20 research outputs found
Fractional Newton-Raphson Method Accelerated with Aitken's Method
The Newton-Raphson (N-R) method is characterized by the fact that generating
a divergent sequence can lead to the creation of a fractal, on the other hand
the order of the fractional derivatives seems to be closely related to the
fractal dimension, based on the above, a method was developed that makes use of
the N-R method and the fractional derivative of Riemann-Liouville (R-L) that
has been named as the Fractional Newton-Raphson (F N-R) method.
In the following work we present a way to obtain the convergence of the F N-R
method, which seems to be at least linearly convergent for the case where the
order of the derivative is different from one, a simplified way to
construct the fractional derivative and fractional integral operators of R-L is
presented, an introduction to the Aitken's method is made and it is explained
why it has the capacity to accelerate the convergence of iterative methods to
finally present the results that were obtained when implementing the Aitken's
method in F N-R method.Comment: Newton-Raphson Method, Fractional Calculus, Fractional Derivative of
Riemann-Liouville, Method of Aitken. arXiv admin note: substantial text
overlap with arXiv:1710.0763
Introducci\'on al C\'alculo Fraccional
The following material was created with the idea of being used for an
introductory fractional calculus course. A recapitulation of the history of
fractional calculus is presented, as well as the different attempts at
fractional derivatives that existed before current definitions. Properties of
the gamma function, beta function and the Mittag-Leffler function are
presented, which are fundamental pieces in the fractional calculus. The basic
properties of Riemann-Liouville and Caputo fractional derivatives are
presented, as well as their implementation to different functions. It also
presents the Laplace transform of a fractional operator and an application to
the fractional free fall problem.Comment: in Spanis
Sets of Fractional Operators and Some of Their Applications
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