Short Memory Effect

In fractional derivative(also in the fractional integral), we already know the derivative of a fixed point depends on its history which is also called the Non-Local property. In Grunwald Letnikov sense, when we want to calculate the fractional derivative at a fixed point far away from its starting point, we can neglect its faraway lower terminal, and focus on its recent history instead.

Which means in the interval $[t-L, t]$, $L$ is the "memory length".

\[_aD^\alpha_t f(t)\approx _{t-L}D^\alpha_t f(t)\]

By deploying the Short Memory Effect, we can reduce our numerical cost while retaining the precision in a way.