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 themethod is referred to as the piecewise spectral homotopy analysismethod (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 scillatory. 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.

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