Generalized fractional derivatives and fourier transforms in tempered distributions with applications

The purpose of this paper is to define and prove that the Riemann–Liouville and Caputo fractional derivatives can be computed for tempered distributions, such that the fractional derivative of a tempered distribution remains a tempered distribution. The fact that the Fourier transform operator is an isomorphism in the dual of the Schwartz space is used, and we found that the fractional Riemann–Liouville and Caputo derivatives can be written as a Fourier transform composition and inverse. In this way, we are able to generalize both fractional Riemann–Liouville and Caputo derivatives for the tempered distributions. Moreover, certain examples of fractional derivatives for some tempered distributions are provided, such as the distribution of Dirac, the distribution of Heaviside, and the distribution of principal value.