arXiv subjects

A. Torres-Hernandez

Publications and source records attributed to A. Torres-Hernandez.

13 recordsLinked to original sources

arXiv2024

Reduction of a nonlinear system and its numerical solution using a fractional iterative method

A nonlinear algebraic equation system of 5 variables is numerically solved, which is derived from the application of the Fourier transform to a differential equation system that allows modeling the behavior of the temperatures and the efficiencies of a hybrid solar receiver, which in simple terms is the combination of a photovoltaic system with a thermoelectric system. In addition, a way to reduce the previous system to a nonlinear system of only 2 variables is presented. Naturally, reducing algebraic equation systems of dimension N to systems of smaller dimensions has the main advantage of reducing the number of variables involved in a problem, but the analytical expressions of the systems become more complicated. However, to minimize this disadvantage, an iterative method that does not explicitly depend on the analytical complexity of the system to be solved is used. A fractional iterative method, valid for one and several variables, that uses the properties of fractional calculus, in particular the fact that the fractional derivatives of constants are not always zero, to find solutions of nonlinear systems is presented.

math.NA↗

arXiv2024

A nonlinear system related to investment under uncertainty solved using the fractional pseudo-Newton method

A nonlinear algebraic equation system of two variables is numerically solved, which is derived from a nonlinear algebraic equation system of four variables, that corresponds to a mathematical model related to investment under conditions of uncertainty. The theory of investment under uncertainty scenarios proposes a model to determine when a producer must expand or close, depending on his income. The system mentioned above is solved using a fractional iterative method, valid for one and several variables, that uses the properties of fractional calculus, in particular the fact that the fractional derivatives of constants are not always zero, to find solutions of nonlinear systems.

math.NA↗

arXiv2024

Fractional flow equations. A model for pressure deficit in an oil well

This article presents a novel system of flow equations that models the pressure deficit of a reservoir considered as a triple continuous medium formed by the rock matrix, vugular medium and fracture. In non-conventional reservoirs, the velocity of the fluid particles is altered due to physical and chemical phenomena caused by the interaction of the fluid with the medium, this behavior is defined as anomalous. A more exact model can be obtained with the inclusion of the memory formalism concept that can be expressed through the use of fractional derivatives. Using Laplace transform of the Caputo fractional derivative and Bessel functions, a semi-analytical solution is reached in the Laplace space.

math.AP↗

arXiv2021

Simulation of the behavior of the Leopardus guigna using random walkers

Considering that there is very little information on the behavior habits of guigna cats, as well as investigations in which small populations are captured to place radiocollars on them and then release them in the place where they were captured, which is done with the intention of collecting data on their positions in a territory and thus make estimates of the mean distances they usually travel. Under the hypothesis that guignas maintain a sedentary behavior in a specific area of a given territory, this paper shows one way to simulate a distribution of points in a territory using random walkers to emulate the distribution of the data that would be obtained by placing radiocollars in a population of guignas, with which it is possible to make estimates of the mean distances that move away from a certain fixed position, and the interactions they can have with points in the territory that represent a high probability of lethality, such as farms, packs of dogs, roads, urban areas, etc. It is necessary to mention that by estimating the possible interactions that a guignas population may have with possible predators in a territory with the help of a satellite image, it is possible to evaluate the points of a territory that represent a potentially lethal risk for the guignas, and thus generate relocation strategies that help preserve them.

physics.soc-ph↗

arXiv2021

Acceleration of the order of convergence of a family of fractional fixed point methods and its implementation in the solution of a nonlinear algebraic system related to hybrid solar receivers

This paper presents a way to define, classify and accelerate the order of convergence of an uncountable family of fractional fixed point methods, which may be useful to continue expanding the applications of fractional operators. The proposed method to accelerate convergence is used in a fractional iterative method, and with the obtained method are solved simultaneously two nonlinear algebraic systems that depend on time-dependent parameters, and that allow obtaining the temperatures and efficiencies of a hybrid solar receiver. Finally, two uncountable families of fractional fixed point methods are presented, in which the proposed method to accelerate convergence can be implemented.

