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

  • Mahmoud A. ZakyEmail author
  • Ibrahem G. Ameen


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.


Multi-term fractional differential equations Volterra integral equations Collocation method Convergence analysis 

Mathematics Subject Classification

58C40 65N35 34L10 34K28 41A25 



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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Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsNational Research CentreGizaEgypt
  2. 2.Department of Mathematics, Faculty of ScienceAl-Azhar UniversityCairoEgypt

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