The solution to the third-kind Volterra integral equation (VIE3) usually has unbounded derivatives near the original point $t=0,$ which brings difficulties to numerical computation. In this paper, we analyze two kinds of modified multistep collocation methods for VIE3: collocation boundary value method with the fractional interpolation (FCBVM) and that with Lagrange interpolation (CBVMG). The former is developed based on the non-polynomial interpolation which is particularly feasible for approximating functions in the form of $t^{\eta }$ with the real number $\eta \ge 0.$ The latter is devised by using classical polynomial interpolation. The application of the boundary value technique enables both approaches to efficiently solve long-time integration problems. Moreover, we investigate the convergence properties of these two kinds of algorithms by Grönwall’s inequality.

第三类沃尔泰拉积分方程（VIE3）的解通常在原点附近有无界导数 $吨=0,$给数值计算带来了困难。在本文中，我们分析了VIE3的两种改进的多步搭配方法：分数插值法（FCBVM）和拉格朗日插值法（CBVMG）的搭配边界值法。前者是基于非多项式插值开发的，它特别适用于以下列形式逼近函数${吨}^{\eta }$ 用实数 $\eta \ge 0.$后者是通过使用经典多项式插值设计的。边界值技术的应用使这两种方法都能有效地解决长时间积分问题。此外，我们通过Grönwall 不等式研究了这两种算法的收敛性。