Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme

Abstract

In this study, we present a new higher-order scheme without memory for simple zeros which has two major advantages. The first one is that each member of our scheme is derivative free and the second one is that the present scheme is capable of producing many new optimal family of eighth-order methods from every 4-order optimal derivative free scheme (available in the literature) whose first substep employs a Steffensen or a Steffensen-like method. In addition, the theoretical and computational properties of the present scheme are fully investigated along with the main theorem, which demonstrates the convergence order and asymptotic error constant. Moreover, the effectiveness of our scheme is tested on several real-life problems like Van der Waal’s, fractional transformation in a chemical reactor, chemical engineering, adiabatic flame temperature, etc. In comparison with the existing robust techniques, the iterative methods in the new family perform better in the considered test examples. The study of dynamics on the proposed iterative methods also confirms this fact via basins of attraction applied to a number of test functions.

What Can Students Learn While Solving Colebrook’s Flow Friction Equation?

“…An example of an approximation of the Colebrook equation based on internal cycles is given by Serghides [51]. Serghides' methods belong to Steffensen types of methods, which do not require derivatives (other Steffensen types of methods can be used in addition [52,53]).…”

Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction

“…Let us start a short review of the literature with some of the existing methods with or without memory before proceeding to the proposed idea. Behl et al [5] presented an optimal scheme that does not need any derivative evaluations. In addition, the given scheme is capable of generating new optimal eighth-order methods from the earlier optimal fourth-order schemes in which the first sub-step employs Steffensen's or a Steffensen-type method.…”

A New Optimal Family of Schröder’s Method for Multiple Zeros

“…It is almost inaccessible to find the solution by an analytical approach. Therefore, we worry about the iterative schemes to find the multiple solution r m with known multiplicity m > 1 of the uni-variate function in Equation (1).…”

“…Construction of the iterative schemes [1][2][3] (that do not essentially require the value of second-order derivative of Θ) having optimal convergence order [4] is the foremost and most difficult issue in the region of numerical analysis. It is also interesting with practical perspective because sometimes it is very difficult to compute the second-order derivative.…”