Abstract
A new Definition of Fractional Derivative and Fractional Integral
Highlights
The fractional calculus is a field of mathematics that studies the integration and differentiation of functions of any order [1, 2, 3]
In order to comprehension the fractional derivative of the power function, we review some important theorems related to our work
Application and Result : In this subsection, in order to show the high importance of Caputo Expansion formula, we apply it to solve fractional derivative and fractional integral when the order of the functions is negative integer and we have obtained approximate values for derivatives and integrations: as in the examples below
Summary
The fractional calculus is a field of mathematics that studies the integration and differentiation of functions of any order [1, 2, 3]. The order of the function ( ) must be positive[16] In this new definition of Caputo the fractional derivative of a constant equal zero. According to Riemann-Liouville and Caputo fractional derivatives, derivative is assumed to be zero when the order of the function negative value i.e. 0 when 0 ∝ 1, 0 , Since the gamma function is undefined for arguments whose real part is negative integer. 0 when 0 ∝ 1, 0 , Since the gamma function is undefined for arguments whose real part is negative integer For this reason we reformulated the Caputo fractional derivative of the power function, in order to find the new formula Caputo Expansion formula.
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