  Fractional differential equations (FDEs) describe subdiffusion behavior of
dynamical systems. Its non-local structure requires taking into account the
whole evolution history during the time integration, which then possibly causes
additional memory use to store the history, growing in time. An alternative to
a quadrature for the history integral is to approximate the fractional kernel
with the sum of exponentials, which is equivalent to considering the FDE
solution as a sum of solutions to a system of ODEs. One possibility to
construct this system is to approximate the Laplace spectrum of the fractional
kernel with a rational function. In this paper, we use the adaptive
Antoulas--Anderson (AAA) algorithm for the rational approximation of the kernel
spectrum which yields only a small number of real valued poles. We propose a
numerical scheme based on this idea and study its stability and convergence
properties. In addition, we apply the algorithm to a time-fractional
Cahn-Hilliard problem.
