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The so-called fractional relaxor-oscillator, whose time evolution is characterized by an index of fractional order,α, exhibits interesting relaxationoscillation characteristics. [ABSTRACT FROM AUTHOR]

Recognizing the importance of proper initialization of a system, which is evolving in time according to a differential equation of fractional order, Lorenzo and Hartley developed the method of properly incorporating the effect of the past (history) by means of an initialization function for the Riemann-Liouville and the Grunwald formulations of fractional calculus. The present work addresses this issue for the Caputo fractional derivative and cautions that the commonly held belief that the Caputo formulation of fractional derivatives properly accounts for the initialization effects is not generally true when applied to the solution of fractional differential equations. Key words Caputo fractional derivatives, initialization issues. [ABSTRACT FROM AUTHOR]

The Mittag-Leffler function Eα(z), which is a generalization of the exponential function, arises frequently in the solutions of physical problems described by differential and/or integral equations of fractional order. Consequently, the zeros of Eα (z) and their distribution are of fundamental importance and play a significant role in the dynamic solutions. The Mittag- Leffler function E (z) is known to have a finite number of real zeros in the range 1 < α > 2 which is applicable for many physical problems. What has not been known is the exact number of real zeros of Eα (z) for a given value of α in this range. An iteration formula is derived for calculating the number of real zeros of Eα(z) for any value of α in the range 1 < α > 2 and some specific results are tabulated. Key words Mittag-Leffler functions, zeros, fractional calculus. [ABSTRACT FROM AUTHOR]