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.

 

 

TABLE OF CONTENTS

Title Page                                                                                                                   i

Certificate of Approval                                                                                             ii

Dedication                                                                                                                  iii

Acknowledgment                                                                                                      iv

Abstract                                                                                                                      v

Table of contents                                                                                                      vi

1. FRACTIONAL ORDER CALCULUS
2. FUNCTIONS OF FRACTIONAL CALCULUS

2.1 The Gamma function                                                                                          3

2.2 The Beta Function                                                                                              5

2.3 The Mittag-Leffler Function                                                                              5

2.4 Laplace Transform                                                                                             7

2.5 The Convolution Theorem                                                                                9

2.6 Riemann-Liouville Fractional integral                                                              10

2.7 Riemann-Liouville Fractional derivative                                                          13

2.8 The Caputo’s Fractional Derivative                                                                  14

2.9 Laplace Transform of Fractional Integral                                                       19

2.1.1 Laplace Transform of Riemann-Liouville Fractional Derivative                20

2.1.2 Laplace Transform of Caputo’s Fractional Derivative                                21

3. FRACTIONALLY DAMPED LINEAR OSCILLATOR

3.1 Derivation of the Inverse Fractional Laplace Transform                              25

3.2 Solution in Terms of Riemann-Liouville Formulation                                                28

3.3 Series Solution in Terms of one-Parameter Mittag-leffler Function             33

4. CAPUTOS’S FORMULATION AND COMPUTER SIMULATION
5. CONCLUSION         

 

 

CHAPTER ONE

FRACTIONAL ORDER CALCULUS

Those with the knowledge of elementary calculus will unanimously agree that in any context the n^th derivative  (shortened to  throughout this work) or n^th 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.

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 [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   in his publication. L’Hopital posed the question what would the result be if  ? Leibniz replied: “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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