A. Torres-Hernandez

By Benito F. Martínez-Salgado, Rolando Rosas-Sampayo, Anthony Torres-Hernández and Carlos Fuentes

In this chapter, we present a discussion about the practical application of the fractal properties of the medium in the mathematical model through the use of fractional partial derivatives. We present one of the known models for the flow in saturated media and its generalization in fractional order derivatives. In the middle section, we present one of the main arguments that motivate the use of fractional derivatives in the porous media models, this is the Professor Nigmatullin’s work. The final part describes the process for obtaining the coupled system of three equations for the monophase flow with triple porosity and triple permeability, briefly mentioning the method used for the first solutions of the system.

Part of the book: Fractal Analysis

By A. Torres-Hernandez, F. Brambila-Paz and R. Ramirez-Melendez

This chapter presents one way to define Abelian groups of fractional operators isomorphic to the group of integers under addition through a family of sets of fractional operators and a modified Hadamard product, as well as one way to define finite Abelian groups of fractional operators through sets of positive residual classes less than a prime number. Furthermore, it is presented one way to define sets of fractional operators which allow generalizing the Taylor series expansion of a vector-valued function in multi-index notation, as well as one way to define a family of fractional fixed-point methods and determine their order of convergence analytically through sets.

Part of the book: Operator Theory