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Mathematical Problems in Engineering
Volume 2016, Article ID 4356371, 11 pages
http://dx.doi.org/10.1155/2016/4356371
Research Article

Fractional Mathematical Operators and Their Computational Approximation

Escuela Técnica Superior de Ingeniería Aeronáutica y del Espacio, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros 3, 28040 Madrid, Spain

Received 27 January 2016; Revised 30 July 2016; Accepted 9 August 2016

Academic Editor: Manuel Doblaré

Copyright © 2016 José Crespo and Francisco Javier Montáns. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Usual applied mathematics employs three fundamental arithmetical operators: addition, multiplication, and exponentiation. However, for example, transcendental numbers are said not to be attainable via algebraic combination with these fundamental operators. At the same time, simulation and modelling frequently have to rely on expensive numerical approximations of the exact solution. The main purpose of this article is to analyze new fractional arithmetical operators, explore some of their properties, and devise ways of computing them. These new operators may bring new possibilities, for example, in approximation theory and in obtaining closed forms of those approximations and solutions. We show some simple demonstrative examples.