On Deterministic Fractional Models

Abstract

This chapter is dedicated to presenting some aspects of the so-called Ordinary and/or Partial Fractional Differential Equations. During last 20 years the main underground reason that explain the interest of the applied researchers in the fractional models have been the known close link that exists between such kind of models and the so-called “Jump” stochastic models, such as the CTRW (Continuous Time Random Walk). During the second half of the twentieth century (until the 1990s), the CTRW method was practically the main tool available to describe subdiffusive and/or superdiffusive phenomena associated with Complex Systems for the researches that work in applied fields. The fractional operators are non-local, while the ordinary derivative is a local operator, and on the other hand, the dynamics of many anomalous processes depend of certain memory of its own dynamics. Therefore, the fractional models linear and/or non-linear look as a good alternative to the ordinary models. Note that fractional operators also provide an alternative method to the classical models including dilate terms. The main of this chapter is dedicated to do a first approach to show how introduce Fractional Models only under a deterministic basement. We will considerate Fractional Dynamics Systems with application to study anomalous growing of populations, and on the other hand, the Fractional Diffusive Equation.