Abstract
In this article, we propose some novel computational methods in the form of iteration schemes for computing the roots of non-linear scalar equations in a new way. The construction of these iteration schemes is purely based on exponential series expansion. The convergence criterion of the suggested schemes is also given and certified that the newly developed iteration schemes possess quartic convergence order. To analyze the suggested schemes numerically, several test examples have been given and then solved. These examples also include some real-world problems such as van der Wall's equation, Plank's radiation law and kinetic problem equation whose numerical results showing the better performance, applicability and efficiency of these iteration schemes against the other similar-nature two-step iteration schemes in the literature. Finally, a detailed graphical analysis of the suggested iteration schemes has been given in the form of polynomiographs for the different complex polynomials with the aid of computer technology that reveals the convergence characteristics and other dynamical features of the presented iteration schemes.