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Mathematical Problems in Engineering
Volume 2013, Article ID 583193, 13 pages
Research Article

A New Piecewise-Spectral Homotopy Analysis Method for Solving Chaotic Systems of Initial Value Problems

1Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
2School of Mathematics, Computer Science and Statistics, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa
3Department of Mathematics, University of Venda, P Bag X5050, Thohoyandou 0950, South Africa

Received 30 September 2012; Revised 14 February 2013; Accepted 25 February 2013

Academic Editor: Trung Nguyen Thoi

Copyright © 2013 H. Saberi Nik et al. 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.


An accurate algorithm for solving initial value problems (IVPs) which are highly oscillatory is proposed. The proposed method is based on a novel technique of extending the standard spectral homotopy analysis method (SHAM) and adapting it to a sequence of multiple intervals. In this new application the method is referred to as the piecewise spectral homotopy analysis method (PSHAM). The applicability of the proposed method is examined on the differential equation system modeling HIV infection of CD4+ T cells and the Genesio-Tesi system which is known to be chaotic and highly oscillatory. Also, for the first time, we present here a convergence proof for SHAM. We treat in detail Legendre collocation and Chebyshev collocation. The method is compared to MATLAB’s ode45 inbuilt solver as a measure of accuracy and efficiency.