A New Diffusive Representation for Fractional Derivatives and its Application

TL;DR

This paper introduces a novel diffusive representation for fractional derivatives that features faster-decaying integrands, enabling more efficient and accurate numerical approximation of fractional differential equations.

Contribution

The paper proposes a new diffusive representation with a rapidly decaying integrand, improving numerical efficiency over existing methods.

Findings

Faster decay of the integrand improves quadrature convergence.

The new representation reduces computational cost.

Enhanced accuracy in numerical solutions of fractional derivatives.

Abstract

Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known variants of this approach is that they require the numerical approximation of some integrals over an unbounded integral whose integrand decays rather slowly which implies that their numerical handling is difficult and costly. We present a novel variant of such a diffusive representation. This form also requires the numerical approximation of an integral over an unbounded domain, but the integrand decays much faster. This allows to use well established quadrature rules with much better convergence properties.