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Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. (English) Zbl 0584.65090

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

MSC:

65R20 Numerical methods for integral equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
45G05 Singular nonlinear integral equations
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