Abstract
This paper proposes the definition of fractional definite integral and analyses the corresponding fundamental theorem of fractional calculus. In this context, we studied the relevant properties of the fractional derivatives that lead to such a definition. Finally, integrals on R2 R 2 and R3 R 3 are also proposed.
Highlights
Fractional Calculus has evolved considerably during the last 30 years and has become popular in many scientific and technical areas [1,2,3,4]
The corresponding “fractional fundamental theorem of calculus”. For this purpose we study the admissibility of fractional derivatives and we find those that are suitable for achieving the proposed integral
These results show that not all the fractional derivative (FD) can be used to generalise the notion of fractional integral (FI)
Summary
Fractional Calculus has evolved considerably during the last 30 years and has become popular in many scientific and technical areas [1,2,3,4]. The concepts of fractional derivative (FD) and fractional integral (FI) assume various forms not always equivalent and not compatible with each other Notwithstanding this development a singular situation exists, since there is no definition of “fractional definite integral” [5,6]. Starting from a revision of classic results, an approach based on a generalisation of the Barrow formula is used to propose the “definite fractional integral” and, from it, to formulate the “fractional fundamental theorem of calculus”. These developments allowed the definition of double and triple integrals on rectangular spaces. Fractal Fract. 2017, 1, 2; doi:10.3390/fractalfract1010002 www.mdpi.com/journal/fractalfract
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