Abstract
The Liouville-Caputo and Riemann-Liouville fractional derivatives have been two of the most useful operators for modeling nonlocal behaviors by fractional differential equations. In terms of Mittag-Leffler function and convolution product, using the Laplace transform, we give the exact values of the solutions of the Liouville-Caputo and Riemann-Liouville time fractional evolution equations associated with the Quantum fractional number operator. Therefore, we study the attractiveness of these solutions.