A new Definition of Fractional Derivative and Fractional Integral

Ahmed M. Kareem
Department of Mathematics, College of Science, University of Diyala, Diyala, Iraq.
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ABSTRACT

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.

Keywords: Gamma function, Beta function, Mittag -Leffler Function, Complementary Error Function, Fractional integration, Riemann-Liouville and Caputo fractional derivatives.


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