Fractional pseudo-Newton method and its use in the solution of a nonlinear system that allows the construction of a hybrid solar receiver
A. Torres-Hernandez, F. Brambila-Paz, P. M. Rodrigo, E. De-la-Vega, C. C. Calabrese
TL;DR
The paper tackles the problem of finding zeros of a nonlinear, potentially unstable system arising from a hybrid CPV-TEG solar receiver. It introduces a fractional pseudo-Newton method based on a fractional derivative-informed iteration that avoids matrix inversions, producing a matrix-free, linearly convergent solver. The approach is demonstrated both on abstract nonlinear/linear examples and on the five-variable hybrid solar receiver model, showing that real initial conditions can lead to complex roots and that convergence can be achieved with carefully chosen parameters and initial bracketing. While convergence is only linear in general, the method provides a practical, easy-to-implement tool for challenging systems with multiple or infinite solutions, useful in engineering contexts where classical Newton methods struggle due to instability.
Abstract
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.