An adaptation of adomian decomposition for numeric–analytic integration of strongly nonlinear and chaotic oscillators S. Ghosh, A. Roy, D. Roy * Department of Civil Engineering, Indian Institute of Science, Bangalore 560012, India Received 11 August 2005; received in revised form 11 August 2006; accepted 15 August 2006 Abstract A novel form of an explicit numeric-analytic technique is developed for solving strongly nonlinear oscillators of engineering interest. The analytic part of this technique makes use of Adomian Decomposition Method (ADM), but unlike other analytical solutions it does not rely on the functional form of the solution over the whole domain of the independent variable. Instead it discretizes the domain and solves multiple IVPs recursively. ADM uses a rearranged Taylor series expansion about a function and finds a series of functions which add up to generate the required solution. The present method discretizes the axis of the independent variable and only collects lower powers of the chosen step size in series solution. Each function constituting the series solution is found analytically. It is next shown that the modified ADM can be used to obtain the analytical solution,in a piecewise form. For nonlinear oscillators such a piecewise solution is valid only within a chosen time step. An attempt has been made to address few issues like the order of local error and convergence of the method. Emphasis has been on the application of the present method to a number of well known oscillators. The method has the advan- tage of giving a functional form of the solution within each time interval thus one has access to finer details of the solution over the interval. This is not possible in purely numerical techniques like the Runge–Kutta method, which provides solution only at the two ends of a given time interval, provided that the interval is chosen small enough for convergence. It is shown that the present technique suc- cessfully overcomes many limitations of the conventional form of ADM. The present method has the versatility and advantages of numerical methods for being applied directly to highly nonlinear problems and also have the elegance and other benefits of analytical techniques. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Adomian decomposition method; Nonlinear oscillators; Chaos 1. Introduction In many practical situations a system of coupled, pos- sibly damped, nonlinear ordinary differential equations model the dynamical behavior of mechanical systems. For example, these equations arise (following some discret- ization procedure) while studying the mechanical response of systems such as strings, beams, absorbers, plates, and so on. In general, exact solutions of such equations are unknown and thus numerical integration, perturbation techniques or geometrical methods (see [1–4] and references there in) have been applied to obtain their approximate solutions. However, in many of the analytical techniques, it becomes necessary to resort to linearization techniques or assumption of weak nonlinearity, except for a small class of low-dimensional problems which can be trans- formed to linear equations. This so-called weak non- linearity or small parameter assumption greatly restricts applications of perturbation techniques known that an overwhelming majority of nonlinear problems have no small parameters at all. Therefore such analytic routes may not be able to treat strongly nonlinear problems. Recently there are few attempts to overcome this restric- tion of weak nonlinearity (see, for instance [5–7]), but they 0045-7825/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2006.08.010 * Corresponding author. Tel.: +91 80 22933129; fax: +91 80 23600404. E-mail address: royd@civil.iisc.ernet.in (D. Roy). www.elsevier.com/locate/cma Comput. Methods Appl. Mech. Engrg. 196 (2007) 1133–1153