Optimal eighth-order solver for nonlinear equations with applications in chemical engineering

A new iterative technique for nonlinear equations is proposed in this work. The new scheme is of three steps, of which the first two steps are based on the sixth-order modified Halley’s method presented by the authors, and the last is a Newton step, with suitable approximations for the first derivatives appeared in the new scheme. The eighth-order of convergence of the new method is proved via Mathematica code. Every iteration of the presented scheme needs the evaluation of three functions and one first derivative. Therefore, the scheme is optimal in the sense of Kung-Traub conjecture. Several test nonlinear problems are considered to compare the performance of the proposed method according to other optimal methods of the same order. As an application, we apply the new scheme to some nonlinear problems from the field of chemical engineering, such as a chemical equilibrium problem (conversion in a chemical reactor), azeotropic point of a binary solution, and volume from van der Waals equation. Comparisons and examples show that the presented method is efficient and comparable to the existing techniques of the same order.