This work is to analyze a Legendre collocation approximation for third-kind Volterra integro-differential equations. The rigorous error analysis in the \(L^{\infty }\) and \(L_{\omega ^{0,0}}^{2}\)-norms is provided for the proposed method. In fact when converting the original equation to an equivalent second kind one, the integral operator of the obtained equation contains two singularities and may become non-compact under certain conditions. In addition, in order to avoid the low-order accuracy caused by the singularity of the solution at the initial point, we adopted the idea of smooth transformation at the beginning to convert the original equation into a new equation with a more regular solution. Finally, the validity and applicability of the method are verified by several numerical experiments.

这项工作是分析第三类Volterra积分微分方程的Legendre搭配近似。\（L ^ {\ infty} \）和\（L _ {\ omega ^ {0,0}} ^ {2} \中的严格错误分析为所建议的方法提供了-norms。实际上，当将原始方程式转换为等效的第二种方程式时，所获得方程式的积分算子包含两个奇点，并且在某些条件下可能变得不紧致。另外，为了避免在初始点由解的奇异性引起的低阶精度，我们在开始时采用了平滑变换的思想，将原始方程式转换为具有更规则解的新方程式。最后，通过几个数值实验验证了该方法的有效性和适用性。