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

A. Torres-Hernandez, F. Brambila-Paz, P. M. Rodrigo

TL;DR

The paper tackles the challenge of solving nonlinear algebraic systems arising from a Fourier-transformed coupled thermal-electrical model of a hybrid solar receiver. It introduces a fractional iterative framework, the fractional pseudo-Newton method based on Riemann-Liouville derivatives, which preconditions the nonlinear residuals to achieve robust convergence without requiring explicit analytic complexity. Additionally, it demonstrates a dimension-reduction strategy from 5 variables to 2, facilitating faster initialization and real-time interpretation of temperatures and efficiencies under varying DNI and ambient conditions. Through 5D and 2D examples, the approach shows promise for efficient, real-time analysis of complex energy-harvesting systems where traditional Newton-type methods struggle due to analytical intractability or unfavorable initial-condition landscapes.

Abstract

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.

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

TL;DR

The paper tackles the challenge of solving nonlinear algebraic systems arising from a Fourier-transformed coupled thermal-electrical model of a hybrid solar receiver. It introduces a fractional iterative framework, the fractional pseudo-Newton method based on Riemann-Liouville derivatives, which preconditions the nonlinear residuals to achieve robust convergence without requiring explicit analytic complexity. Additionally, it demonstrates a dimension-reduction strategy from 5 variables to 2, facilitating faster initialization and real-time interpretation of temperatures and efficiencies under varying DNI and ambient conditions. Through 5D and 2D examples, the approach shows promise for efficient, real-time analysis of complex energy-harvesting systems where traditional Newton-type methods struggle due to analytical intractability or unfavorable initial-condition landscapes.

Abstract

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.

Paper Structure

This paper contains 9 sections, 3 theorems, 61 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Fixed Point Method
  3. Order of Convergence
  4. Riemann-Liouville Fractional Derivative
  5. Fractional Pseudo-Newton Method
  6. Some Examples
  7. Equations of a Hybrid Solar Receiver
  8. Reducing the Number of Equations of a Hybrid Solar Receiver
  9. Conclusions

Key Result

Proposition 2.4

Let $f:\Omega \subset \mathbb{R} ^n\to \mathbb{R} ^n$ be a function with a value $\xi\in \Omega$ such that $\left\lVert f(\xi) \right\rVert =0$, and let $\Phi: \mathbb{R} ^n \to \mathbb{R} ^n$ be an iteration function as follows with $A(x)$ a matrix. If the following condition is fulfilled then $\Phi$ satisfies a necessary (but not sufficient) condition to be (locally) convergent of order (at le

Theorems & Definitions (11)

  • Definition 2.1
  • Definition 2.2
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • Definition 3.1
  • Proposition 3.2
  • Example 4.1
  • Example 4.2
  • ...and 1 more