Fractional Calculus Stability, Fractional-Oscillators, and Fractional-Filters: Theories and Applications

Abstract

The concept of the differentiation operator D = _! is familiar to all who have dx

studied the elementary calculus. In 1695 L'Hopital inquired in his correspondence to Leibniz what the meaning of D'f was, if n was a fraction. Since that time, fractional calculus has drawn the attention of many famous mathematicians, such as Euler, Laplace, Fourier, Abel, Liouville, Riemann, and Laurent. Later the theory of fractional calculus extended to include operators D", where u could be rational or irrational, positive or negative, real or complex. Thus the name fractional calculus . became somewhat of a misnomer. Some scientists believe that, a better description

might be differentiation and integration to an arbitrary order. The applications of fractional calculus spread in different fields of engineering, science, and medicine. The study of fractional calculus became very important because it is the generalization of the classical calculus.

In this thesis the researcher would like to present in the first part a historical overview of fractional calculus, mathematical definitions, numerical solutions of fractional-order differential equations and applications. In the second part, the researcher will study the stability conditions for any fractional-order differential equation with constant coefficients, and poles located in physical s -plane through transforming this equation to new planes F -plane or W -plane. Several examples are presented in this part.

The third part presents the general theorem for the necessary and sufficient condition of oscillation for any Linear state-space fractional-order system or fractional-order differential equation. The most popular eleven examples (seven examples using two fractional elements and four examples using three fractional elements) are discussed to work under fractional order, special cases and PSpice simulations are also introduced. Four of the simplest fractional-order oscillation systems are studied together with their numerical simulations. Look-up tables are presented through the suggested procedure for some cases to overcome the solution of nonlinear fractional order equation. Finally A general theorem for any n-fractional­ order system is given.