Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel (Advances in Applied Mathematics and Mechanics, 2021)
In this paper, a Chebyshev-collocation spectral method is developed forVolterra integral equations (VIEs) of second kind with weakly singular kernel. We firstchange the equation into an equivalent VIE so that the solution of the new equationpossesses better regularity. The integral term in the resulting VIE is approximated byGauss quadrature formulas using the Chebyshev collocation points. The convergenceanalysis of this method is based on the Lebesgue constant for the Lagrange interpolationpolynomials, approximation theory for orthogonal polynomials, and the operatortheory. The spectral rate of convergence for the proposed method is established in theL∞-norm and weighted L2-norm. Numerical results are presented to demonstrate theeffectiveness of the proposed method.