On modified two-step iterative method in the fractional sense: some applications in real world phenomena

ABSTRACT

This study proposes a new two-step iterative scheme for solving nonlinear equations. This scheme is based on the Newton's method, in which the order of convergence is 4. As this scheme requires two function evaluations and one derivative evaluation at each iteration, it is optimal in the sense of the Kung and Traub conjecture [20] and in terms of computational cost, and we show that its efficiency index is $\sqrt[3]{4}\approx 1.587$. Finally, using the properties of a new derivative of arbitrary real order, our approach is extended and the convergence, stability and superiority of our suggested scheme is discussed.

Keywords:

• Nonlinear equations
• two-step methods
• efficiency index
• order of convergence
• simple root
• derivative of arbitrary real order

Mathematics Subject Classifications:

• 34A08
• 49S05

Disclosure statement

No potential conflict of interest was reported by the authors.