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On the rate of convergence of spectral collocation methods for nonlinear multi-order fractional initial value problems

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Abstract

Multi-order fractional differential equations are motivated by their flexibility to describe complex multi-rate physical processes. This paper is concerned with the convergence behavior of a spectral collocation method when used to approximate solutions of nonlinear multi-order fractional initial value problems. The collocation scheme and its convergence analysis are developed based on the novel spectral collocation method which has been recently presented by Wang et al. (J Sci Comput 76(1):166–188, 2018) for single fractional-order boundary value problems. The proposed method is an indirect approach since we act on the equivalent Volterra integral equation of the second kind. More precisely, the spectral rate of convergence for the proposed method is established in the \(L^2 \)- and \( L^{\infty } \)-norms. The method enjoys high accuracy for problems with smooth solutions. Exponentially rapid convergence is observed with a small number of degree of freedoms and for all samples of fractional orders. Numerical examples are presented to support the theoretical finding.

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Acknowledgements

We thank the five reviewers for their valuable comments and for bringing to our attention some important references, which helped improve our paper greatly.

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Authors and Affiliations

  1. Department of Applied Mathematics, National Research Centre, Dokki, Giza, 12622, Egypt

    Mahmoud A. Zaky

  2. Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt

    Ibrahem G. Ameen

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  1. Mahmoud A. Zaky
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  2. Ibrahem G. Ameen
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Correspondence to Mahmoud A. Zaky.

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Communicated by Vasily E. Tarasov.

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Zaky, M.A., Ameen, I.G. On the rate of convergence of spectral collocation methods for nonlinear multi-order fractional initial value problems. Comp. Appl. Math. 38, 144 (2019). https://doi.org/10.1007/s40314-019-0922-5

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  • DOI: https://doi.org/10.1007/s40314-019-0922-5

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