  Computing numerical solutions to fractional differential equations can be
computationally intensive due to the effect of non-local derivatives in which
all previous time points contribute to the current iteration. In general,
numerical approaches that depend on truncating part of the system history while
efficient, can suffer from high degrees of error and inaccuracy. Here we
present an adaptive time step memory method for smooth functions applied to the
Grunwald-Letnikov fractional diffusion derivative. This method is
computationally efficient and results in smaller errors during numerical
simulations. Sampled points along the system history at progressively longer
intervals are assumed to reflect the values of neighboring time points. By
including progressively fewer points backward in time, a temporally weighted
history is computed that includes contributions from the entire past of the
system, maintaining accuracy, but with fewer points actually calculated,
greatly improving computational efficiency.
