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

## TL;DR

This paper introduces novel fractional derivative-based numerical methods for solving non-linear systems in complex space, improving convergence properties and addressing discontinuities present in traditional fractional Newton-Raphson methods.

## Contribution

It proposes new fractional Newton-Raphson variants with variable derivative orders, enhancing convergence and stability for complex non-linear systems.

## Key findings

- Methods achieve at least quadratic convergence.
- New approaches avoid discontinuities in fractional derivatives.
- Applicable to systems with real initial conditions.

## Abstract

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.