋯𝜕𝐽₁𝜕𝐶_1𝑚=𝜕𝐽₂𝜕𝐶₂₁=𝜕𝐽₂𝜕𝐶₂₂=⋯𝜕𝐽₂𝜕𝐶_2𝑚=𝜕𝐽₃𝜕𝐶₃₁=𝜕𝐽₃𝜕𝐶₃₂=⋯=𝜕𝐽₃𝜕𝐶_3𝑚=0. (16)(17)

The 𝑚th order approximate solution can be obtained by these constants so-obtained. The more general auxiliary functions 𝐻₁(𝑝), 𝐻₂(𝑝), 𝐻₃(𝑝) are useful for convergence, which depends upon constants 𝐶₁₁, 𝐶₁₂, 𝐶₂₁, 𝐶₂₂, 𝐶₃₁, 𝐶₃₂,…, can be optimally identified by (17), and is useful in error minimization.

3. Application of Extended OHAM to Coupled Schrödinger-KdV Equation

To demonstrate the effectiveness of the extended OHAM for coupled Schrödinger- KdV equation taken from [6], we have 𝜕𝛽(𝑥,𝑡)−𝜕𝜕𝑡²𝜂(𝑥,𝑡)𝜕𝑥²−𝜂(𝑥,𝑡)𝜇(𝑥,𝑡)=0,𝜕𝜂(𝑥,𝑡)+𝜕𝜕𝑡²𝛽(𝑥,𝑡)𝜕𝑥²()𝜇()()+𝛽𝑥,𝑡𝑥,𝑡=0,𝜕𝜇𝑥,𝑡()()𝜕𝑡+6𝜇𝑥,𝑡𝜕𝜇𝑥,𝑡+𝜕𝜕𝑥³𝜇(𝑥,𝑡)𝜕𝑥³()()−2𝛽𝑥,𝑡𝜕𝜇𝑥,𝑡()()𝜕𝑥−2𝜂𝑥,𝑡𝜕𝜂𝑥,𝑡𝜕𝑥=0, (18)(19) with boundary conditions 𝛽()(𝑥),𝜂()(𝑥),𝜇()=3𝑥,0=cos𝑥,0=sin𝑥,04. (20)

The exact solution of (19) for −3≤𝑥≤3 and 0≤𝑡≤1 is given by 𝛽()𝑡𝑥,𝑡=cos𝑥+4,𝜂()𝑡𝑥,𝑡=sin𝑥+4,𝜇()=3𝑥,𝑡4. (21)

Applying the extended OHAM technique discussed in Section 2,()1−𝑝𝜕𝛽𝑥,𝑡𝜕𝑡−𝐻₁𝑝𝜕𝛽(𝑥,𝑡)−𝜕𝜕𝑡²𝜂(𝑥,𝑡)𝜕𝑥²()𝜇(()−𝜂𝑥,𝑡𝑥,𝑡)=0,1−𝑝𝜕𝜂𝑥,𝑡𝜕𝑡−𝐻₂𝑝𝜕𝜂(𝑥,𝑡)+𝜕𝜕𝑡²𝛽(𝑥,𝑡)𝜕𝑥²()𝜇(()+𝛽𝑥,𝑡𝑥,𝑡)=0,1−𝑝𝜕𝜇𝑥,𝑡𝜕𝑡−𝐻₃𝑝𝜕𝜇(𝑥,𝑡)()()𝜕𝑡+6𝜇𝑥,𝑡𝜕𝜇𝑥,𝑡+𝜕𝜕𝑥³𝜇(𝑥,𝑡)𝜕𝑥³()()−2𝛽𝑥,𝑡𝜕𝜇𝑥,𝑡()()𝜕𝑥−2𝜂𝑥,𝑡𝜕𝜂𝑥,𝑡𝜕𝑥=0. (22)

We consider 𝛽=𝛽₀+𝑝𝛽₁+𝑝²𝛽₂,𝜂=𝜂₀+𝑝𝜂₁+𝑝²𝜂₂,𝜇=𝜇₀+𝑝𝜇₁+𝑝²𝜇₂,𝐻₁𝑝=𝑝𝐶₁₁+𝑝²𝐶₁₂,𝐻₂𝑝=𝑝𝐶₂₁+𝑝²𝐶₂₂,𝐻₃𝑝=𝑝𝐶₃₁+𝑝²𝐶₃₂. (23)Zeroth Order System. Consider 𝜕𝛽₀𝜕𝑡=0,𝜕𝜂₀𝜕𝑡=0,𝜕𝜇₀𝜕𝑡=0, (24) with initial conditions 𝛽₀()(𝑥),𝜂𝑥,0=cos₀()(𝑥),𝜇𝑥,0=sin₀()=3𝑥,04. (25)

