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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.

**TABLE OF CONTENTS**

Title
Page

Abstract

Table
of contents

1. FRACTIONAL ORDER CALCULUS

2. FUNCTIONS OF FRACTIONAL CALCULUS

2.1
The Gamma function

2.2
The Beta Function

2.3
The Mittag-Leffler Function

2.4
Laplace Transform

2.5
The Convolution Theorem

2.6
Riemann-Liouville Fractional integral

2.7
Riemann-Liouville Fractional derivative

2.8
The Caputo’s Fractional Derivative

2.9
Laplace Transform of Fractional Integral

2.1.1
Laplace Transform of Riemann-Liouville Fractional Derivative

2.1.2
Laplace Transform of Caputo’s Fractional Derivative

3. FRACTIONALLY DAMPED LINEAR OSCILLATOR

3.1
Derivation of the Inverse Fractional Laplace Transform

3.2
Solution in Terms of Riemann-Liouville Formulation

3.3
Series Solution in Terms of one-Parameter Mittag-leffler Function

4. CAPUTOS’S FORMULATION AND COMPUTER
SIMULATION

**CHAPTER ONE**

**FRACTIONAL ORDER CALCULUS**

Those with the knowledge
of elementary calculus will unanimously agree that in any context the n
(shortened to
throughout this work) or n
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
can be expanded as

^{th}derivative^{th}integration**[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

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.

The concept of
fractional calculus developed simultaneously with the theory of integer order
calculus. Unlike many branches of mathematics and other disciplines whose exact
origins are not clear, we can point to a particular date when fractional
calculus was born. This interesting field of study was initiated in a
correspondence
in his publication. L’Hopital posed the
question what would the result be if
? Leibniz replied:

**[2]**between**L’Hopital and Leibniz, the co-inventor of the calculus. In a letter dated 30**^{th}September, 1695, L’Hopital had asked Leibniz the meaning of the notation the latter had used for the n^{th }derivative**“An apparent paradox from which one day useful consequences will be drawn.”**Later, this little conversation between these two mathematical giants caught the attention of other prominent mathematicians like Lacroix, Abel, Euler, Liouville and Riemann e.t.c. Each of these researchers shaped the evolution of the fractional calculus in their own ways.
The utility of the
fractional order calculus is not in doubt judging from recent and current
findings among researchers in biological, physical sciences and engineering. Fractional
differentiation has been used by modelers to study speech signals

**[3],**astronomical image processing**[4, 5, 6, 7],**earthquakes [**8, 9**] and viscoelasticity**[10,11]**.An enthusiastic reader can quickly browse through a catalogue of the applications of fractional order calculus in
[

**12, 13**] So far, we have studied physical systems in terms of integer order calculus. Intuitively, one can argue that fractional calculus is more harmonious with the real world. Nature, we all know, does not always obey the integers. Little wonder fractional calculus has generated so much interest across the mathematical world. Researches that are based on the theory of fractional calculus are ongoing and it is among the expanding frontiers of mathematics. It is obvious that greater applications of calculus to human problems in the future will likely depend on fractional calculus.**MOTIVATION**

This research was
inspired by two journals. Using analytical techniques of classical (discrete)
calculus Oyesanya

**[14]**treated the nonlinear Duffing oscillator

and applied the results
to the phenomenon of earthquake prediction.

Later, Naber

**[15]**treated the fractional oscillator equation considering only the case where the fractional derivative is on the damping term:

Using Laplace transform
technique, he approached the problem through contour integration and found that
there are nine distinct cases as opposed to the usual three cases for discrete
calculus.

The above two equations
are analogues of the Duffing Oscillator. So, this research solves the linear
and unforced analogue of (i) where the derivatives are all fractional. It
extends (ii) by making all derivatives fractional but differs from it by
expressing the solution in terms of the Mittag-leffler function; both, however,
explored the use of Laplace Transform Method. In a nutshell, we seek to investigate the
solution of fractionally damped linear oscillator. Chapter one briefly explains
the meaning of fractional calculus. Chapter two is devoted to the development
of the functions and formulations /definitions of fractional calculus. In
chapter three, we apply the tools developed along the way to the problem of
fractional order oscillation. Chapter four is numerical; computer simulation of
the solution is presented based on Caputo’s formulation of the fractional derivative.
The conclusion is treated in chapter five......

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