FRACTIONAL MECHANICAL OSCILLATOR (STATISTICS PROJECT TOPICS AND MATERIALS)
ABSTRACT
The linearly damped free fractional mechanical oscillator equation is solved by Laplace Transform method and series solution technique. In both methods, the solution is expressed in terms of the Mittag-Leffler function defined by The Rieman-Liouville and Caputo’s formulations of the fractional differentiation are both considered. The parameters   carry over their meanings from discrete calculus as the damping coefficient and circular frequency respectively,  is the order of the fractional derivative. The damping coefficient is a measure of resistive force present in the medium through which the oscillator vibrates while the resonant frequency is its natural frequency in the absence of external excitations.
CHAPTER ONE
FRACTIONAL ORDER CALCULUS
Those with the knowledge of elementary calculus will unanimously agree that in any context the nth derivative  (shortened to  throughout this work) or nth integration  of a function f is mentioned, n is automatically construed as a positive integer. Consequently, we can talk about the second  and third  derivatives of a specified function f. The theory of fractional calculus is concerned with the generalization of the concepts of differentiation and integration to arbitrary orders. It is an outgrowth of the traditional definitions of the derivative and integral operators in much the same way as the fractional exponent is the natural extension of exponents with integer values [1]. We were all taught that exponents are a short mathematical notation for a repeated multiplication of a number by itself a given number of times. Therefore, a quantity like  can be expanded as
This operation, however, strains the imagination when one attempts to expand or interpret an indicial quantity with a rational index the same way. For instance, going by the definition of exponentiation, a quantity like  literally means to multiply the base 8 by itself  times. This problem is hard to interpret or represent physically but we are certain that it has solutions that do not require much ingenuity to obtain. The argument is that, presently, physical conceptualization of fractional order calculus is breathtaking but its sound foundation is consistent with the logic of other branches of mathematics.