Its solution is 𝛽₀()(𝑥),𝜂𝑥,𝑡=cos₀()(𝑥),𝜇𝑥,𝑡=sin₀()=3𝑥,𝑡4. (26)First Order System. Consider 𝜕𝛽₁(𝑥,𝑡)=𝜕𝑡1+𝐶₁₁𝜕𝛽₀𝜕𝑡−𝐶₁₁𝜂₀𝜇₀−𝐶₁₁𝜕²𝜂₀𝜕𝑥²,𝜕𝜂₁(𝑥,𝑡)=𝜕𝑡1+𝐶₂₁𝜕𝜂₀𝜕𝑡+𝐶₂₁𝛽₀𝜇₀+𝐶₂₁𝜕²𝛽₀𝜕𝑥²,𝜕𝜇₁(𝑥,𝑡)=𝜕𝑡1+𝐶₃₁𝜕𝜇₀𝜕𝑡−2𝐶₃₁𝛽₀𝜕𝛽₀𝜕𝑥+𝜂₀𝜕𝜂₀𝜕𝑥−3𝜇₀𝜕𝜇₀𝜕𝑥+𝐶₃₁𝜕³𝜇₀𝜕𝑥³ (27) with 𝛽₁(𝑥,0)=0,𝜂₁(𝑥,0)=0,𝜇₁(𝑥,0)=0. (28)

Its solution is 𝛽₁𝑥,𝑡,𝐶₁₁=𝐶₁₁4𝜂𝑡sin𝑥,₁𝑥,𝑡,𝐶₂₁𝐶=−²¹4𝜇𝑡cos𝑥,₁𝑥,𝑡,𝐶₃₁=0. (29)Second Order System. Consider 𝜕𝛽₂(𝑥,𝑡)=𝜕𝑡1+𝐶₁₁𝜕𝛽₁𝜕𝑡−𝐶₁₁𝜂₁𝜇₀+𝜂₀𝜇₁+𝐶₁₂𝛽₀−𝜂₀𝜇₀−𝐶₁₂𝜕²𝜂₀𝜕𝑥²−𝐶₁₁𝜕²𝜂₁𝜕𝑥²,𝜕𝜂₂(𝑥,𝑡)=𝜕𝑡1+𝐶₂₁𝜕𝜂₀𝜕𝑡+𝐶₂₁𝛽₁𝜇₀+𝛽₀𝜇₁+𝐶₂₂𝜂₀+𝛽₀𝜇₀+𝐶₂₁𝜕²𝛽₁𝜕𝑥²+𝐶₂₂𝜕²𝜇₀𝜕𝑥²,𝜕𝜇₂(𝑥,𝑡)=𝜕𝑡1+𝐶₃₁𝜕𝜇₁𝜕𝑡−2𝐶₃₁×𝛽₁𝜕𝛽₀𝜕𝑥+𝛽₀𝜕𝛽₁𝜕𝑥+𝜂₁𝜕𝜂₀𝜕𝑥+𝜂₀𝜕𝜂₁𝜕𝑥−3𝜇₁𝜕𝜇₀𝜕𝑥−3𝜇₀𝜕𝜇₁𝜕𝑥−2𝐶₃₂𝛽₀𝜕𝛽₀𝜕𝑥+𝜂₀𝜕𝜂₀𝜕𝑥−3𝜇₀𝜕𝜇₀𝜕𝑥+𝐶₃₁𝜕𝜇₀𝜕𝑥+𝐶₃₂𝜕³𝜇₀𝜕𝑥³+𝐶₃₁𝜕³𝜇₁𝜕𝑥³ (30) with 𝛽₂(𝑥,0)=0,𝜂₂(𝑥,0)=0,𝜇₂(𝑥,0)=0. (31)

The solution of second order system is 𝛽₂𝑥,𝑡,𝐶₁₁,𝐶₁₂=1328𝐶₁₁𝑡sin𝑥+8𝐶²₁₁𝑡sin𝑥+8𝐶₁₂𝑡sin𝑥−𝑡²𝐶₁₁𝐶₂₁,𝜂cos𝑥₂𝑥,𝑡,𝐶₂₁,𝐶₂₂1=−328𝐶₂₁cos𝑥+8𝐶²₂₁𝑡cos𝑥+𝐶₁₁𝐶₂₁𝑡²sin𝑥+𝐶₂₂,𝜇𝑡cos𝑥₂𝑥,𝑡,𝐶₃₁=14𝐶₂₁𝐶₃₁𝑡²cos(2𝑥)−𝐶₁₁𝐶₃₁𝑡²cos(2𝑥). (32)

