A MAPLE package of new ADM-Padé approximate solution for nonlinear problems

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Abstract

Based on the new definition of four classes of Adomian polynomials proposed by Rach in 2008, a MAPLE package of new Adomian-Padé approximate solution for solving nonlinear problems is presented. This package combines the merits of the Adomian decomposition method and the diagonal Padé technique, and may give more accurate solutions of nonlinear problems with strong nonlinearity. Besides, the package is user-friendly and efficient, one only needs to input the initial conditions, governing equation and four optional parameters, then our package will output the analytic approximate solution within a few seconds, where the equation is decomposed into three parts, they are the linear term R, nonlinear term NN and source function g, which are all in functional form. Meanwhile, several graphs generated from the above solutions are displayed and demonstrate a favorable comparison. In this paper, several different types of examples are given to illustrate the validity and promising flexibility of the package. This package provides us with a convenient and useful tool for dealing with nonlinear problems, as well as its electronic version is free to download via the journal website.

Introduction

In various fields of science and engineering, many nonlinear phenomena can be modeled by nonlinear differential or integral equations. Investigating analytic solutions of these equations has attracted considerable interest and intensive research, for these solutions may give more insight into internal aspects of nonlinear problems. However, there are only a few nonlinear differential equations whose exact solutions have been discovered. Therefore, it is very important and necessary to construct analytic approximate solutions of these problems. Various methods and algorithms have been proposed to construct analytic approximate solutions of nonlinear problems. One of the most effective methods is the Adomian decomposition method (ADM) [1], [2], [3]. The ADM has been applied to solve a rather wide range of nonlinear equations. The key point of the method is to decompose the nonlinear operator into a special series of polynomials n=0An, where An represents the Adomian polynomials [4], [5]. Many researchers have focused their attention on this method. Apart from the standard Adomian decomposition method, some revised algorithms have been successively proposed, such as Cherruault, Yang, and Fang have developed different algorithms to rapidly construct the Adomian polynomials [6], respectively. In 2004, Chen and Lu introduced a new algorithm [7] and programmed it in MAPLE. Babolian and Javadi also presented another new method [8] to calculate Adomian polynomials quickly, but the method cannot calculate Adomian polynomials for systems of nonlinear equations. In 2007 Gu and Li extended the above operator and applied it to systems of nonlinear equations [9]. In 2010, by introducing the index vectors of the Adomian polynomials, Duan [10] discovered the recurrence relations of the index vectors and presented a new and simple algorithm for the Adomian polynomials.

In 2008, Rach redefined and unified the family of Adomian polynomials. Furthermore, he has proved that these new types of Adomian polynomials cover all of the above mentioned ones, but these new algorithms are much more efficient. In this paper, we develop the MAPLE package NAPA, which is a complete implementation of Rach’s algorithms together with the Padé technique. NAPA can automatically output analytic approximate solutions for nonlinear differential equations. The effectiveness and validity of the package is demonstrated by applying it to different kinds of examples. It can be seen that this package is effective and promising to obtain accurate analytic approximate solutions for nonlinear equations.

With the above facts in mind, our paper is organized as follows. In Section 2, we briefly describe the basic idea of the new ADM-Padé technique and the new definition of the Adomian polynomials. In Section 3, a MAPLE package is presented. In Section 4, several different examples with analytic solutions are given to demonstrate the effectiveness of the package. In conclusion, we present a discussion and summary.

Section snippets

The new ADM-Padé technique

As we know, the ADM has been widely used to solve nonlinear differential equations in mathematical physics. Intensive research and considerable interest have focused on the study of analytic approximate solutions for nonlinear differential equations. However, it has been demonstrated that the ADM is a valuable basic method for nonlinear problems. In order to overcome some open problems of the ADM and obtain more accurate analytic approximate solutions, in most cases, the basic method should be

The package NAPA

Although the new ADM-Padé approximate method described in the above section is relatively simple in principle, it will be very complicated and time-consuming by manual computation. Fortunately, the new ADM-Padé approximate (NAPA) method is an algorithmic method. Therefore, in this paper we present the MAPLE package NAPA, which can automatically derive analytic approximate solutions of nonlinear differential equations. In our package u is the default dependent variable and t is the independent

The application of NAPA

In this section, some examples are given to show the effectiveness of our package. And it is noted that the elapsed times which are reported in the following is recorded throng a desktop with the processing speed of its CPU in 2.41 GHz and RAM available in 2 GB. For the limitation of the length of paper, we just consider four examples including four different types of nonlinear differential equations and two cases of f : (f  {1, 2, 3, 4} and not f  {1, 2, 3, 4}), and for each example we omit displaying

Conclusions

In various fields of science and engineering, more and more mathematical models are characterized by nonlinear differential equations. Meanwhile, the Adomian decomposition method has been shown to be a straightforward and effective method to solve these equations. Since calculation of the Adomian polynomials plays a paramount role in the Adomian decomposition method, we mechanize the computing process of the decomposition method based on different classes of the Adomian polynomials using MAPLE

Acknowledgments

This work was supported by National Nature Science Fundation of China (10771072) and (11071274).

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