Diffusive Representations for the Numerical Evaluation of Fractional Integrals
Kai Diethelm
TL;DR
The paper addresses the computational challenge of numerically evaluating the non-local Riemann-Liouville integral $J_a^\alpha f(t)$ by introducing diffusive representations that express the integral as a superposition $J_a^\alpha f(t)=\int_\Omega \phi(t,\omega)\,d\omega$, where each $\phi(\cdot,\omega)$ solves a simple ordinary differential equation in $t$. It establishes a general framework with admissible transformations $\psi$, derives regularity and end-behavior properties of $\phi$, and provides Eulerian-type numerical schemes that leverage quadrature over $\omega$ and ODE solvers to achieve $O(N)$ time and $O(1)$ memory for large-scale multiple-evaluation problems. The key contributions include rigorous representations, smoothness and asymptotic results for $\phi$, and detailed proofs (Theorems on the diffusive representation, IVP, and its smoothness) that underpin the proposed algorithms. The work lays the groundwork for fast solvers for fractional integral equations and offers a path to extend these ideas to Caputo derivatives, potentially enabling efficient fractional differential equation solvers in practice.
Abstract
Diffusive representations of fractional differential and integral operators can provide a convenient means to construct efficient numerical algorithms for their approximate evaluation. In the current literature, many different variants of such representations have been proposed. Concentrating on Riemann-Liouville integrals whose order is in (0,1), we here present a general approach that comprises most of these variants as special cases and that allows a detailed investigation of the analytic properties of each variant. The availability of this information allows to choose concrete numerical methods for handling the representations that exploit the specific properties, thus allowing to construct very efficient overall methods.