Stability and Uniqueness of Fractional Order Newton-Raphson’s Method for Nonlinear Equations

Abstract

The study proposes a fractional-order derivative-based Newton-Raphson algorithm for solving nonlinear fractional-order systems. The algorithm focuses on polynomial, exponential, and trigonometric functions and demonstrates its 2α^th convergence with a Gamma operation damping bound. The method is effective in solving complex nonlinear differential equations of fractional orders, with rigorous mathematical results consistent across both partial subordinates. The technique is particularly useful for complex roots and precise fractional calculus, making it an effective substitute for the traditional Newton-Raphson method. This method is useful for solving complex nonlinear differential equations in engineering and physics due to its more precise computation than most usual methods. The method’s effectiveness is influenced by the specific issue, precision level, and computing power available.