Abstract

In this article the approximate solutions of nonlinear Hirota Satsuma coupled Korteweg De- Vries (KDV) and modified coupled KDV equations have been obtained by using reliable algorithm of New Iterative Method (NIM). The results obtained give higher accuracy than that of homotopy analysis method (HAM). The obtained solutions show that NIM is effective, simpler, easier, and explicit and gives a suitable way to control the convergence of the approximate solution.

1. Introduction

The nonlinear coupled system of partial differential equations has variety of applications in different fields of mechanics, biology, hydrodynamics, and plasma physics. The exact solution of nonlinear partial differential equations cannot be found easily. To find the approximate solutions of these types of equations many researchers have adopted different approaches. The well-known approaches for approximate solutions of coupled system of differential equations are numerical methods, iterative methods, perturbation methods, homotopy based methods, etc. Each approach has its own advantages and disadvantages. In numerical methods discretization is used which affect the accuracy. They also required most computational work and time. In the case of strong nonlinear problems the numerical methods do not give us more accurate result. Some numerical methods are collocation methods (CMs) [1, 2], finite difference method (FDM) [3, 4], finite element method (FEM) [5, 6], radial basis function (RBF) method [7, 8], etc. Perturbation techniques use perturbation quantities to transform a nonlinear problem into infinite number of linear subproblems. Perturbation methods have some limitations as well, such as nonavailability of perturbative quantity and strong nonlinearity. To avoid this difficulty, there do exist homotopy based methods as well [9–11]. Adomian Decomposition Method (ADM) [11] and Differential Transformation Method (DTM) [12] deal with the nonlinearity but their main drawback is that the convergence region of obtained solution is generally small. Iterative methods are mathematical techniques that produce a sequence of approximate solutions. They used initial guess to create sequential approximations. Since these methods involve repetition of the same process many times, computers can act well for finding solutions of equations iteratively. A definite way of operation of an iterative method is termination criteria. Iterative methods have important role in finding approximate solution of differential equations. Some of the well-known iterative methods are variational iteration method (VIM) [13, 14], modified variational iteration method (MVIM) [15, 16], optimal variational iteration method (OVIM) [17], etc.

Recently Daftardar-Gejji and Jafari [18] have proposed an iterative method for solving linear and nonlinear functional equations called the New Iterative Method (NIM). This method has proven useful for solving a variety of nonlinear equations such as algebraic equations, ODEs, PDEs, evolution equations, and system of nonlinear dynamical systems [19–33].

The KDV equation arises in the study of nonlinear dispersive waves [34]. It was derived by Korteweg and de Vries in 1895 for modeling of shallow water waves in canal [35]. This equation plays important role in diverse areas of engineering and scientific applications and, therefore, enormous amount of research work has been invested in the study of such equations. Wu et al. derived a new corresponding hierarchy of nonlinear evolution equations [35]. Fan worked on the extended tanh-function method and symbolic computation to obtain, respectively, four kinds of Soliton solutions for a new coupled MKDV and new generalized Hirota Satsuma coupled KDV equations [36]. Raslan et al. worked on the decomposition method to obtain the Soliton solutions for generalized Hirota Satsuma coupled KDV and coupled MKDV equations [37]. Zhong et al. applied the VIM to obtain approximate analytic solutions of coupled MKDV and generalized Hirota Satsuma coupled KDV equations [38]. Kangalgil et al. demonstrated the feasibility and validity of DTM for proposed equations [39]. Arife et al. presented the numerical solution of systems of Hirota Satsuma coupled KDV and coupled MKDV equations by means of HAM [9]. Ghoreishi et al. applied HAM for obtaining the approximate solution of MKDV system [40].

Our aim in this paper is to extend the application of NIM for finding the approximate solutions of nonlinear Hirota Satsuma coupled KDV and modified coupled KDV equations. The results obtained are in close agreement with the exact solutions than that of HAM. The accuracy of NIM can further be improved by increasing the number of iterations.

2. The Governing Coupled System

Recently, by introducing a 4 × 4 matrix spectral problem with three potentials, Wu et al. derived a new corresponding hierarchy of nonlinear evolution equations [35]. Two typical equations in the hierarchy are new generalized Hirota Satsuma coupled KDV and new coupled MKDV equations.

We considered the following new generalized Hirota Satsuma coupled KDV equation,and modified coupled KDV equation

With (1)-(2) reduce to new complex coupled Hirota Satsuma KDV and modified coupled complex KDV equations, respectively [35, 41]. For , (2) reduces to generalized KDV equation and, for , it reduces to modified KDV equation.

2.1. Basic Idea of NIM

The basic mathematical theory of New Iterative Method is described as follows.

Let us consider the nonlinear equation of the form ofwhere is known function and denotes nonlinear operator from a Banach space BB. According to the basic idea of NIM solution of the above equation has the series formThe nonlinear operator can be decomposed as Hence general equation of (1) can be written as From (6) we have and in general

The k-term approximate solution of the general equation (1) is

2.2. Convergence Criteria of NIM

Theorem 1. If is in a neighborhood of and for any and for some real , then the series is absolutely convergent and Sufficient condition for convergence is as follows.

Theorem 2. If is , then the series is absolutely convergent.

The reader is referred to [18, 22] to get more insight of convergence analysis.

3. Implementation of NIM to Hirota Satsuma Coupled KDV and Coupled MKDV Equations

Problem 3. Consider the coupled Hirota Satsuma KDV equation given by (1) together with following initial conditions [35].where , . By using the initial conditions, the given equation (1) is equivalent to the following integral equations:Appling the basic idea of NIM and using the recursive relation (6) we have the following.

Zeroth Order SolutionsFirst Order SolutionsSecond Order Solutions The NIM yields the 2nd order solutions as follows: