Positive time fractional derivative

In mathematical modeling of the non-squared frequency-dependent diffusions, also known as the anomalous diffusions, it is desirable to have a positive real Fourier transform for the time derivative of arbitrary fractional or odd integer order. The Fourier transform of the fractional time derivative in the Riemann-Liouville and Caputo senses, however, involves a complex power function of the fractional order. In this study, a positive time derivative of fractional or odd integer order is introduced to respect the positivity in modeling the anomalous diffusions.