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Fractional Calculus: Definitions and Applications Joseph M Western Kentucky University TopSCHOLAR® Masters Theses & Specialist Projects Graduate School 4-2009 Fractional Calculus: Definitions and Applications Joseph M. Kimeu Western Kentucky University, [email protected] Follow this and additional works at: http://digitalcommons.wku.edu/theses Part of the Algebraic Geometry Commons, and the Numerical Analysis and Computation Commons Recommended Citation Kimeu, Joseph M., "Fractional Calculus: Definitions and Applications" (2009). Masters Theses & Specialist Projects. Paper 115. http://digitalcommons.wku.edu/theses/115 This Thesis is brought to you for free and open access by TopSCHOLAR®. It has been accepted for inclusion in Masters Theses & Specialist Projects by an authorized administrator of TopSCHOLAR®. For more information, please contact [email protected]. FRACTIONAL CALCULUS: DEFINITIONS AND APPLICATIONS A Thesis Presented to The Faculty of the Department of Mathematics Western Kentucky University Bowling Green, Kentucky In Partial Fulfillment Of the Requirements for the Degree Master of Science By Joseph M. Kimeu May 2009 FRACTIONAL CALCULUS: DEFINITIONS AND APPLICATIONS Date Recommended 04/30/2009 Dr. Ferhan Atici, Director of Thesis Dr. Di Wu Dr. Dominic Lanphier _________________________________________ Dean, Graduate Studies and Research Date A C K N O W L E D G E M E N TS I would like to express my deepest gratitude to my advisor, Dr. Ferhan Atici, for her thoughtful suggestions and excellent guidance, without which the completion of this thesis would not have been possible. I would also like to extend my sincere appreciation to Dr. Wu and Dr. Lanphier for their serving as members of my thesis committee; and to Dr. Chen and Dr. Magin for their insightful discussions on the Mittag-Leffler function. Lastly, I would like to thank Penzi Joy, and my family for their encouragement and support. iii TABLE OF C O N T E N TS TABLE OF CONTENTS ................................................................................................... iv ABSTRACT ........................................................................................................................ v CHAPTER 1: INTRODUCTION ....................................................................................... 1 1.1 The Origin of Fractional Calculus ................................................................................ 1 1.2 Definition of Fractional Calculus.................................................................................. 3 1.3 Useful Mathematical Functions .................................................................................... 3 CHAPTER 2: THE RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL .................... 9 2.1 Definition of the Riemann-Liouville Fractional Integral .............................................. 9 2.2 Examples of Fractional Integrals ................................................................................ 11 2.3 Dirichlet’s Formula ..................................................................................................... 15 2.4 The Law of Exponents for Fractional Integrals .......................................................... 16 2.5 Derivatives of the Fractional Integral and the Fractional Integral of Derivatives ...... 17 CHAPTER 3: THE RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE .............. 19 3.1 Definition of the Riemann-Liouville Fractional Derivative ....................................... 19 3.2 Examples of Fractional Derivatives ............................................................................ 19 CHAPTER 4: FRACTIONAL DIFFERENTIAL EQUATIONS ..................................... 23 4.1 The Laplace Transform ............................................................................................... 23 4.2 Laplace Transform of the Fractional Integral ............................................................. 24 4.3 Laplace Transform of the Fractional Derivative ......................................................... 24 4.4 Examples of Fractional Differential Equations........................................................... 27 CHAPTER 5: APPLICATIONS AND THE PUTZER ALGORITHM ........................... 31 5.1 A Mortgage Problem................................................................................................... 31 5.2 A Decay-Growth Problem .......................................................................................... 35 5.3 The Matrix Exponential Function ............................................................................... 36 5.4 The Matrix Mittag-Leffler Function ........................................................................... 40 CONCLUSION AND FUTURE SCOPE ......................................................................... 52 APPENDIX ....................................................................................................................... 53 BIBLIOGRAPHY ............................................................................................................. 55 iv FR A C T I O N A L C A L C U L US: D E FINI T I O NS A ND APPL I C A T I O NS Joseph M. Kimeu May 2009 55Pages Directed by Dr. Ferhan Atici Department of Mathematics Western Kentucky University A BST R A C T Using ideas of ordinary calculus, we can differentiate a function, say, to the or order. We can also establish a meaning or some potential applications of the results. However, can we differentiate the same function to, say, the order? Better still, can we establish a meaning or some potential applications of the results? We may not achieve that through ordinary calculus, but we may through fractional calculus—a more generalized form of calculus. This thesis, consisting of five chapters, explores the definition and potential applications of fractional calculus. The first chapter gives a brief history and definition of fractional calculus. The second and third chapters, respectively, look at the Riemann-Liouville definitions of the fractional integral and derivative. The fourth chapter looks at some fractional differential equations with an emphasis on the Laplace transform of the fractional integral and derivative. The last chapter considers two application problems—a mortgage problem and a decay-growth problem. The decay-growth problem prompts the use of an extended Putzer algorithm to evaluate , where is a matrix, and is the matrix Mittag-Leffler function. It is shown that when , the value of equals that of the matrix exponential function, , which is essential in solving the integer order differential equation . The new function, , is then used in solving the fractional differential equation , where Finally, a brief comparison is made between the fractional and integer order solutions. v C H APT E R 1 IN T R O DU C T I O N The concept of integration and differentiation is familiar to all who have studied elementary calculus. We know, for instance, that if , then integrating to the order results in , and integrating the same function to the order results in . Similarly, , and . However, what if we wanted to integrate our function to the order, or find its order derivative? How could we define our operations? Better still, would our results have a meaning or an application comparable to that of the familiar integer order operations? 1.1 The Origin of Fractional Calculus Fractional calculus owes its origin to a question of whether the meaning of a derivative to an integer order could be extended to still be valid when is not an integer. This question was first raised by L’Hopital on September 30th, 1695. On that day, in a letter to Leibniz, he posed a question about , Leibniz’s notation for the derivative of the linear function . L’Hopital curiously asked what the result would be if Leibniz responded that it would be “an apparent paradox, from which one day useful consequences will be drawn,” [5]. 1 2 Following this unprecedented discussion, the subject of fractional calculus caught the attention of other great mathematicians, many of whom directly or indirectly contributed to its development. They included Euler, Laplace, Fourier, Lacroix, Abel, Riemann and Liouville. In 1819, Lacroix became the first mathematician to publish a paper that mentioned a fractional derivative [3]. Starting with where m is a positive integer, Lacroix found the nth derivative, (1.1.1) And using Legendre’s symbol , for the generalized factorial, he wrote (1.1.2) Finally by letting and , he obtained (1.1.3) However, the first use of fractional operations was made not by Lacroix, but by Abel in 1823 [3]. He applied fractional calculus in the solution of an integral equation that arises in the formulation of the tautochrone problem—the problem of determining the shape of 3 the curve such that the time of descent of an object sliding down that curve under uniform gravity is independent of the object’s starting point. 1.2 Definition of Fractional Calculus Over the years, many mathematicians, using their own notation and approach, have found various definitions that fit the idea of a non-integer order integral or derivative. One version that has been popularized in the world of fractional calculus is the Riemann- Liouville definition. It is interesting to note that the Riemann-Liouville definition of a fractional derivative gives the same result as that obtained by Lacroix in equation (1.1.3). Since most of the other definitions of fractional calculus are largely variations of the Riemann-Liuoville version, it is this version that will be mostly addressed in this work. 1.3 Useful Mathematical
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