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faculty of mathematics
and natural sciences
Fractional Calculus
Bachelor Project Mathematics
October 2015
Student: D.E. Koning
First supervisor: Dr. A.E. Sterk
Second supervisor: Prof. dr. H.L. Trentelman
Abstract
This thesis introduces fractional derivatives and fractional integrals, shortly
differintegrals. After a short introduction and some preliminaries the
Grun¨ wald-Letnikov and Riemann-Liouville approaches for defining a
differintegral will be explored. Then some basic properties of differintegrals,
such as linearity, the Leibniz rule and composition, will be proved. Thereafter
the definitions of the differintegrals will be applied to a few examples. Also
fractional differential equations and one method for solving them will be
discussed. The thesis ends with some examples of fractional differential
equations and applications of differintegrals.
CONTENTS
Contents
1 Introduction 4
2 Preliminaries 5
2.1 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Change the Order of Integration . . . . . . . . . . . . . . . . . . 6
2.4 The Mittag-Leffler Function . . . . . . . . . . . . . . . . . . . . . 6
3 Fractional Derivatives and Integrals 7
3.1 The Grun¨ wald-Letnikov construction . . . . . . . . . . . . . . . . 7
3.2 The Riemann-Liouville construction . . . . . . . . . . . . . . . . 8
3.2.1 The Riemann-Liouville Fractional Integral . . . . . . . . . 9
3.2.2 The Riemann-Liouville Fractional Derivative . . . . . . . 9
4 Basic Properties of Fractional Derivatives 11
4.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Zero Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.3 Product Rule & Leibniz’s Rule . . . . . . . . . . . . . . . . . . . 12
4.4 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.4.1 Fractional integration of a fractional integral . . . . . . . 12
4.4.2 Fractional differentiation of a fractional integral . . . . . . 13
4.4.3 Fractional integration and differentiation of a fractional
derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5 Examples 15
5.1 The Power Function . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.2 The Exponential Function . . . . . . . . . . . . . . . . . . . . . . 16
5.3 The Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 17
6 Fractional Linear Differential Equations 18
6.1 The Laplace Transforms of Fractional Derivatives . . . . . . . . . 18
6.1.1 Laplace Transform of the Riemann-Liouville Differintegral 19
6.1.2 Laplace Transform of the Grun¨ wald-Letnikov Fractional
Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.2 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . 21
6.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7 Applications 26
7.1 Economic example . . . . . . . . . . . . . . . . . . . . . . . . . . 26
7.1.1 Concrete example . . . . . . . . . . . . . . . . . . . . . . 27
8 Conclusions 29
9 References 31
Bachelor Project Fractional Calculus 3
1 INTRODUCTION
1 Introduction
Fractional calculus explores integrals and derivatives of functions. However, in
this branch of Mathematics we are not looking at the usual integer order but
at the non-integer order integrals and derivatives. These are called fractional
derivatives and fractional integrals, which can be of real or complex orders and
therefore also include integer orders. In this thesis we refer to differintegrals if
we are talking about the combination of these fractional derivatives and inte-
grals.
So if we consider the function f(t) = 1x2, the well-known integer first-order
′ 2 ′′
and second-order derivatives are f (t) = x and f (t) = 1, respectively. But
what if we would like to take the 1-th order derivative or maybe even the q1-
2 2
th order derivative? This question was already mentioned in a letter from the
mathematician Leibniz to L’Hˆopital in 1695. Since then several famous math-
ematicians, such as Grun¨ wald, Letnikov, Riemann, Liouville and many more,
have dealt with this problem. Some of them came up with an approach on how
to define such a differentiation operator. For a very interesting more detailed
history of Fractional Calculus we refer to [1, p. 1-15]
First in chapter 2 we shall give some basic formulas and techniques which
are necessary to better understand the rest of the thesis. Then in chapter 3
two definitions for a differintegral will be given. The Grun¨ wald-Letnikov and
the Riemann-Liouville approach will be explored. These are the two most fre-
quently used differintegrals. Afterwards in chapter 4 some basic properties of
these differintegrals will be given and proved. Then in chapter 5 we shall ex-
plore a few examples. In chapter 6 we will take a look at fractional differential
equations (FDE’s). Therefore we also need to explore the Laplace transforms of
fractional derivatives. Chapter 6 ends with some examples of FDE’s. Thereafter
chapter 7 deals with a few applications of differintegrals which is followed by a
conclusion in chapter 8.
Bachelor Project Fractional Calculus 4
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