Adding (26), (29), and (32), we obtain 𝛽𝑥,𝑡,𝐶₁₁,𝐶₂₁(𝑥)+1=cos4𝐶₁₁(𝑥)+1𝑡sin328𝐶₁₁𝑡sin𝑥+8𝐶²₁₁𝑡sin𝑥+8𝐶₁₂𝑡sin𝑥−𝑡²𝐶₁₁𝐶₂₁,𝜂cos𝑥𝑥,𝑡,𝐶₂₁,𝐶₂₂(𝑥)−1=sin4𝐶₂₁(𝑥)−1𝑡sin328𝐶₂₁cos𝑥+8𝐶²₂₁𝑡cos𝑥+𝐶₁₁𝐶₂₁𝑡²sin𝑥+𝐶₂₂,𝜇𝑡cos𝑥𝑥,𝑡,𝐶₃₁=34+14𝐶₂₁𝐶₃₁𝑡²cos(2𝑥)−𝐶₁₁𝐶₃₁𝑡²cos(2𝑥). (33)

For the calculation of the constants 𝐶₁₁, 𝐶₁₂, 𝐶₂₁, 𝐶₂₂, and 𝐶₃₁ using (33) in (19) and applying the method of least square mentioned in (16)-(17) by taking, we get 𝐶₁₁=−3.041182429907255×10⁻¹⁴,𝐶₁₂𝐶=−1.1871110474593864₂₁=−3.041182429907255×10⁻¹⁴,𝐶₂₂𝐶=−0.999258572471839,₃₁=−8.101774168020832×10⁻¹⁵.𝛽(𝑥,𝑡)=cos(𝑥)−7.602956074768137×10⁻¹⁵(𝑥)+1𝑡sin32−9.248790571976596×10⁻²⁸𝑡²(𝑥)(𝑥),𝜂()=cos−9.496888379675335𝑡sin𝑥,𝑡7.602956074768137×10⁻¹⁵(𝑥)(𝑥)+1𝑡cos+sin(𝑥)327.994064457977714𝑡cos−9.248790571976596×10⁻²⁸𝑡²(𝑥),𝜇()=3sin𝑥,𝑡4. (34)(35)

The solution obtained by variational iteration method [6] is as follows: 𝛽()(𝑥)−1𝑥,𝑡=cos4(𝑥)−1𝑡sin𝑡32²(𝑥),𝜂()(𝑥)+1cos𝑥,𝑡=sin4(𝑥)−1𝑡cos𝑡32²(𝑥),𝜇()=3sin𝑥,𝑡4. (36)

The solution obtained by homotopy perturbation method [2] is as follows: 𝛽()(𝑥)−1𝑥,𝑡=cos4(𝑥)−1𝑡sin𝑡32²(𝑥)+1cos𝑡384³(𝑥),𝜂()(𝑥)+1sin𝑥,𝑡=sin4(𝑥)−1𝑡cos𝑡32²(𝑥)−1sin𝑡384³(𝑥),𝜇()=3cos𝑥,𝑡4. (37)

4. Results and Discussions

The extended mathematical theory presented in Section 2 gives highly accurate solutions for the BVP presented in Section 3. We have used Mathematica 7 for most of our computational work. In Tables 1 and 2, the absolute errors of extended OHAM results corresponding to the exact solution of the problem given in (19) are given for 𝛽(𝑥,𝑡), 𝜂(𝑥,𝑡) at 𝑡=0.5, 0.2, 0.1, 0.01, 0.001. In Table 3, the extended OHAM results are compared with exact solution for 𝜇(𝑥,𝑡). In Figures 1, 2, 3, 4, 5, and 6, 3D plots of the results obtained by extended OHAM and exact solutions are plotted for 𝛽(𝑥,𝑡), 𝜂(𝑥,𝑡), 𝜇(𝑥,𝑡) at fixed value of 𝑡=0.1. While the 3D plots of the residuals at 𝑡=0.1 are given in Figures 7, 8, and 9, Figures 10, 11, 12, 13, 14, and 15 show the 2D plots of extended OHAM and exact solutions for 𝛽(𝑥,𝑡), 𝜂(𝑥,𝑡), 𝜇(𝑥,𝑡) at 𝑡=0.1. The 2D residual plots are given in Figures 16, 17, and 18 at 𝑡=0.1. In Figures 19, 20, and 21 the 2D plots of ordered solutions are given at 𝑡=0.1 for 𝛽(𝑥,𝑡), 𝜂(𝑥,𝑡), 𝜇(𝑥,𝑡). The extended OHAM solutions are compared with VIM and HPM solution in 2D, Figures 22, 23, and 24. Here we observed from Figures 19–21 that the extended OHAM solution converges rapidly with the increase in the order of approximation and produced better results. From Figures 22–24 it is evident that the extended OHAM results are nearly identical to the exact, VIM, and HPM solutions. Here the results are very consistent with the decreasing time as evident from Tables 1-2.

