Theoretical Aspects of Nonlinear Systems

**Chapter 1**

**Abstract**

**1. Introduction**

**3**

A Review on Fractional

*María I. Troparevsky, Silvia A. Seminara*

value problems and boundary value problems.

wavelet decomposition, numerical approximation

**Keywords:** fractional derivatives, fractional differential equations,

Fractional calculus is the theory of integrals and derivatives of arbitrary real (and even complex) order and was first suggested in works by mathematicians such as Leibniz, L'Hôpital, Abel, Liouville, Riemann, etc. The importance of fractional derivatives for modeling phenomena in different branches of science and engineering is due to their nonlocality nature, an intrinsic property of many complex systems. Unlike the derivative of integer order, fractional derivatives do not take into account only local characteristics of the dynamics but considers the global evolution of the system; for that reason, when dealing with certain phenomena, they provide

To illustrate this fact, we will retrieve an example from [1]. Recall the relationship between stress *σ*ð Þ*t* and strain *ε*ð Þ*t* in a material under the influence of external forces:

more accurate models of real-world behavior than standard derivatives.

*and Marcela A. Fabio*

Differential Equations and

a Numerical Method to Solve

Some Boundary Value Problems

Fractional differential equations can describe the dynamics of several complex and nonlocal systems with memory. They arise in many scientific and engineering areas such as physics, chemistry, biology, biophysics, economics, control theory, signal and image processing, etc. Particularly, nonlinear systems describing different phenomena can be modeled with fractional derivatives. Chaotic behavior has also been reported in some fractional models. There exist theoretical results related to existence and uniqueness of solutions to initial and boundary value problems with fractional differential equations; for the nonlinear case, there are still few of them. In this work we will present a summary of the different definitions of fractional derivatives and show models where they appear, including simple nonlinear systems with chaos. Existing results on the solvability of classical fractional differential equations and numerical approaches are summarized. Finally, we propose a numerical scheme to approximate the solution to linear fractional initial

#### **Chapter 1**
