  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.
