In this work we develop a means to rapidly and accurately compute the Caputo fractional derivative of a function, using fast convolution. The key element to this approach is the compression of the fractional kernel into a sum of M decaying exponentials, where M is minimal. Specifically, after N time steps we find M= O (log N) leading to a scheme with O (N log N) complexity. We illustrate our method by solving the fractional differential equation representing the Kelvin-Voigt model of viscoelasticity, and the partial differential equations that model the propagation of electromagnetic pulses in the Cole-Cole model of induced dielectric polarization.
Causley, Matthew F. and Petropoulus, Peter G., "Efficient high-order methods for solving fractional differential equations of order α ∈ (0, 1) using fast convolution and applications in wave propagation" (2015). Mathematics Publications. 98.