Several Results of Fractional Derivatives in Dʹ(R +)

Abstract

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 ' ( R + ) , as well as the symbolic solution for the following differential equation by Babenko's method y ( x ) + λ Γ ( - α ) ∫ 0 x y ( σ ) ( x - σ ) α + 1 d σ = δ ( x ) , where Re α > 0