Adomian decomposition method with Chebyshev polynomials

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Abstract

In this paper an efficient modification of the Adomian decomposition method is presented by using Chebyshev polynomials. The proposed method can be applied to linear and non-linear models. The scheme is tested for some examples and the obtained results demonstrate reliability and efficiency of the proposed method.

Introduction

The Adomian decomposition method and its modifications [9], [12], [14] have efficiently used to solve the ordinary differential equations. It is the purpose of this paper to introduce a new reliable modification of Adomian decomposition method. For this reason, at the beginning of implementation of Adomian method, Chebyshev orthogonal polynomials are used to expand functions. In addition, the proposed modified Adomian decomposition method is numerically performed through Maple programming. The obtained results show the advantage using the proposed modified Adomian decomposition method.

It is well known that the eigenfunctions of certain singular Sturm–Liouville problems allow the approximation of functions C[a, b] where truncation error approaches zero faster than any negative power of the number of basic functions used in the approximation, as that number (order of truncation M) tends to infinity [4]. This phenomenon is usually referred to as “spectral accuracy” [5]. The accuracy of derivatives obtained by direct, term-by-term differentiation of such truncated expansion naturally deteriorates [4], but for low-order derivatives and sufficiently high order truncations this deterioration is negligible, compared to the restrictions in accuracy introduced by typical difference approximations (for more details, refer to [2], [7]). Throughout, we are using first kind orthogonal Chebyshev polynomials {Tk}k=0+, which are eigenfunctions of singular Sturm–Liouville problem1-x2T(x)+k21-x2Tk(x)=0.

Section snippets

Modified Adomian decomposition method

Here, the review of the standard Adomian decomposition method is presented. For this reason, consider the differential equation,Lu+Ru+Nu=g(x),where N is a non-linear operator, L is the highest-order derivative which is assumed to be invertible, R is a linear differential operator of less order than L and g is the source term.

The method is based on applying the operator L−1 formally to the expressionLu=g-Ru-Nu,so, by using the given conditions we obtainu=f-L-1(Ru)-L-1(Nu),where the function f

Test problems

In this section, two initial ordinary differential equations are considered and these problems are solved by Adomian decomposition method (4a), (4b), (5), uT(x), and proposed Adomian decomposition method (7), uC(x). The algorithms are performed by Maple 8 with 10 digits precision.

Example 1

Consider for 0  x  1u+xu+x2u3=(2+6x2)ex2+x2e3x2,u(0)=1,u(0)=0,with the exact solution u(x) = ex2. According to 1 we have,Lu+Ru+Nu=g(x),where L=d2dx2, R=xddx, Nu = x2u3 and g(x) = (2 + 6x2)ex2 + x2e3x2. In addition, F(u) = Nu = x2u3

References (14)

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