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In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of the second kind with a weakly singular kernel. We use some func- tion transformation and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval (−1,1), so that the solution of the new equation possesses better regularity and the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high order accuracy for the approximation, the integral term in the resulting equation is approximated by using Jacobi spectral quadrature rules. The convergence analysis of this novel method is based on the Lebesgue constants corresponding to the Lagrange interpolation polyno- mials, polynomials approximation theory for orthogonal polynomials and the operator theory. The spectral rate of convergence for the proposed method is established in the L1-norm and weighted L2-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel