A Fast Time Stepping Method for Evaluating Fractional Integrals

Abstract. We evaluate the fractional integral Iα[f ](t) = 1 Γ(α) ∫ t 0 (t− τ)α−1 f(τ) dτ , 0 < α < 1, at time steps t = Δt, 2Δt, . . . , NΔt by making use of the integral representation of the convolution kernel tα−1 = 1 Γ(1−α) ∫∞ 0 e −ξ t ξ−α dξ. We construct an efficient Q-point quadrature of this integral representation and use it as a part of a fast time stepping method. The new method has algorithmic complexity O(NQ) and storage requirementO(Q). The number of quadrature nodesQ is independent of N and grows like O ((− log − logΔt)2), where is the quadrature error tolerance and Δt is the size of the time step. The (possible) singularity of f near τ = 0 is taken into account. This new method is particularly well-suited for long time simulations.