A perturbative study of fractional relaxation phenomena

Fractional differential equations provide a convenient mathematical framework to discuss many important physical processes in the complex media. An expansion method has been proposed [V.E. Tarasov, G.M. Zaslavsky, Physica A 368 (2006) 399–415] to discuss the dynamics in the media where the order of the fractional derivative α is close to an integer number. This expansion is over the small parameter ε=n−α with small positive ε and positive integer n. They also found that this expansion in not uniform with respect to t 1. We extend the formalism to the values of α=n+ε. we also show that in certain cases this expansion is uniform. We apply this uniform expansion to the fractional relaxation, composite fractional relaxation and to the composite fractional oscillation phenomena.