Fractional Derivatives with Variable Memory

Recently, fractional derivatives (FDs) and fractional differential equations are extensively used in modeling of most dynamic processes which generally involve memory factors. Normally, the dynamic models involving full memory effect can be solved with the help of global fractional derivatives, whereas those involves with local memory effect can be solved by using only fractional derivatives with short memory. The prime objective of this work is to define new fractional derivatives of Riemann–Liouville and Caputo types, which can be used for both short and global memory effects. Some general and dynamic properties of the proposed differential operators in comparison to existing formulas of FDs are discussed.