A Fast Time Stepping Method for Evaluating Fractional Integrals
SIAM Journal on Scientific Computing
Journal of Computational and Applied Mathematics
Numerical solution of linear Volterra integral equations of the second kind with sharp gradients
Journal of Computational and Applied Mathematics
A Bootstrap Method for Sum-of-Poles Approximations
Journal of Scientific Computing
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To approximate convolutions which occur in evolution equations with memory terms, a variable-step-size algorithm is presented for which advancing $N$ steps requires only $O(N\log N)$ operations and $O(\log N)$ active memory, in place of $O(N^2)$ operations and $O(N)$ memory for a direct implementation. A basic feature of the fast algorithm is the reduction, via contour integral representations, to differential equations which are solved numerically with adaptive step sizes. Rather than the kernel itself, its Laplace transform is used in the algorithm. The algorithm is illustrated on three examples: a blowup example originating from a Schrödinger equation with concentrated nonlinearity, chemical reactions with inhibited diffusion, and viscoelasticity with a fractional order constitutive law.