Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

A. Torres-Hernandez, F. Brambila-Paz, R. Montufar-Chaveznava

TL;DR

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.

Abstract

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.

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

TL;DR

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.

Abstract

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.

Paper Structure

This paper contains 8 sections, 2 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Fixed Point Method
  3. Riemann-Liouville Fractional Operators
  4. Fractional Fixed Point Method
  5. Acceleration in the Order of Convergence of the Set $\mathop{\mathrm{Ord}}\nolimits_2^1(\xi)$
  6. Equations of a Hybrid Solar Receiver
  7. Solutions of the Equations of a Hybrid Solar Receiver
  8. Conclusions

Key Result

Corollary 2.1

Let $\Phi: \mathbb{R} ^n \to \mathbb{R} ^n$ be an iteration function. If $\Phi$ defines a sequence $\left\{x_i\right\} _{\geq 1}$ such that $x_i\to \xi\in \mathbb{R} ^n$. So, $\Phi$ has an order of convergence of order (at least) $p$ in $B(\xi;\delta)$, where

Figures (2)

  • Figure 1: Illustrations of some trajectories generated by the fractional Newton-Raphson method for the same initial condition $x_0$ but with different orders $\alpha$ of the fractional operator used torres2021fracnewrap.
  • Figure 2: Histogram and density curve of the efficiency of a hybrid solar receiver obtained from a simulation corresponding to a period of thirty days, which is equivalent to $2410$ pairs of parameters ($DNI,T_{air}$) randomly generated on the domain $[12,958]\times [11,45]$. The selected domain is based on data measured in real-time at the Center for Advanced Studies in Energy and Environment (CEAEMA) rodrigo2019performancede2021fractional. The values generated for the simulation presented the mean values $mean(DNI)=662.35$ and $mean(T_{air})=31.28$ with sample standard deviations $std(DNI)=257.83$ and $std(T_{air})=6.11$, while the values of the efficiencies were obtained through the solutions of the function \ref{['eq:5-005']} using the fractional quasi-Newton method accelerated.

Theorems & Definitions (5)

  • Corollary 2.1
  • Corollary 4.1
  • Example 1
  • Example 2
  • Example 3