A convergence analysis of the ADM and an application

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Abstract

In this paper by considering the decomposition scheme, we solve the modified Korteweg–de Vries equation (MKdV for short) for the initial condition. We prove the convergence of Adomian decomposition method (ADM) applied to the MKdV equation.

Introduction

Consider the solution u(x,t) of the nonlinear partial differential equationut+cumux3ux3=0,where c and ν are constants and m=0,1,2. If m=0 and m=1, Eq. (1) becomes linear KdV and nonlinear KdV equations, respectively. The nonlinear KdV equation possesses steady progressing wave solutions. It is well known that KdV equation can be written down on the basis that both nonlinearity and dispersion might occur together. In 1895 Korteweg and de Vries showed that long waves, in water of relatively shallow depth, could be described approximately by a nonlinear equation [1], [2].

In [3], we implemented the Adomian decomposition method (in short ADM) [4], [5] for the exact solution and approximate solution of a linear Korteweg–de Vries like equation, i.e. m=0. Nonlinear phenomena play a crucial role in applied mathematics and physics. The nonlinear problem are solved easily and elegantly without linearizing the problem by using the ADM. The nonlinear KdV (i.e., m=1 in Eq. (1)) equation has been the focus of considerable recent studies by [6], [7], [8], [9], [10]. Gardner [9] developed a variational and its Hamiltonian formulation to handle this problem and also Gardner et al. [10] introduced various methods for explicit solutions of Eq. (1). Explicit solutions to the nonlinear equations are of fundamental importance. Various methods for obtaining explicit solutions to nonlinear evolution equations have been proposed. Among them are Hirota’s dependent variable transformation, the inverse scattering transform, and the Bäcklund transformation. All these methods are described in [1], [8] and the references therein. A feature common to all these methods is that they are using the transformations to reduce the equation into more simple equation then solve it. Unlike classical techniques, the nonlinear equations are solved easily and elegantly without transforming the equation by using the ADM. The technique has many advantages over the classical techniques, mainly, it avoids linearization and perturbation in order to find solutions of a given nonlinear equations. It is providing an efficient explicit solution with high accuracy, minimal calculation, avoidance of physically unrealistic assumptions.

In [11], we implemented the decomposition method for solving the nonlinear KdV equation by using the ADM. In this paper, we will consider various MKdV equations to find explicit solutions and numerical solutions of these equations by using the ADM rather than the traditional methods. The decomposition scheme will be illustrated by studying suitable MKdV examples, homogeneous or inhomogeneous, to compute explicit and numerical solutions. Furthermore, we will also illustrate self-canceling phenomena for homogeneous as well as inhomogeneous the MKdV equation using the modified decomposition method (in short MDM) [12].

The method is useful for obtaining both approximate and numerical approximations of linear or nonlinear differential equations and it is also quite straightforward to write computer codes in any symbolic languages. If the numerical solutions are necessary to compute, the rapid convergence is obvious. Furthermore, as the decomposition method does not require discretization of the variables, it is not effected by computation round off errors and one is not faced with necessity of large computer memory and time. There are some rather significant advantages over methods which must assume linearity, “smallness”, deterministic behavior, etc. The method has features in common with many other methods, but it is distinctly different on close examination, and one should not be mislead by apparent simplicity into superficial conclusions [4], [5].

Section snippets

Analysis of the method

In the preceding section we have discussed particular devices of the general type of the MKdV equation. For purposes of illustration of the ADM, in this study we shall consider Eq. (1) in its standard formut+cu2ux+3ux3=f(x,t),where c is a constant, the solution of which is to be obtained subject to the initial condition u(x,0)=g(x).

Following [4], [5] we define for the above equation the linear operators L̷t=t, L̷x=3x3 and the definite integration inverse operator L̷−1t. Therefore the

Application and results

Example 1

For purposes of illustration of the decomposition method for solving homogenous a general form of the MKdV equation (see [8, p. 35]). We considered, the first equation has the rational solution of which is to be obtained subject to the initial conditionut+6u2ux+uxxx=0,u(x,0)=c−4c4c2x2+1,where c is any real constant.

The solution of this equation, we simply taken the equation in a operator form exactly in the same manner as the form of Eq. (3) and using (5) to find the zeroth component of u0 asu0

Numerical experiments and convergence analysis of the ADM

For numerical comparisons purposes, we consider two MKdV equations. The first one has the rational and the second one has the traveling-wave solutions. Based on the ADM, we constructed the solution u(x,t) aslimn→∞φn=u(x,t),whereφn(x,t)=∑nk=0uk(x,t),n⩾0,and the recurrence relation is given as in (6). Moreover, the decomposition series (26) solutions are generally converge very rapidly in real physical problems [4]. The convergence of the decomposition series have investigated by several authors

References (19)

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