A new definition of fractional derivative and fractional integral

الملخص EN

In this paper, we introduce three different definitions of fractional derivatives, namely Riemann-Liouville derivative, Caputo derivative and the new formula Caputo expansion formula, and some basics properties of these derivatives are discussed.

The difference between Caputo and Riemann–Liouville formulas for the fractional derivatives also mentioned.

The paper focuses on find approximate values for function derivatives, when the function order is a negative integer, illustrated by some theorems and examples.