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In this paper we propose and analyze a fractional Jacobi-collocation spectralmethod for the second kind Volterra integral equations (VIEs) with weaklysingular kernel $(x-s)^{-\mu},0<\mu<1$. First we develop a family of fractionalJacobi polynomials, along with basic approximation results for some weightedprojection and interpolation operators defined in suitable weighted Sobolevspaces. Then we construct an efficient fractional Jacobi-collocation spectralmethod for the VIEs using the zeros of the new developed fractional Jacobipolynomial. A detailed convergence analysis is carried out to derive errorestimates of the numerical solution in both $L^{\infty}$- and weighted$L^{2}$-norms. The main novelty of the paper is that the proposed method ishighly efficient for typical solutions that VIEs usually possess. Precisely, itis proved that the exponential convergence rate can be achieved for solutionswhich are smooth after the variable change $x\rightarrow x^{1/\lambda}$ for asuitable real number $\lambda$. Finally a series of numerical examples arepresented to demonstrate the efficiency of the method.

A M\"untz-Collocation spectral method for weakly singular volterra integral equations