Analysis, Modeling and Stability of Fractional Order Differential Systems 1

1The Fractional Integrator

1.1. Introduction

A fundamental topic of mathematics concerns the integration of ordinary differential equations (ODEs). Except particular cases where analytical solutions exist, the search for general solutions for any type of ODE and excitation has been an important issue either theoretically or numerically for engineering applications. The quest for practical solutions has motivated the formulation of approximate numerical solutions (Euler, Runge–Kutta, etc.) and the realization of mechanical and electrical computers based on a physical analogy. The interest of an historical reminder concerning analog computers is to highlight the fundamental role played by the integration operator, which is also present in an implicit form in numerical ODE solvers.

As in the integer order case, the integration of fractional differential equations (FDEs) requires a fractional integration operator, which is the topic of this chapter. After a reminder of the integer order ODE simulation, the Riemann–Liouville integration or fractional order integration is presented as a generalization of repeated integer order integration. Then, the fractional integration operator is defined and applied to the FDE simulation. The realization of the fractional integrator will be addressed in Chapter 2.

1.2. Simulation and modeling of integer order ordinary differential equations

1.2.1. Simulation with analog computers

Analog computers are nowadays obsolete, but they present a ...