  Anomalous relaxation and diffusion processes have been widely characterized
by fractional derivative models, where the definition of the fractional-order
derivative remains a historical debate due to the singular memory kernel that
challenges numerical calculations. This study first explores physical
properties of relaxation and diffusion models where the fractional derivative
was defined recently using an exponential kernel. Analytical analysis shows
that the fractional derivative model with an exponential kernel cannot
characterize non-exponential dynamics well-documented in anomalous relaxation
and diffusion. A legitimate extension of the previous fractional derivative is
then proposed by replacing the exponential kernel with a stretched exponential
kernel. Numerical tests show that the fractional derivative model with the
stretched exponential kernel can describe a much wider range of anomalous
diffusion than the exponential kernel, implying the potential applicability of
the new fractional derivative in quantifying real-world, anomalous relaxation
and diffusion processes.
