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In this work, a mechanical quadrature method based on modified trapezoid formula is used for solving weakly singular Volterra integral equation with proportional delays. An improved Gronwall inequality is testified and adopted to prove the existence and uniqueness of the solution of the original equation. Then, we study the convergence and the error estimation of the mechanical quadrature method. Moreover, Richardson extrapolation based on the asymptotic expansion of error not only possesses adoi:10.1155/2019/4813802 fatcat:dnp2etyhlbfcldakxvgrotaqsu