  Fractional differential equations (FDEs) describe subdiffusion behavior of
dynamical systems. Its non-local structure requires to take 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 of the history integral is to approximate the fractional kernel
with the sum of exponentials, which is equivalent to consider 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. Moreover, we
apply the algorithm to a time-fractional Cahn-Hilliard problem.
