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# Fractional Calculus and Applied Analysis

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Volume 18, Issue 1

# Several Results of Fractional Derivatives in Dʹ(R+)

Chenkuan Li
• Corresponding author
• Department of Mathematics and Computer Science Brandon University, Brandon, Manitoba, RγA 6A9-CANADA
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Published Online: 2015-02-10 | DOI: https://doi.org/10.1515/fca-2015-0013

## Abstract

In this paper, we define fractional derivative of arbitrary complex order of the distributions concentrated on ${R}_{+}$, based on convolutions of generalized functions with the supports bounded on the same side. Using distributional derivatives, which are generalizations of classical derivatives, we present a few interesting results of fractional derivatives in $D\text{'}\left({R}_{+}\right)$, as well as the symbolic solution for the following differential equation by Babenko's method

$y\left(x\right)+\frac{\lambda }{\Gamma \left(-\alpha \right)}{\int }_{0}^{x}\frac{y\left(\sigma \right)}{{\left(x-\sigma \right)}^{\alpha +1}}d\sigma =\delta \left(x\right)$,

where Re $\alpha >0$

MSC 2010: 46F10; 26A33

## References

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Published Online: 2015-02-10

Citation Information: Fractional Calculus and Applied Analysis, Volume 18, Issue 1, Pages 192–207, ISSN (Online) 1314-2224,

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