On optimization and solution of roots of a function using Taylor's expansion and fractional derivatives

TL;DR

This paper introduces a novel method for finding roots of functions using Taylor's expansion and fractional derivatives, offering faster convergence than Newton's method, and extends it to multivariable functions for optimization and solving nonlinear systems.

Contribution

The paper presents a new root-finding and optimization method based on fractional derivatives, generalizing classical techniques and demonstrating improved convergence speed over traditional methods.

Findings

The proposed method converges faster than Newton's method.

It effectively solves nonlinear systems with fewer iterations.

The method can be applied to optimize functions using fractional derivatives.

Abstract

A method is given for finding roots of a one-variable function using Taylor's expansion of that function and fractional derivative calculated at a suitable tangent point without using Newton's method, but is regarded as a variant of Halley and Newton's one. Several examples regarding polynomials are stated as well. Then, the given method is generalized to functions of several variables belonging to an $n$-dimensional space and one example is given for optimization and solution of a nonlinear system of equations by both our method and Gradient Descent one. A comparison of our method is made with Gradient descent one for a system of the functions of three variables. Our given method seems to be much more rapidly than the Newton's one since by finding a suitable point on the function's curve, the number of iterations is to be much less than Newton's iterative steps. We also find order of fractional derivative, which corresponds to equation's found root and compare tangent lines drawn at the root by both fractional and classical derivatives. The methods given in this paper can be used for optimization of function via fractional derivatives of order $\beta$.