Convergence and accuracy of Adomian's decomposition method for the solution of Lorenz equations Peter Vadasz a, * ,1 , Shmuel Olek b a Department of Mechanical Engineering, University of Durban-Westville, Private Bag X54001, Durban 4000, South Africa b Planning, Development and Technology Division, Israel Electric Corporation Ltd., P.O. Box 10, Haifa 31000, Israel Received 20 April 1999; received in revised form 12 August 1999 Abstract The convergence and accuracy of Adomian's decomposition method of solution is analysed in the context of its application to the solution of Lorenz equations which govern at lower order the convection in a porous layer (or respectively in a pure ¯uid layer) heated from below. Adomian's decomposition method provides an analytical solution in terms of an in®nite power series and is applicable to a much wider range of heat transfer problems. The practical need to evaluate the solution and obtain numerical values from the in®nite power series, the consequent series truncation, and the practical procedure to accomplish this task, transform the analytical results into a computational solution evaluated up to a ®nite accuracy. The analysis indicates that the series converges within a suciently small time domain, a result that proves to be signi®cant in the derivation of the practical procedure to compute the in®nite power series. Comparison of the results obtained by using Adomian's decomposition method with corresponding results obtained by using a numerical Runge±Kutta±Verner method show that both solutions agree up to 12±13 signi®cant digits at subcritical conditions, and up to 8±9 signi®cant digits at certain supercritical conditions, the critical conditions being associated with the loss of linear stability of the steady convection solution. The dierence between the two solutions is presented as projections of trajectories in the state space, producing similar shapes that preserve under scale reduction or magni®cation, and are presumed to be of a fractal form. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Lorenz equations; Free convection; Weak turbulence; Chaos; Adomian's decomposition 1. Introduction The application of Adomian's [1,2] decomposition method as an alternative solution method to a wide variety of heat transfer problems motivates this study. The method is applicable to any heat transfer problem that can be reduced to a ®nite set of non-linear (or lin- ear) ordinary dierential equations, transforming an initial-boundary value problem which consists of par- tial dierential equations, with their initial and bound- ary conditions, governing the heat transfer process, into an initial value (or boundary value) problem. While the applicability of the method to the problem of heat convection in a ¯uid layer heated from below was demonstrated by Vadasz [3], and its application to the corresponding problem in porous media by Vadasz International Journal of Heat and Mass Transfer 43 (2000) 1715±1734 0017-9310/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0017-9310(99)00260-4 www.elsevier.com/locate/ijhmt 1 This work was undertaken while P. Vadasz was on Sabba- tical leave as Visiting Professor at University of Natal-Dur- ban, South Africa. * Corresponding author. Tel.: +27-31-204-4873; fax: +27- 31-204-4002. E-mail address: vadasz@pixie.udw.ac.za (P. Vadasz).