math.NA↗

arXiv2021

Fractional Newton-Raphson Method Accelerated with Aitken's Method

In the following document, we present a way to obtain the order of convergence of the Fractional Newton-Raphson (F N-R) method, which seems to have an order of convergence at least linearly for the case in which the order $α$ of the derivative is different from one. A simplified way of constructing the Riemann-Liouville (R-L) fractional operators, fractional integral and fractional derivative, is presented along with examples of its application on different functions. Furthermore, an introduction to the Aitken's method is made and it is explained why it has the ability to accelerate the convergence of the iterative methods, to finally present the results that were obtained when implementing the Aitken's method in the F N-R method.

math.NA↗

arXiv2021

An approximation to zeros of the Riemann zeta function using fractional calculus

In this document, as far as the authors know, an approximation to the zeros of the Riemann zeta function has been obtained for the first time using only derivatives of constant functions, which was possible only because a fractional iterative method was used. This iterative method, valid for one and several variables, uses the properties of fractional calculus, in particular the fact that the fractional derivatives of constants are not always zero, to find multiple zeros of a function using a single initial condition. This partly solves the intrinsic problem of iterative methods that if we want to find N zeros it is necessary to give N initial conditions. Consequently, the method is suitable for approximating nontrivial zeros of the Riemann zeta function when the absolute value of its imaginary part tends to infinity. The deduction of the iterative method is presented, some examples of its implementation, and finally 53 different values near to the zeros of the Riemann zeta function are shown.

math.NA↗

arXiv2021

Fractional Newton-Raphson Method

The Newton-Raphson (N-R) method is useful to find the roots of a polynomial of degree n. However, this method is limited since it diverges for the case in which polynomials only have complex roots if a real initial condition is taken. In the present work, we explain an iterative method that is created using the fractional calculus, which we will call the Fractional Newton-Raphson (F N-R) Method, which has the ability to enter the space of complex numbers given a real initial condition, which allows us to find both the real and complex roots of a polynomial unlike the classical Newton-Raphson method.

math.NA↗

arXiv2020

Numerical solution using radial basis functions for multidimensional fractional partial differential equations of type Black-Scholes

The aim of this paper is to solve numerically, using the meshless method via radial basis functions, time-space-fractional partial differential equations of type Black-Scholes. The time-fractional partial differential equation appears in several diffusion problems used in physics and engineering applications, and models subdiffusive and superdiffusive behavior of the prices at the stock market. This work shows the flexibility of the radial basis function scheme to solve multidimensional problems with several types of nodes and it also shows how to reduce the condition number of the matrices involved.

math.NA↗

arXiv2020

Introducción al Cálculo 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.

math.GM↗

arXiv2020

Fractional pseudo-Newton method and its use in the solution of a nonlinear system that allows the construction of a hybrid solar receiver

The following document presents a possible solution and a brief stability analysis for a nonlinear system, which is obtained by studying the possibility of building a hybrid solar receiver; It is necessary to mention that the solution of the aforementioned system is relatively difficult to obtain through iterative methods since the system is apparently unstable. To find this possible solution is used a novel numerical method valid for one and several variables, which using the fractional derivative, allows us to find solutions for some nonlinear systems in the complex space using real initial conditions, this method is also valid for linear systems. The method described above has an order of convergence (at least) linear, but it is easy to implement and it is not necessary to invert some matrix for solving nonlinear systems and linear systems.

math.NA↗

arXiv2020

Fractional Newton-Raphson Method and Some Variants for the Solution of Non-linear Systems

The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow to find solutions for some non-linear systems in the complex space using real initial conditions. The origin of these methods is the fractional Newton-Raphson method but unlike the latter, the orders of fractional derivatives proposed here are functions. In the first method, a function is used to guarantee an order of convergence (at least) quadratic, and in the others, a function is used to avoid the discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible that the methods have at most an order of convergence (at least) linear.

math.NA↗

arXiv2019

Proposal for Use the Fractional Derivative of Radial Functions in Interpolation Problems

In this document we present the construction of a radial functions that have the objective of emulating the behavior of the radial basis function thin plate spline (TPS), which we will name as function TPS, we propose a way to partially and totally apply the fractional derivative to these functions to be used in interpolation problems, a proposal is presented to precondition the matrices generated in the interpolation problem using the $QR$ decomposition and finally is proposed the form of a radial interpolant to be used when solving differential equations using the asymmetric collocation method.

math.NA↗