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Solving the third-kind Volterra integral equation via the boundary value technique: Lagrange polynomial versus fractional interpolation
Applied Mathematics and Computation  (IF4.091),  Pub Date : 2021-10-03, DOI: 10.1016/j.amc.2021.126685
Hao Chen, Junjie Ma

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 tη with the real number η0. 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.