  A fractional derivative is a temporally nonlocal operation which is
computationally intensive due to inclusion of the accumulated contribution of
function values at past times. In order to lessen the computational load while
maintaining the accuracy of the fractional derivative, a novel numerical method
for the Caputo fractional derivative is proposed. The present adaptive memory
method significantly reduces the requirement for computational memory for
storing function values at past time points and also significantly improves the
accuracy by calculating convolution weights to function values at past time
points which can be non-uniformly distributed in time. The superior accuracy of
the present method to the accuracy of the previously reported methods is
identified by deriving numerical errors analytically. The sub-diffusion process
of a time-fractional diffusion equation and the creeping response of a
fractional viscoelastic model are simulated to demonstrate the accuracy as well
as the computational efficiency of the present method.
