Lubich, Ch. Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. (English) Zbl 0584.65090 Math. Comput. 45, 463-469 (1985). This article treats the application of fractional linear multistep methods as introduced by the author in Discretized fractional calculus, SIAM J. Math. Anal. 17, 704-719 (1986) to weakly singular Volterra integral equations of the second kind. It is shown that the proposed methods are convergent of the same order as the underlying multistep method. This is a remarkable result, since the exact solution is in general not smooth at the initial value. A nice stability analysis is presented, which generalizes the concept of A-stability (for stiff ordinary differential equations) to weakly singular linear integral equations. It is mentioned that fast Fourier transform techniques can be used for an efficient implementation. This new class of numerical methods looks very promising and has several advantages over product integration methods. Reviewer: E.Hairer Cited in 3 ReviewsCited in 110 Documents MSC: 65R20 Numerical methods for integral equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 45G05 Singular nonlinear integral equations Keywords:Abel integral equation; convergence; comparison of methods; fractional linear multistep methods; weakly singular Volterra integral equations of the second kind; A-stability; fast Fourier transform; product integration methods PDFBibTeX XMLCite \textit{Ch. Lubich}, Math. Comput. 45, 463--469 (1985; Zbl 0584.65090) Full Text: DOI