Table 1 

Absolute error of OHAM solution 𝛽(𝑥,𝑡) corresponding to the exact solution.

Table 2 

Absolute error of OHAM solution 𝜂(𝑥,𝑡) corresponding to the exact solution.

Table 3 

Comparison of 𝜇(𝑥,𝑡) solutions obtained by OHAM to the exact solution.

Figure 1 

3D, OHAM solution of 𝛽(𝑥,𝑡) at 𝑡=0.1.

Figure 2 

3D, exact solution of 𝛽(𝑥,𝑡) at 𝑡=0.1.

Figure 3 

3D, OHAM solution of 𝜂(𝑥,𝑡) at 𝑡=0.1.

Figure 4 

3D, exact solution of 𝜂(𝑥,𝑡) at 𝑡=0.1.

Figure 5 

3D, OHAM solution of 𝜇(𝑥,𝑡) at 𝑡=0.1.

Figure 6 

3D, exact solution of 𝜇(𝑥,𝑡) at 𝑡=0.1.

Figure 7 

3D, residual of 𝛽(𝑥,𝑡) at 𝑡=0.1.

Figure 8 

3D, residual of 𝜂(𝑥,𝑡) at 𝑡=0.1.

Figure 9 

3D, residual of 𝜇(𝑥,𝑡) at 𝑡=0.1.

Figure 10 

2D, OHAM solution of 𝛽(𝑥,𝑡) at 𝑡=0.1.

Figure 11 

2D, exact solution of 𝛽(𝑥,𝑡) at 𝑡=0.1.

Figure 12 

2D, OHAM solution of 𝜂(𝑥,𝑡) at 𝑡=0.1.

Figure 13 

2D, exact solution of 𝜂(𝑥,𝑡) at 𝑡=0.1.

Figure 14 

2D, OHAM solution of 𝜇(𝑥,𝑡) at 𝑡=0.1.

Figure 15 

2D, exact solution of 𝜇(𝑥,𝑡) at 𝑡=0.1.

Figure 16 

2D, residual of 𝛽(𝑥,𝑡) at 𝑡=0.1.

Figure 17 

2D, residual of 𝜂(𝑥,𝑡) at 𝑡=0.1.

Figure 18 

2D, residual of 𝜇(𝑥,𝑡) at 𝑡=0.1.

Figure 19 

2D, comparison of ordered solutions for 𝛽(𝑥,𝑡) at 𝑡=0.1.

Figure 20 

2D, comparison of ordered solutions for 𝜂(𝑥,𝑡) at 𝑡=0.1.

Figure 21 

2D, comparison of ordered solutions for 𝜇(𝑥,𝑡) at 𝑡=0.1.

Figure 22 

2D, comparison of OHAM, VIM, HPM, and exact solutions for 𝛽(𝑥,𝑡) at 𝑡=0.1.

Figure 23 

2D, comparison of OHAM, VIM, HPM, and exact solutions for 𝜂(𝑥,𝑡) at 𝑡=0.1.

Figure 24 

2D, comparison of OHAM, VIM, HPM, and exact solutions for 𝜇(𝑥,𝑡) at 𝑡=0.1.

5. Conclusion

In this paper, we have seen the effectiveness of extended OHAM to coupled Schrödinger-KdV equation. By applying the basic idea of extended OHAM to coupled Schrödinger-KdV equation, we found it simpler in applicability and more convenient to control convergence and that it involved less computational overhead. Therefore, extended OHAM shows its validity and great potential for the solution of nonlinear system of PDEs problems in science and engineering.

Conflict of Interests

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