A new algorithm for calculating adomian polynomials for nonlinear operators

https://doi.org/10.1016/S0096-3003(99)00063-6Get rights and content

Abstract

In this paper, a reliable technique for calculating Adomian polynomials for nonlinear operators will be developed. The new algorithm offers a promising approach for calculating Adomian polynomials for all forms of nonlinearity. The algorithm will be illustrated by studying suitable forms of nonlinearity. A nonlinear evolution model will be investigated.

Introduction

The Adomian decomposition method 1, 2, 3 has been applied to a wide class of stochastic and deterministic problems in physics, biology and chemical reactions. For nonlinear models, the method has shown reliable results in supplying analytical approximation that converges very rapidly. It is well known by many that the decomposition method decomposes the linear term u(x,t) into an infinite sum of components un(x,t) defined byu(x,t)=n=0un(x,t).Moreover, the decomposition method identifies the nonlinear term F(u(x,t)) by the decomposition seriesF(u(x,t))=n=0An,where An are the so-called Adomian polynomials. Adomian 1, 2, 3 formally introduced formulas that can generate Adomian polynomials for all forms of nonlinearity.

Recently, a great deal of interest has been focused by 4, 5, 6 among others to develop a practical method for the calculation of Adomian polynomials An. The concern was to develop an alternative technique that will calculate Adomian polynomials in a practical way without any need for the formulas introduced by Adomian 1, 2, 3. However, the methods developed by 4, 5, 6 require a huge size of calculations and employ several formulas identical in spirit to that used by Adomian. We believe, as Adomian and others do, that a simple and reliable technique can be established to make the calculations less dependable on the formulas.

The objective of this paper is to establish a promising algorithm that can be used to calculate Adomian polynomials for nonlinear terms in an easy way. The newly developed method depends mainly on algebraic and trigonometric identities and on Taylor expansions as well.

Section snippets

Analysis of the method

It seems reasonable to summarize the essential steps introduced by Adomian 1, 2, 3 to calculate Adomian polynomials. Given a nonlinear operator F(u(x,t)), the first few polynomials are given byA0=F(u0),A1=u1F(u0),A2=u2F(u0)+12!u12F(u0),A3=u3F(u0)+u1u2F(u0)+13!u13F(u0),A4=u4F(u0)+12!u22+u1u3F(u0)+12!u12u2F(u0)+14!u14F(iv)(u0).Other polynomials can be generated in a similar manner. For more details about the Adomian approach, see Refs. 1, 2, 3, 4, 5, 6, 7, 8, 9.

Two important observations

Nonlinear polynomials

Case 1. F(u)=u2.

We first setu=n=0un.Substituting (5) into F(u)=u2 givesF(u)=(u0+u1+u2+u3+u4+u5+⋯)2.Expanding the expression at the right-hand side givesF(u)=u02+2u0u1+2u0u2+u12+2u0u3+2u1u2+⋯.The expansion in (7) can be rearranged by grouping all terms with the sum of the subscripts of the components of un is the same. This means that we can rewrite (7) as This gives Adomian polynomials for F(u)=u2 byA0=u02,A1=2u0u1,A2=2u0u2+u12,A3=2u0u3+2u1u2,A4=2u0u4+2u1u3+u22,A5=2u0u5+2u1u4+2u2u3.

Case 2. F(u

Application

Example. Solve the homogeneous nonlinear problemut+u2ux=0,u(x,0)=3x.

Solution. In Ref. [10], this evolution equation was handled by using the characteristics method. Applying Lt−1, to both sides of Eq. (51)givesu(x,t)=3x−Lt−1(u2ux).Set u=∑n=0un and equate u2ux=∑n=0An we obtainn=0un(x,t)=3x−Lt−1n=0An,where An are Adomian polynomials that we derived before. The components un(x,t) of the solution u(x,t) can be computed by using the recursive relationu0(x,t)=3x,uk+1(x,t)=−Lt−1Ak,k⩾0,which gives

Discussion and conclusion

The Adomian decomposition method is a powerful method which has provided an efficient potential for the solution of physical applications modeled by nonlinear differential equations. The rise of nonlinear terms is vital to progress in many applied sciences involving dynamical systems. Although the calculation of Adomian polynomials for all forms of nonlinearity was discussed by many, the developed techniques suffered from the numerous formulas needed and from certain computational difficulties.

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