*Article* **Derivative Free Fourth Order Solvers of Equations with Applications in Applied Disciplines**

### **Ramandeep Behl 1, Ioannis K. Argyros 2, Fouad Othman Mallawi 1 and J. A. Tenreiro Machado 3,\***


 Correspondence: jtm@isep.ipp.pt

Received: 27 March 2019; Accepted: 17 April 2019; Published: 23 April 2019

**Abstract:** This paper develops efficient equation solvers for real- and complex-valued functions. An earlier study by Lee and Kim, used the Taylor-type expansions and hypotheses on higher than first order derivatives, but no derivatives appeared in the suggested method. However, we have many cases where the calculations of the fourth derivative are expensive, or the result is unbounded, or even does not exist. We only use the first order derivative of function Ω in the proposed convergence analysis. Hence, we expand the utilization of the earlier scheme, and we study the computable radii of convergence and error bounds based on the Lipschitz constants. Furthermore, the range of starting points is also explored to know how close the initial guess should be considered for assuring convergence. Several numerical examples where earlier studies cannot be applied illustrate the new technique.

**Keywords:** divided difference; radius of convergence; Kung–Traub method; local convergence; Lipschitz constant; Banach space

**MSC:** 47J05; 47J25; 65H10; 65G99
