Mathematical Modeling with Multidisciplinary Applications

CHAPTER 15

FRACTIONAL CALCULUS AND ITS APPLICATIONS

IVO PETRÁŠ

Technical University of Košice, Slovakia

15.1 INTRODUCTION

It is well-known that an important part of mathematical modeling of objects and processes is a description of their dynamics. In this manner we obtain a dynamical mathematical model, usually in the form of differential equations. In such equations we are able to use a mathematical phenomenon, so-called “fractional calculus.”

The term fractional calculus is more than 300 years old. It is a generalization of the ordinary differentiation and integration to noninteger (arbitrary) order. The subject is as old as the calculus of differentiation and goes back to times when Leibniz, Gauss, and Newton invented this kind of calculation. In a letter to L’Hopital in 1695 Leibniz raised the following question: “Can the meaning of derivatives with integer order be generalized to derivatives with noninteger orders?” The story goes that L’Hopital was somewhat curious about that question and replied with another question to Leibniz. “What if the order will be 1/2?” Leibniz in a letter dated September 30, 1695 replied: “It will lead to a paradox, from which one day useful consequences will be drawn.” The question raised by Leibniz for a fractional derivative was an ongoing topic for the last 300 years. Several mathematicians contributed to this subject over the years. People like Liouville, Riemann, and Weyl made major contributions to the theory of fractional calculus. The story ...