Fractional Mechanical Oscillator

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