Fractional Calculus

The following sections are included:

• Dedication
• Preface
• Preface to the Second Edition
• Contents

This chapter is preliminary in character and contents. We give here the definitions and some properties of several fractional integrals and fractional derivatives of different types. Also we give the definition and properties of some special functions which will be used in this book…

When working with problems stemming from “real-world” applications, it is only rarely possible to evaluate the solution of a given fractional differential equation in closed form, and even if such an analytic solution is available, it is typically too complicated to be used in practice. Therefore it is indispensable to have a number of numerical algorithms at hand so that one is able to compute numerical solutions with a sufficient accuracy in reasonable time. Thus, this and the following chapter will be devoted to a study of such algorithms. To be precise, in this chapter we will give a survey of the standard numerical methods that should give the reader an impression of what he or she may expect from today's state-of-the-art algorithms. In Chapter 3 we will then look at the most important algorithms more closely, giving a detailed account of their respective strengths and weaknesses…

In the previous chapter we had introduced a number of numerical methods for (ordinary and partial) fractional differential equations. Our goal now is to look more closely at some especially important methods from this selection. In particular we want to provide the reader with some basic knowledge that allows to decide for a concrete algorithm for a specific problem at hand. Moreover we want to mention a few techniques that are generally applicable to many classes of algorithms and that allow to obtain slightly modified versions that retain the convergence behavior but that require less computational effort.

The source codes of some very significant special algorithms will be listed in Appendix A.

In this Chapter, we will present the latest and most complete extension of the Stirling numbers using the framework of the fractional calculus that we are presently aware of.

The calculus of variations is dealing with the problem of extremising functionals. As it is known this problem represents a generalization of the problem of finding extremes of functions of several variables, or it is the problem of finding extremes of functions of an infinite number of variables. The roots of the calculus of variations can be seen in the works of the Greek thinkers…

The following sections are included:

• Introduction
• The Definition of Continuous-Time Random Walks
• Fractional Diffusion and Limit Theorems

The following sections are included:

• Introduction to Financial Markets
• Models of Price Fluctuations in Financial Markets
• Simulation
• Option Pricing
• Other Applications
  • Application to insurance and economics
  • Applications to physics

Following the discussion of applications of fractional calculus in finance and economics given in Chapter 7, we now turn our attention towards the use of fractional calculus techniques for modeling three concrete processes arising in a completely different area, namely in the life sciences. Specifically, Section 8.1 will be devoted to the mathematical description of the evolution of an epidemic, Section 8.2 is aimed at the modeling of a biological reaction connected to a fermentation process, and in Section 8.3 we shall present an application from pharmacology…

In this appendix, we have collected the source codes of a number of the most important numerical methods for fractional differential equations. This appendix is by no means meant to be comprehensive or complete in any sense: our goal is to provide the reader with a selection of generally applicable easy-to-use programs that should be helpful in a large number of potential applications…

The following sections are included:

• Bibliography
• Index