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ISRN Applied Mathematics
Volume 2013 (2013), Article ID 806056, 3 pages
http://dx.doi.org/10.1155/2013/806056
Comment on “Adomian Decomposition Method for a Class of Nonlinear Problems”
1Mathematic Department of Islamic Azad University, Branch of Arak, Arak, Iran
2Technical and Vocational University, Ahwaz, Iran
Received 29 May 2013; Accepted 19 August 2013
Academic Editors: Z. Huang and X.-S. Yang
Copyright © 2013 S. Dalvandpour and A. Motamedinasab. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Sánchez Cano in his paper “Adomian Decomposition Method for a Class of Nonlinear Problems” in application part pages 8, 9, and 10 had made some mistakes in context; in this paper we correct them.
1. Introduction
Adomian [1, 2] proposed a powerful method for solving nonlinear functional equation. The technique uses a decomposition of the nonlinear operator as a series of functions; each term of this series is a generalized polynomial called Adomian polynomial.
We will see that using the Adomian decomposition method together with some properties of the nested integral [3, 4] the solution of nonlinear ordinary differential equations system is obtained.
2. Correct Equations
In page 8, he showed that the solution is given by By rearranging he obtained And similarly for we will have By rearranging he obtained Continuing in this fashion, he concluded the following formulas: But the correct formulas are given by Writing (2) and (4) as a single integral, he had Similarly, In page 9, he uses With and , he obtained In fact, the correct solution is given by In page 10, he showed two cases for the solutions and using (2), (4), and .
Case 1 (). In this case, he obtained the solutions But the correct method is the following: Similarly,
Case 2 (). He arrived at the formulas and as follows: But the correct formulas are given by
Similarly,
References
- G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic, Dordrecht, The Netherlands, 1994. View at Zentralblatt MATH · View at MathSciNet
- G. Adomian, Nonlinear Stochastic Systems and Applications to Physics, Kluwer Academic, Dordrecht, The Netherlands, 1989.
- A. M. Wazwaz, “Exact solutions to nonlinear diffusion equations obtained by the decomposition method,” Applied Mathematics and Computation, vol. 123, no. 1, pp. 109–122, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
- A. M. Wazwaz, “A new algorithm for calculating Adomian polynomials for nonlinear operators,” Applied Mathematics and Computation, vol. 111, no. 1, pp. 53–69, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet