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1. Introduction
This paper introduces a new method for solving highly oscillatory and chaotic initial value problems (IVPs). The new method extends, for the first time, the application of the spectral homotopy analysis method [1, 2] to IVPs. The spectral homotopy analysis method was developed by Motsa et al. [1, 2] for solving nonlinear boundary value problems (BVPs) over finite intervals. It has been successfully been applied to other BVPs arising mainly in fluid mechanics-related problems [3–6]. In the previously cited applications the SHAM method was applied to problems which possess smooth solutions over small regions. For rapidly oscillating chaotic systems over very large regions, the SHAM may not give accurate results. The current work seeks to develop a new method that will be valid for rapidly changing solutions over all regions, small, medium-, and large sized. A simple way of ensuring the validity of the approximations for large intervals and for all functions is to determine the solution in a sequence of equal intervals, which are subject to continuity conditions at the end points of each interval.
Recently, in an effort to increase the radius of convergence of some analytical methods of approximations, multistage or piecewise approximations have been developed for solving IVPs over general intervals. This multistage approach seeks to implement the standard approximation method on sequences of subintervals whose union makes up the domain of the underlying problem. The effect of this piece-wise (multistep) approach is to accelerate the convergence of the approximate solution over a large region and to improve the...
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