Dynamics of Some Discretized Fractional-Order Differential Equations

Dynamics of Some Discretized Fractional-Order Differential Equations

Sanaa Moussa Salman (Alexandria University, Egypt) and Ahmed M. A. El-Sayed (Alexandria University, Egypt)
DOI: 10.4018/978-1-7998-3122-8.ch004

Abstract

This chapter deals with fractional-order differential equations and their discretization. First of all, a discretization process for discretizing ordinary differential equations with piecewise constant arguments is presented. Secondly, a discretization method is proposed for discretizing fractional-order differential equations. Stability of fixed points of the discretized equations are investigated. Numerical simulations are carried out to show the dynamic behavior of the resulting difference equations such as bifurcation and chaos.
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Brief Historical Background Of Discretization Methods

Generally, there are two discretization methods: direct discretization and indirect discretization. In the direct method one can directly develop discrete-time operators to discretize the fractional-order continuous-time operator. In the indirect discretization method, a rational continuous-time operator should be developed which is then discretized using any of the well-known discretization techniques.

There are several approaches to direct discretization

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