Computational Fractional Dynamical Systems

1Introduction to Fractional Calculus

1.1 Introduction

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary (non‐integer) order. It is also an area of mathematics that investigates the possibilities of using real or even complex numbers as powers of the differential operator. This area is three centuries old compared to conventional calculus, but initially, it was not very popular. Fractional derivatives and integrals are not local in nature, so the nonlocal distributed effects are considered. The subject of fractional calculus has gained considerable popularity and importance during the past three decades, mainly due to its validated applications, dynamic nature, and comprehensive representation of complex nonlinear phenomena in various fields of science and engineering. The mathematical models in electromagnetics, rheology, viscoelasticity, electrochemistry, control theory, fluid dynamics, financial mathematics, and material science are well defined by fractional‐order differential equations.

1.2 Birth of Fractional Calculus

In a letter to L'Hospital in 1695, Leibniz asked the following question: “Can the meaning of integer‐order derivatives be generalized to non‐integer‐order derivatives?” L’Hospital was very curious about that question and replied to Leibniz by asking what would happen to the term if . In order to explain the ...