Short Memory Effect

In fractional derivative, 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 employing the Short Memory Effect, we can reduce our numerical cost while retain the precision in a way.