FRACTIONAL-ORDER NEWTON-RAPHSON METHOD FOR NONLINEAR EQUATION WITH CONVERGENCE AND STABILITY ANALYSES

Abstract

Fractional-order techniques have many applications in real-life problems nowadays. The utilization of fragmentary math in many parts of science and engineering is wide and somewhat recent. There are various types of subsidiaries that can be valuable in various issues. In this paper, we focus on the effect of this kind of fractional derivative in the search for roots of nonlinear equations and its dependence on the initial estimations. We will check the convergence and stability analyses of the fractional Newton-Raphson (FNR) method for the proposed definitions. We will apply fractional Riemann-Liouville and Caputo-derivatives in the standard Newton root-finding method for different examples and also verify convergence and stability for these problems. Finding a root is very important for nonlinear equations which are used to solve chemical, biological, and engineering problems.