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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 ∞ -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.

While the approximate solutions of one-dimensional nonlinear Volterra–Fredholm integral equations with smooth kernels are now well understood, no systematic studies of the numerical solutions of their multi-dimensional counterparts exist. In this paper, we provide an efficient numerical approach for the multi-dimensional nonlinear Volterra–Fredholm integral equations based on the multi-variate Legendre-collocation approach. Spectral collocation methods for multi-dimensional nonlinear integral equations are known to cause major difficulties from a convergence analysis point of view. Consequently, rigorous error estimates are provided in the weighted Sobolev space showing the exponential decay of the numerical errors. The existence and uniqueness of the numerical solution are established. Numerical experiments are provided to support the theoretical convergence analysis. The results indicate that our spectral collocation method is more flexible with better accuracy than the existing ones.